{"assistant":{"explanations":[{"calculationRefs":["contractible-order"],"text":"На стягиваемом пространстве первая когомология тривиальна: порядок 1."},{"calculationRefs":["contractible-dimension"],"text":"Тривиальной группе отвечает размерность 0."},{"calculationRefs":["contractible-not-two"],"text":"Утверждение о порядке 2 здесь не подтверждено и не опровергнуто: модель считает порядок, а не отрицает чужие значения."}],"origin":"assistant","summary":"На стягиваемом конечном пространстве Ȟ¹(X, Z/2) имеет порядок 1 и размерность 0 над полем из двух элементов: нетривиальных классов нет. Предположение о порядке 2 не подтверждено и не опровергнуто: модель считает порядок сама и не высказывается о чужом значении. Всё считается из точек пространства и значений функций на покрытии: сечениями признаны локально постоянные функции, согласованные пары пересчитаны, и порядок получен из равенства |Ȟ¹| · |F(U₁₂)| = |F(U₁)| · |F(U₂)| · |Ȟ⁰|."},"calculations":[{"answer":{"answer":{"evaluationStatus":"COMPUTED","kind":"TRUTH","meaning":"установлено","missingInputs":[],"truthStatus":"TRUE_ONLY"},"closedEditionRules":[],"derived":["section_over(urn:case:stacks:cech:fn-W-1, urn:case:stacks:cech:F, urn:case:stacks:cech:W)","section_over(urn:case:stacks:cech:fn-U2-00, urn:case:stacks:cech:F, urn:case:stacks:cech:U2)","section_over(urn:case:stacks:cech:fn-W-0, urn:case:stacks:cech:F, urn:case:stacks:cech:W)","section_over(urn:case:stacks:cech:fn-U2-11, urn:case:stacks:cech:F, urn:case:stacks:cech:U2)","section_over(urn:case:stacks:cech:fn-X-000, urn:case:stacks:cech:F, urn:case:stacks:cech:X)","section_over(urn:case:stacks:cech:fn-U1-11, urn:case:stacks:cech:F, 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functions agree at a point when their values there coincide"},{"language":"ru","status":"unofficial","text":"две функции согласованы в точке, когда их значения там совпадают"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#CechC0Order","kind":"rule","labels":[{"language":"en","status":"official","text":"01FI: the order of \\(C^0\\) is the product of the orders of \\(F(U_1)\\) and \\(F(U_2)\\), all functions presented"},{"language":"ru","status":"unofficial","text":"01FI: порядок \\(C^0\\) — произведение порядков \\(F(U_1)\\) и \\(F(U_2)\\), все функции предъявлены"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#CechC1Order","kind":"rule","labels":[{"language":"en","status":"official","text":"01FI: the order of \\(C^1\\) is the order of \\(F(U_12)\\), all functions presented"},{"language":"ru","status":"unofficial","text":"01FI: порядок \\(C^1\\) — порядок \\(F(U_12)\\), все функции предъявлены"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#CechH0MatchesSections","kind":"rule","labels":[{"language":"en","status":"official","text":"01EG on a covering of \\(U\\): \\(|F(U)| = |Ȟ^0(𝒰, F)|\\) with all functions on \\(U\\) presented"},{"language":"ru","status":"unofficial","text":"01EG на покрытии \\(U\\): \\(|F(U)| = |Ȟ^0(𝒰, F)|\\) при всех предъявленных функциях на \\(U\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#CechH0Order","kind":"rule","labels":[{"language":"en","status":"official","text":"01EF: \\(Ȟ^0 = ker d^0\\); its order is the number of pairs \\((s_1, s_2)\\) agreeing on \\(U_12\\)"},{"language":"ru","status":"unofficial","text":"01EF: \\(Ȟ^0 = ker d^0\\); его порядок — число пар \\((s_1, s_2)\\), согласованных на \\(U_12\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#CechH1Dimension","kind":"rule","labels":[{"language":"en","status":"official","text":"the \\(Z/2\\)-dimension of \\(Ȟ^1\\) is the exponent of its order"},{"language":"ru","status":"unofficial","text":"\\(Z/2\\)-размерность \\(Ȟ^1\\) — показатель её порядка"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#CechH1Order","kind":"rule","labels":[{"language":"en","status":"official","text":"01EF with 01FI: over the field \\(Z/2\\), \\(|im d^0| = |C^0| / |ker d^0|\\) and \\(Ȟ^1 = C^1 / im d^0\\), so the order \\(n\\) of \\(Ȟ^1\\) satisfies \\(n · |C^0| = |C^1| · |ker d^0|\\)"},{"language":"ru","status":"unofficial","text":"01EF с 01FI: над полем \\(Z/2\\) \\(|im d^0| = |C^0| / |ker d^0|\\) и \\(Ȟ^1 = C^1 / im d^0\\), потому порядок \\(n\\) группы \\(Ȟ^1\\) удовлетворяет \\(n · |C^0| = |C^1| · |ker d^0|\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#CompatiblePair","kind":"rule","labels":[{"language":"en","status":"official","text":"01FI, \\(d^0(s_1, s_2) = s_2|U_12 − s_1|U_12\\): the pair is a 0-cocycle when both restrict to the same section of \\(U_12\\)"},{"language":"ru","status":"unofficial","text":"01FI, \\(d^0(s_1, s_2) = s_2|U_12 − s_1|U_12\\): пара — 0-коцикл, когда обе ограничиваются в одно сечение над \\(U_12\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#Covering","kind":"type_decl","labels":[{"language":"en","status":"official","text":"open covering \\(U = ∪ U_i\\)"},{"language":"ru","status":"unofficial","text":"открытое покрытие \\(U = ∪ U_i\\)"}],"name":"Covering","package":"urn:stacks:clir:sheaf-cohomology"},{"id":"urn:stacks:clir:sheaf-cohomology#CoversByPoints","kind":"rule","labels":[{"language":"en","status":"official","text":"the members cover \\(U\\) when each member lies in \\(U\\) and every point of \\(U\\) lies in some member"},{"language":"ru","status":"unofficial","text":"члены покрывают \\(U\\), когда каждый член лежит в \\(U\\) и всякая точка \\(U\\) лежит в некотором члене"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#IntersectionByPoints","kind":"rule","labels":[{"language":"en","status":"official","text":"\\(W = U ∩ V\\) when \\(W ⊂ U, W ⊂ V\\) and every point common to \\(U\\) and \\(V\\) lies in \\(W\\)"},{"language":"ru","status":"unofficial","text":"\\(W = U ∩ V\\), когда \\(W ⊂ U, W ⊂ V\\) и всякая общая точка \\(U\\) и \\(V\\) лежит в \\(W\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#LocallyConstantAlongSpecialization","kind":"rule","labels":[{"language":"en","status":"official","text":"006W with 0061: a function on \\(U\\) is locally constant when \\(f(p) = f(g)\\) for every point \\(p\\) of \\(U\\) and every generalization \\(g\\) of \\(p\\) in \\(U\\)"},{"language":"ru","status":"unofficial","text":"006W с 0061: функция на \\(U\\) локально постоянна, когда \\(f(p) = f(g)\\) для всякой точки \\(p\\) из \\(U\\) и всякой генерализации \\(g\\) точки \\(p\\) в \\(U\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#Open","kind":"type_decl","labels":[{"language":"en","status":"official","text":"open subset of a topological space"},{"language":"ru","status":"unofficial","text":"открытое подмножество топологического пространства"}],"name":"Open","package":"urn:stacks:clir:sheaf-cohomology"},{"id":"urn:stacks:clir:sheaf-cohomology#OpenByGeneralizationStability","kind":"rule","labels":[{"language":"en","status":"official","text":"0062 (2) on a finite space: a subset of \\(X\\) stable under generalization is open"},{"language":"ru","status":"unofficial","text":"0062 (2) на конечном пространстве: подмножество \\(X\\), устойчивое под генерализацией, открыто"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#Point","kind":"type_decl","labels":[{"language":"en","status":"official","text":"point of a topological space"},{"language":"ru","status":"unofficial","text":"точка топологического пространства"}],"name":"Point","package":"urn:stacks:clir:sheaf-cohomology"},{"id":"urn:stacks:clir:sheaf-cohomology#RestrictionByValues","kind":"rule","labels":[{"language":"en","status":"official","text":"006E: for \\(V ⊂ U\\) the restriction of \\(s\\) to \\(V\\) is the function \\(t\\) on \\(V\\) agreeing with \\(s\\) at every point of \\(V\\)"},{"language":"ru","status":"unofficial","text":"006E: при \\(V ⊂ U\\) ограничение \\(s\\) на \\(V\\) — функция \\(t\\) на \\(V\\), совпадающая с \\(s\\) в каждой точке \\(V\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#SameValueAtTwoPoints","kind":"rule","labels":[{"language":"en","status":"official","text":"\\(f(p) = f(g)\\) when the values of \\(f\\) at \\(p\\) and at \\(g\\) coincide"},{"language":"ru","status":"unofficial","text":"\\(f(p) = f(g)\\), когда значения \\(f\\) в \\(p\\) и в \\(g\\) совпадают"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#Section","kind":"type_decl","labels":[{"language":"en","status":"official","text":"section of a presheaf over an open subset"},{"language":"ru","status":"unofficial","text":"сечение предпучка над открытым подмножеством"}],"name":"Section","package":"urn:stacks:clir:sheaf-cohomology"},{"id":"urn:stacks:clir:sheaf-cohomology#SectionsOfConstantSheaf","kind":"rule","labels":[{"language":"en","status":"official","text":"006W: a locally constant function \\(U → Z/2\\) on an open \\(U\\) of \\(X\\) is a section of \\((Z/2)_X\\) over \\(U\\)"},{"language":"ru","status":"unofficial","text":"006W: локально постоянная функция \\(U → Z/2\\) на открытом \\(U ⊂ X\\) — сечение \\((Z/2)_X\\) над \\(U\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#Space","kind":"type_decl","labels":[{"language":"en","status":"official","text":"topological space"},{"language":"ru","status":"unofficial","text":"топологическое пространство"}],"name":"Space","package":"urn:stacks:clir:sheaf-cohomology"},{"id":"urn:stacks:clir:sheaf-cohomology#SubsetByPoints","kind":"rule","labels":[{"language":"en","status":"official","text":"\\(A ⊂ B\\) when every point of \\(A\\) lies in \\(B\\)"},{"language":"ru","status":"unofficial","text":"\\(A ⊂ B\\), когда всякая точка \\(A\\) лежит в \\(B\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#agree_at","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the two functions take the same value at the point"},{"language":"ru","status":"unofficial","text":"две функции принимают одно значение в точке"}],"name":"agree_at","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#agree_at/arg/s","labels":[],"name":"s","type":{"name":"urn:stacks:clir:sheaf-cohomology#Section"}},{"id":"urn:stacks:clir:sheaf-cohomology#agree_at/arg/t","labels":[],"name":"t","type":{"name":"urn:stacks:clir:sheaf-cohomology#Section"}},{"id":"urn:stacks:clir:sheaf-cohomology#agree_at/arg/p","labels":[],"name":"p","type":{"name":"urn:stacks:clir:sheaf-cohomology#Point"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#all_functions_presented","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"every function \\(U → Z/2\\) is presented as a candidate section of \\(F\\) over \\(U\\)"},{"language":"ru","status":"unofficial","text":"всякая функция \\(U → Z/2\\) предъявлена кандидатом в сечения \\(F\\) над \\(U\\)"}],"name":"all_functions_presented","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#all_functions_presented/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#all_functions_presented/arg/u","labels":[],"name":"u","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#candidate_subset","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(U\\) is presented as a subset of \\(X\\) (possibly empty), to be tested for openness"},{"language":"ru","status":"unofficial","text":"\\(U\\) предъявлено как подмножество \\(X\\) (возможно, пустое) для проверки открытости"}],"name":"candidate_subset","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#candidate_subset/arg/u","labels":[],"name":"u","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}},{"id":"urn:stacks:clir:sheaf-cohomology#candidate_subset/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#cech_c0_order","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"01FI: \\(|C^0| = |F(U_1)| · |F(U_2)|\\)"},{"language":"ru","status":"unofficial","text":"01FI: \\(|C^0| = |F(U_1)| · |F(U_2)|\\)"}],"name":"cech_c0_order","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#cech_c0_order/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#cech_c0_order/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:sheaf-cohomology#Covering"}},{"id":"urn:stacks:clir:sheaf-cohomology#cech_c0_order/arg/n","labels":[],"name":"n","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#cech_c1_order","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"01FI: \\(|C^1| = |F(U_12)|\\)"},{"language":"ru","status":"unofficial","text":"01FI: \\(|C^1| = |F(U_12)|\\)"}],"name":"cech_c1_order","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#cech_c1_order/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#cech_c1_order/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:sheaf-cohomology#Covering"}},{"id":"urn:stacks:clir:sheaf-cohomology#cech_c1_order/arg/n","labels":[],"name":"n","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#cech_h0_matches_sections","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"01EG: the natural map \\(F(U) → Ȟ^0(𝒰, F)\\) is bijective on this covering: the orders coincide"},{"language":"ru","status":"unofficial","text":"01EG: естественное отображение \\(F(U) → Ȟ^0(𝒰, F)\\) на этом покрытии биективно: порядки совпадают"}],"name":"cech_h0_matches_sections","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#cech_h0_matches_sections/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#cech_h0_matches_sections/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:sheaf-cohomology#Covering"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#cech_h0_order","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"01EF: \\(|Ȟ^0(𝒰, F)| = |ker d^0|\\), the number of compatible pairs"},{"language":"ru","status":"unofficial","text":"01EF: \\(|Ȟ^0(𝒰, F)| = |ker d^0|\\) — число согласованных пар"}],"name":"cech_h0_order","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#cech_h0_order/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#cech_h0_order/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:sheaf-cohomology#Covering"}},{"id":"urn:stacks:clir:sheaf-cohomology#cech_h0_order/arg/n","labels":[],"name":"n","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#cech_h1_dimension","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(dim_{Z/2} Ȟ^1(𝒰, F) = k\\), i.e. \\(|Ȟ^1| = 2^k\\)"},{"language":"ru","status":"unofficial","text":"\\(dim_{Z/2} Ȟ^1(𝒰, F) = k\\), то есть \\(|Ȟ^1| = 2^k\\)"}],"name":"cech_h1_dimension","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#cech_h1_dimension/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#cech_h1_dimension/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:sheaf-cohomology#Covering"}},{"id":"urn:stacks:clir:sheaf-cohomology#cech_h1_dimension/arg/k","labels":[],"name":"k","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#cech_h1_order","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"01EF: \\(|Ȟ^1(𝒰, F)|\\) for a two-member covering: \\(C^2 = 0\\), so \\(Ȟ^1 = C^1 / im d^0\\) and \\(|Ȟ^1| · |C^0| = |C^1| · |ker d^0|\\)"},{"language":"ru","status":"unofficial","text":"01EF: \\(|Ȟ^1(𝒰, F)|\\) для двучленного покрытия: \\(C^2 = 0\\), потому \\(Ȟ^1 = C^1 / im d^0\\) и \\(|Ȟ^1| · |C^0| = |C^1| · |ker d^0|\\)"}],"name":"cech_h1_order","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#cech_h1_order/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#cech_h1_order/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:sheaf-cohomology#Covering"}},{"id":"urn:stacks:clir:sheaf-cohomology#cech_h1_order/arg/n","labels":[],"name":"n","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#compatible_pair","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"01FI: \\((s_1, s_2) ∈ C^0 = F(U_1) × F(U_2)\\) lies in \\(ker d^0\\): \\(s_1\\) and \\(s_2\\) agree on \\(U_12\\)"},{"language":"ru","status":"unofficial","text":"01FI: \\((s_1, s_2) ∈ C^0 = F(U_1) × F(U_2)\\) лежит в \\(ker d^0\\): \\(s_1\\) и \\(s_2\\) согласованы на \\(U_12\\)"}],"name":"compatible_pair","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#compatible_pair/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#compatible_pair/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:sheaf-cohomology#Covering"}},{"id":"urn:stacks:clir:sheaf-cohomology#compatible_pair/arg/s1","labels":[],"name":"s1","type":{"name":"urn:stacks:clir:sheaf-cohomology#Section"}},{"id":"urn:stacks:clir:sheaf-cohomology#compatible_pair/arg/s2","labels":[],"name":"s2","type":{"name":"urn:stacks:clir:sheaf-cohomology#Section"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#contains","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the point lies in the subset \\(U\\)"},{"language":"ru","status":"unofficial","text":"точка лежит в подмножестве \\(U\\)"}],"name":"contains","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#contains/arg/u","labels":[],"name":"u","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}},{"id":"urn:stacks:clir:sheaf-cohomology#contains/arg/p","labels":[],"name":"p","type":{"name":"urn:stacks:clir:sheaf-cohomology#Point"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#covers","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the covering is an open covering \\(U = ∪ U_i\\) of \\(U\\)"},{"language":"ru","status":"unofficial","text":"покрытие — открытое покрытие \\(U = ∪ U_i\\) множества \\(U\\)"}],"name":"covers","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#covers/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:sheaf-cohomology#Covering"}},{"id":"urn:stacks:clir:sheaf-cohomology#covers/arg/u","labels":[],"name":"u","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#defined_on","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the candidate is a function on the points of \\(U\\)"},{"language":"ru","status":"unofficial","text":"кандидат — функция на точках \\(U\\)"}],"name":"defined_on","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#defined_on/arg/s","labels":[],"name":"s","type":{"name":"urn:stacks:clir:sheaf-cohomology#Section"}},{"id":"urn:stacks:clir:sheaf-cohomology#defined_on/arg/u","labels":[],"name":"u","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#finite_space","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(X\\) is a finite space with the specialization (Alexandrov) topology: the open subsets are exactly the subsets stable under generalization"},{"language":"ru","status":"unofficial","text":"\\(X\\) — конечное пространство с топологией специализации (Александрова): открытые подмножества — в точности устойчивые под генерализацией"}],"name":"finite_space","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#finite_space/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#first_member","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"01FI: \\(U_1\\), the first member in the total ordering of the covering"},{"language":"ru","status":"unofficial","text":"01FI: \\(U_1\\) — первый член в полном упорядочении покрытия"}],"name":"first_member","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#first_member/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:sheaf-cohomology#Covering"}},{"id":"urn:stacks:clir:sheaf-cohomology#first_member/arg/u","labels":[],"name":"u","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#generalizes","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"0061: \\(g ⇝ p\\) — \\(p\\) is a specialization of \\(g\\), \\(g\\) a generalization of \\(p\\): \\(p\\) ∈ closure of \\(\\{g\\}\\)"},{"language":"ru","status":"unofficial","text":"0061: \\(g ⇝ p\\) — \\(p\\) есть специализация \\(g\\), \\(g\\) — генерализация \\(p\\): \\(p\\) ∈ замыканию \\(\\{g\\}\\)"}],"name":"generalizes","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#generalizes/arg/g","labels":[],"name":"g","type":{"name":"urn:stacks:clir:sheaf-cohomology#Point"}},{"id":"urn:stacks:clir:sheaf-cohomology#generalizes/arg/p","labels":[],"name":"p","type":{"name":"urn:stacks:clir:sheaf-cohomology#Point"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#intersection","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(W = U_i ∩ U_j\\)"},{"language":"ru","status":"unofficial","text":"\\(W = U_i ∩ U_j\\)"}],"name":"intersection","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#intersection/arg/w","labels":[],"name":"w","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}},{"id":"urn:stacks:clir:sheaf-cohomology#intersection/arg/u1","labels":[],"name":"u1","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}},{"id":"urn:stacks:clir:sheaf-cohomology#intersection/arg/u2","labels":[],"name":"u2","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#locally_constant","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"006W: the function on \\(U\\) is locally constant: constant along every generalization \\(g ⇝ p\\) inside \\(U\\), hence on connected components"},{"language":"ru","status":"unofficial","text":"006W: функция на \\(U\\) локально постоянна: постоянна вдоль всякой генерализации \\(g ⇝ p\\) внутри \\(U\\), то есть на компонентах связности"}],"name":"locally_constant","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#locally_constant/arg/s","labels":[],"name":"s","type":{"name":"urn:stacks:clir:sheaf-cohomology#Section"}},{"id":"urn:stacks:clir:sheaf-cohomology#locally_constant/arg/u","labels":[],"name":"u","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#member","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(U_i\\) is a member of the covering"},{"language":"ru","status":"unofficial","text":"\\(U_i\\) — член покрытия"}],"name":"member","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#member/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:sheaf-cohomology#Covering"}},{"id":"urn:stacks:clir:sheaf-cohomology#member/arg/ui","labels":[],"name":"ui","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#open_of","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(U\\) is an open subset of \\(X\\)"},{"language":"ru","status":"unofficial","text":"\\(U\\) — открытое подмножество \\(X\\)"}],"name":"open_of","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#open_of/arg/u","labels":[],"name":"u","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}},{"id":"urn:stacks:clir:sheaf-cohomology#open_of/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#point_of","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the point belongs to \\(X\\)"},{"language":"ru","status":"unofficial","text":"точка принадлежит \\(X\\)"}],"name":"point_of","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#point_of/arg/p","labels":[],"name":"p","type":{"name":"urn:stacks:clir:sheaf-cohomology#Point"}},{"id":"urn:stacks:clir:sheaf-cohomology#point_of/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#pow2-0","kind":"assertion","labels":[{"language":"en","status":"official","text":"orders of finite \\(Z/2\\)-vector spaces: \\(2^k\\) for \\(k = 0 … 6\\)"},{"language":"ru","status":"unofficial","text":"порядки конечных \\(Z/2\\)-пространств: \\(2^k\\) при \\(k = 0 … 6\\)"}],"package":"urn:stacks:clir:sheaf-cohomology"},{"id":"urn:stacks:clir:sheaf-cohomology#power_of_two","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(n = 2^k\\), the order of a \\(k\\)-dimensional \\(Z/2\\)-vector 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\\(V\\)"}],"name":"restricts_to","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#restricts_to/arg/s","labels":[],"name":"s","type":{"name":"urn:stacks:clir:sheaf-cohomology#Section"}},{"id":"urn:stacks:clir:sheaf-cohomology#restricts_to/arg/v","labels":[],"name":"v","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}},{"id":"urn:stacks:clir:sheaf-cohomology#restricts_to/arg/t","labels":[],"name":"t","type":{"name":"urn:stacks:clir:sheaf-cohomology#Section"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#same_value_at","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the function takes the same value at the two points: \\(f(p) = f(g)\\)"},{"language":"ru","status":"unofficial","text":"функция принимает одно значение в двух точках: \\(f(p) = f(g)\\)"}],"name":"same_value_at","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#same_value_at/arg/s","labels":[],"name":"s","type":{"name":"urn:stacks:clir:sheaf-cohomology#Section"}},{"id":"urn:stacks:clir:sheaf-cohomology#same_value_at/arg/p","labels":[],"name":"p","type":{"name":"urn:stacks:clir:sheaf-cohomology#Point"}},{"id":"urn:stacks:clir:sheaf-cohomology#same_value_at/arg/g","labels":[],"name":"g","type":{"name":"urn:stacks:clir:sheaf-cohomology#Point"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#second_member","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"01FI: \\(U_2\\), the second member in the total ordering of the covering"},{"language":"ru","status":"unofficial","text":"01FI: \\(U_2\\) — второй член в полном упорядочении покрытия"}],"name":"second_member","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#second_member/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:sheaf-cohomology#Covering"}},{"id":"urn:stacks:clir:sheaf-cohomology#second_member/arg/u","labels":[],"name":"u","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#section_over","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(s ∈ F(U)\\), a section of \\(F\\) over \\(U\\)"},{"language":"ru","status":"unofficial","text":"\\(s ∈ F(U)\\), сечение \\(F\\) над \\(U\\)"}],"name":"section_over","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#section_over/arg/s","labels":[],"name":"s","type":{"name":"urn:stacks:clir:sheaf-cohomology#Section"}},{"id":"urn:stacks:clir:sheaf-cohomology#section_over/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#section_over/arg/u","labels":[],"name":"u","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#subset_of","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(A ⊂ B\\) as sets of points"},{"language":"ru","status":"unofficial","text":"\\(A ⊂ B\\) как множества точек"}],"name":"subset_of","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#subset_of/arg/a","labels":[],"name":"a","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}},{"id":"urn:stacks:clir:sheaf-cohomology#subset_of/arg/b","labels":[],"name":"b","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#target","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the covering is aimed at \\(U\\): its members are proposed to cover \\(U\\)"},{"language":"ru","status":"unofficial","text":"покрытие нацелено на \\(U\\): его члены предложены как покрытие \\(U\\)"}],"name":"target","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#target/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:sheaf-cohomology#Covering"}},{"id":"urn:stacks:clir:sheaf-cohomology#target/arg/u","labels":[],"name":"u","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#value_at","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the function takes the value \\(v ∈ \\{0, 1\\}\\) at the point"},{"language":"ru","status":"unofficial","text":"функция принимает значение \\(v ∈ \\{0, 1\\}\\) в точке"}],"name":"value_at","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#value_at/arg/s","labels":[],"name":"s","type":{"name":"urn:stacks:clir:sheaf-cohomology#Section"}},{"id":"urn:stacks:clir:sheaf-cohomology#value_at/arg/p","labels":[],"name":"p","type":{"name":"urn:stacks:clir:sheaf-cohomology#Point"}},{"id":"urn:stacks:clir:sheaf-cohomology#value_at/arg/v","labels":[],"name":"v","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#z2_constant_sheaf","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(F\\) is the constant sheaf \\((Z/2)_X\\): sections over \\(U\\) are the locally constant maps \\(U → Z/2\\)"},{"language":"ru","status":"unofficial","text":"\\(F\\) — постоянный пучок \\((Z/2)_X\\): сечения над \\(U\\) — локально постоянные отображения \\(U → Z/2\\)"}],"name":"z2_constant_sheaf","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#z2_constant_sheaf/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#z2_constant_sheaf/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"}],"tables":[]},"provenance":{"acts":[{"contributed":true,"fragmentCount":26,"fragments":["urn:stacks:clir:sheaf-cohomology#ST_0061","urn:stacks:clir:sheaf-cohomology#ST_0062","urn:stacks:clir:sheaf-cohomology#ST_006E","urn:stacks:clir:sheaf-cohomology#ST_006W","urn:stacks:clir:sheaf-cohomology#ST_01EF","urn:stacks:clir:sheaf-cohomology#ST_01EG","urn:stacks:clir:sheaf-cohomology#ST_01FI","urn:stacks:clir:sheaf-cohomology#ST_01FM"],"jurisdiction":"none","namespace":"urn:stacks:clir:sheaf-cohomology","package":"stacks-sheaf-cohomology","title":"Когомологии пучков по The Stacks Project: пучок, 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$X$ be a topological space.\n\\begin{enumerate}\n\\item If $x, x' \\in X$ then we say $x$ is a {\\it specialization} of $x'$,\nor $x'$ is a {\\it generalization} of $x$ if $x \\in \\overline{\\{x'\\}}$.\nNotation: $x' \\leadsto x$.\n\\item A subset $T \\subset X$ is {\\it stable under specialization}\nif for all $x' \\in T$ and every specialization $x' \\leadsto x$ we have\n$x \\in T$.\n\\item A subset $T \\subset X$ is {\\it stable under generalization}\nif for all $x \\in T$ and every generalization $x' \\leadsto x$ we have\n$x' \\in T$.\n\\end{enumerate}\n\\end{definition}"}]},{"contentHash":"sha256:72a4516ed2a111a21ce235698e7b70959c091ee1e696331bfa395377c0199cd0","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_TOPOLOGY_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:sheaf-cohomology#ST_0062","kind":"fragment","locator":"tag/0062","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:881627a4bc7da511bfc2e3d7cd01f971abb8e2f72ae02f0a295a59b98de782f8","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-open-closed-specialization}\nLet $X$ be a topological space.\n\\begin{enumerate}\n\\item Any closed subset of $X$ is stable under specialization.\n\\item Any open subset of $X$ is stable under generalization.\n\\item A subset $T \\subset X$ is stable under specialization\nif and only if\nthe complement $T^c$ is stable under generalization.\n\\end{enumerate}\n\\end{lemma}"}]},{"contentHash":"sha256:5940ad0150140fcc429bfcb66cf91e0303638f57e88332528ed64e0204a1cc00","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_SHEAVES_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:sheaf-cohomology#ST_006E","kind":"fragment","locator":"tag/006E","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:e728b50fec00b5a51ffd85b7518d87d42e03d9b363884124cfca05a3aa077563","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-presheaf}\nLet $X$ be a topological space.\n\\begin{enumerate}\n\\item A {\\it presheaf $\\mathcal{F}$ of sets on $X$} is a rule which\nassigns to each open $U \\subset X$ a set $\\mathcal{F}(U)$ and\nto each inclusion $V \\subset U$ a map\n$\\rho^U_V : \\mathcal{F}(U) \\to \\mathcal{F}(V)$ such that\n$\\rho^U_U = \\text{id}_{\\mathcal{F}(U)}$ and\nwhenever $W \\subset V \\subset U$ we have\n$\\rho^U_W = \\rho^V_W \\circ \\rho ^U_V$.\n\\item A {\\it morphism $\\varphi : \\mathcal{F} \\to \\mathcal{G}$\nof presheaves of sets on $X$} is a rule which assigns to each\nopen $U \\subset X$ a map of sets $\\varphi : \\mathcal{F}(U)\n\\to \\mathcal{G}(U)$ compatible with restriction maps,\ni.e., whenever $V \\subset U \\subset X$ are open the\ndiagram\n$$\n\\xymatrix{\n\\mathcal{F}(U) \\ar[r]^\\varphi \\ar[d]^{\\rho^U_V} &\n\\mathcal{G}(U) \\ar[d]^{\\rho^U_V} \\\\\n\\mathcal{F}(V) \\ar[r]^\\varphi & \\mathcal{G}(V)\n}\n$$\ncommutes.\n\\item The category of presheaves of sets on $X$ will be denoted\n$\\textit{PSh}(X)$.\n\\end{enumerate}\n\\end{definition}"}]},{"contentHash":"sha256:e15b670cf8eaecb2e9774da8a00edf1ff814ee1bfb132f1eb75f8f353d5ac661","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_SHEAVES_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:sheaf-cohomology#ST_006W","kind":"fragment","locator":"tag/006W","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:dd05660d523049b43c908e00357b644b49aecdc06ba7eb53adb2c325e626e2c3","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-constant-sheaf}\nLet $X$ be a topological space. Let $A$ be a set.\nThe {\\it constant sheaf with value $A$} denoted $\\underline{A}$, or\n$\\underline{A}_X$ is the sheaf that assigns to an open $U \\subset X$\nthe set of all locally constant maps $U \\to A$ with restriction mappings\ngiven by restrictions of functions.\n\\end{definition}"}]},{"contentHash":"sha256:44ca688f1b371dc408a7186efb3a770de31e5cf542f3b48926b9929642de76de","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_COHOMOLOGY_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:sheaf-cohomology#ST_01EF","kind":"fragment","locator":"tag/01EF","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:95b540b6925fdab8fa3a53def4a39d102181a21ef47fcce25d076b6a97bc0b1a","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-cech-complex}\nLet $X$ be a topological space.\nLet $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering.\nLet $\\mathcal{F}$ be an abelian presheaf on $X$.\nThe complex $\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})$\nis the {\\it {\\v C}ech complex} associated to $\\mathcal{F}$ and the\nopen covering $\\mathcal{U}$. Its cohomology groups\n$H^i(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}))$ are\ncalled the {\\it {\\v C}ech cohomology groups} associated to\n$\\mathcal{F}$ and the covering $\\mathcal{U}$.\nThey are denoted $\\check H^i(\\mathcal{U}, \\mathcal{F})$.\n\\end{definition}"}]},{"contentHash":"sha256:32257319184ae0e9c5584919035ad193c88002f0f7ce4eded08cfa8acbb48f20","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_COHOMOLOGY_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:sheaf-cohomology#ST_01EG","kind":"fragment","locator":"tag/01EG","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:4c5b22ab222601aaae95952557fde5950661f40d81749b9bb69b4a8f61ff6108","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-cech-h0}\nLet $X$ be a topological space.\nLet $\\mathcal{F}$ be an abelian presheaf on $X$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is an abelian sheaf and\n\\item for every open covering $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$\nthe natural map\n$$\n\\mathcal{F}(U) \\to \\check{H}^0(\\mathcal{U}, \\mathcal{F})\n$$\nis bijective.\n\\end{enumerate}\n\\end{lemma}"}]},{"contentHash":"sha256:100a796ac383e4193e5e93c36072f00b3cbdec059858ee798aa6bb02c78c3997","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_COHOMOLOGY_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:sheaf-cohomology#ST_01FI","kind":"fragment","locator":"tag/01FI","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:4dbf509c03629779d780d60d6502aae3001874670f84eb6bc4cd9e0dab533882","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-ordered-cech-complex}\nLet $X$ be a topological space.\nLet $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering.\nAssume given a total ordering on $I$.\nLet $\\mathcal{F}$ be an abelian presheaf on $X$.\nThe complex $\\check{\\mathcal{C}}_{ord}^\\bullet(\\mathcal{U}, \\mathcal{F})$\nis the {\\it ordered {\\v C}ech complex} associated to $\\mathcal{F}$, the\nopen covering $\\mathcal{U}$ and the given total ordering on $I$.\n\\end{definition}"}]},{"contentHash":"sha256:13ee1524e1e1e43409a1fb17733206a93162331f710fb544c8218b6036a284e1","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_COHOMOLOGY_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:sheaf-cohomology#ST_01FM","kind":"fragment","locator":"tag/01FM","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:783d414a41461bccfb07b242a527657ee8fda1055c33a6ef41abf92ee925f966","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-alternating-usual}\nLet $X$ be a topological space.\nLet $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering.\nAssume $I$ comes equipped with a total ordering.\nThe map $c \\circ \\pi$ is homotopic to the identity on\n$\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})$.\nIn particular the inclusion map\n$\\check{\\mathcal{C}}_{alt}^\\bullet(\\mathcal{U}, \\mathcal{F}) \\to\n\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})$\nis a homotopy equivalence.\n\\end{lemma}"}]}],"text":"cech_h1_order: **TRUE_ONLY** — установлено\nВыведено правом: section_over(urn:case:stacks:cech:fn-W-1, urn:case:stacks:cech:F, urn:case:stacks:cech:W); section_over(urn:case:stacks:cech:fn-U2-00, urn:case:stacks:cech:F, urn:case:stacks:cech:U2); section_over(urn:case:stacks:cech:fn-W-0, urn:case:stacks:cech:F, urn:case:stacks:cech:W); section_over(urn:case:stacks:cech:fn-U2-11, urn:case:stacks:cech:F, urn:case:stacks:cech:U2); section_over(urn:case:stacks:cech:fn-X-000, urn:case:stacks:cech:F, urn:case:stacks:cech:X); section_over(urn:case:stacks:cech:fn-U1-11, urn:case:stacks:cech:F, urn:case:stacks:cech:U1); section_over(urn:case:stacks:cech:fn-U1-00, urn:case:stacks:cech:F, urn:case:stacks:cech:U1); cech_c0_order(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, 4); section_over(urn:case:stacks:cech:fn-X-111, urn:case:stacks:cech:F, urn:case:stacks:cech:X); covers(urn:case:stacks:cech:cov, urn:case:stacks:cech:X); cech_c1_order(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, 2); compatible_pair(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, urn:case:stacks:cech:fn-U1-11, urn:case:stacks:cech:fn-U2-11); compatible_pair(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, urn:case:stacks:cech:fn-U1-00, urn:case:stacks:cech:fn-U2-00); cech_h0_order(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, 2); cech_h0_matches_sections(urn:case:stacks:cech:F, urn:case:stacks:cech:cov); cech_h1_order(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, 1); cech_h1_dimension(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, 0)\n…и ещё 467 выведенных фактов вне предмета вопроса (полный вывод — law_explain)\nПрименены правила: AgreeAtPoint, CechC0Order, CechC1Order, CechH0MatchesSections, CechH0Order, CechH1Dimension, CechH1Order, CompatiblePair, CoversByPoints, IntersectionByPoints, LocallyConstantAlongSpecialization, OpenByGeneralizationStability, RestrictionByValues, SameValueAtTwoPoints, SectionsOfConstantSheaf, SubsetByPoints\nПраво (вне юрисдикции государства): Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — доктрина (programHash sha256:6b62eb59e903…)\nproof-граф: 574 узлов — поле evaluation готово для law_explain"},{"answer":{"answer":{"evaluationStatus":"COMPUTED","kind":"TRUTH","meaning":"установлено","missingInputs":[],"truthStatus":"TRUE_ONLY"},"closedEditionRules":[],"derived":["section_over(urn:case:stacks:cech:fn-W-1, urn:case:stacks:cech:F, urn:case:stacks:cech:W)","section_over(urn:case:stacks:cech:fn-U2-00, urn:case:stacks:cech:F, urn:case:stacks:cech:U2)","section_over(urn:case:stacks:cech:fn-W-0, urn:case:stacks:cech:F, urn:case:stacks:cech:W)","section_over(urn:case:stacks:cech:fn-U2-11, urn:case:stacks:cech:F, urn:case:stacks:cech:U2)","section_over(urn:case:stacks:cech:fn-X-000, urn:case:stacks:cech:F, urn:case:stacks:cech:X)","section_over(urn:case:stacks:cech:fn-U1-11, urn:case:stacks:cech:F, urn:case:stacks:cech:U1)","section_over(urn:case:stacks:cech:fn-U1-00, urn:case:stacks:cech:F, urn:case:stacks:cech:U1)","cech_c0_order(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, 4)","section_over(urn:case:stacks:cech:fn-X-111, urn:case:stacks:cech:F, urn:case:stacks:cech:X)","covers(urn:case:stacks:cech:cov, urn:case:stacks:cech:X)","cech_c1_order(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, 2)","compatible_pair(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, urn:case:stacks:cech:fn-U1-11, urn:case:stacks:cech:fn-U2-11)","compatible_pair(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, urn:case:stacks:cech:fn-U1-00, urn:case:stacks:cech:fn-U2-00)","cech_h0_order(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, 2)","cech_h0_matches_sections(urn:case:stacks:cech:F, 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functions agree at a point when their values there coincide"},{"language":"ru","status":"unofficial","text":"две функции согласованы в точке, когда их значения там совпадают"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#CechC0Order","kind":"rule","labels":[{"language":"en","status":"official","text":"01FI: the order of \\(C^0\\) is the product of the orders of \\(F(U_1)\\) and \\(F(U_2)\\), all functions presented"},{"language":"ru","status":"unofficial","text":"01FI: порядок \\(C^0\\) — произведение порядков \\(F(U_1)\\) и \\(F(U_2)\\), все функции предъявлены"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#CechC1Order","kind":"rule","labels":[{"language":"en","status":"official","text":"01FI: the order of \\(C^1\\) is the order of \\(F(U_12)\\), all functions presented"},{"language":"ru","status":"unofficial","text":"01FI: порядок \\(C^1\\) — порядок \\(F(U_12)\\), все функции предъявлены"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#CechH0MatchesSections","kind":"rule","labels":[{"language":"en","status":"official","text":"01EG on a covering of \\(U\\): \\(|F(U)| = |Ȟ^0(𝒰, F)|\\) with all functions on \\(U\\) presented"},{"language":"ru","status":"unofficial","text":"01EG на покрытии \\(U\\): \\(|F(U)| = |Ȟ^0(𝒰, F)|\\) при всех предъявленных функциях на \\(U\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#CechH0Order","kind":"rule","labels":[{"language":"en","status":"official","text":"01EF: \\(Ȟ^0 = ker d^0\\); its order is the number of pairs \\((s_1, s_2)\\) agreeing on \\(U_12\\)"},{"language":"ru","status":"unofficial","text":"01EF: \\(Ȟ^0 = ker d^0\\); его порядок — число пар \\((s_1, s_2)\\), согласованных на \\(U_12\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#CechH1Dimension","kind":"rule","labels":[{"language":"en","status":"official","text":"the \\(Z/2\\)-dimension of \\(Ȟ^1\\) is the exponent of its order"},{"language":"ru","status":"unofficial","text":"\\(Z/2\\)-размерность \\(Ȟ^1\\) — показатель её порядка"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#CechH1Order","kind":"rule","labels":[{"language":"en","status":"official","text":"01EF with 01FI: over the field \\(Z/2\\), \\(|im d^0| = |C^0| / |ker d^0|\\) and \\(Ȟ^1 = C^1 / im d^0\\), so the order \\(n\\) of \\(Ȟ^1\\) satisfies \\(n · |C^0| = |C^1| · |ker d^0|\\)"},{"language":"ru","status":"unofficial","text":"01EF с 01FI: над полем \\(Z/2\\) \\(|im d^0| = |C^0| / |ker d^0|\\) и \\(Ȟ^1 = C^1 / im d^0\\), потому порядок \\(n\\) группы \\(Ȟ^1\\) удовлетворяет \\(n · |C^0| = |C^1| · |ker d^0|\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#CompatiblePair","kind":"rule","labels":[{"language":"en","status":"official","text":"01FI, \\(d^0(s_1, s_2) = s_2|U_12 − s_1|U_12\\): the pair is a 0-cocycle when both restrict to the same section of \\(U_12\\)"},{"language":"ru","status":"unofficial","text":"01FI, \\(d^0(s_1, s_2) = s_2|U_12 − s_1|U_12\\): пара — 0-коцикл, когда обе ограничиваются в одно сечение над \\(U_12\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#Covering","kind":"type_decl","labels":[{"language":"en","status":"official","text":"open covering \\(U = ∪ U_i\\)"},{"language":"ru","status":"unofficial","text":"открытое покрытие \\(U = ∪ U_i\\)"}],"name":"Covering","package":"urn:stacks:clir:sheaf-cohomology"},{"id":"urn:stacks:clir:sheaf-cohomology#CoversByPoints","kind":"rule","labels":[{"language":"en","status":"official","text":"the members cover \\(U\\) when each member lies in \\(U\\) and every point of \\(U\\) lies in some member"},{"language":"ru","status":"unofficial","text":"члены покрывают \\(U\\), когда каждый член лежит в \\(U\\) и всякая точка \\(U\\) лежит в некотором члене"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#IntersectionByPoints","kind":"rule","labels":[{"language":"en","status":"official","text":"\\(W = U ∩ V\\) when \\(W ⊂ U, W ⊂ V\\) and every point common to \\(U\\) and \\(V\\) lies in \\(W\\)"},{"language":"ru","status":"unofficial","text":"\\(W = U ∩ V\\), когда \\(W ⊂ U, W ⊂ V\\) и всякая общая точка \\(U\\) и \\(V\\) лежит в \\(W\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#LocallyConstantAlongSpecialization","kind":"rule","labels":[{"language":"en","status":"official","text":"006W with 0061: a function on \\(U\\) is locally constant when \\(f(p) = f(g)\\) for every point \\(p\\) of \\(U\\) and every generalization \\(g\\) of \\(p\\) in \\(U\\)"},{"language":"ru","status":"unofficial","text":"006W с 0061: функция на \\(U\\) локально постоянна, когда \\(f(p) = f(g)\\) для всякой точки \\(p\\) из \\(U\\) и всякой генерализации \\(g\\) точки \\(p\\) в \\(U\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#Open","kind":"type_decl","labels":[{"language":"en","status":"official","text":"open subset of a topological space"},{"language":"ru","status":"unofficial","text":"открытое подмножество топологического пространства"}],"name":"Open","package":"urn:stacks:clir:sheaf-cohomology"},{"id":"urn:stacks:clir:sheaf-cohomology#OpenByGeneralizationStability","kind":"rule","labels":[{"language":"en","status":"official","text":"0062 (2) on a finite space: a subset of \\(X\\) stable under generalization is open"},{"language":"ru","status":"unofficial","text":"0062 (2) на конечном пространстве: подмножество \\(X\\), устойчивое под генерализацией, открыто"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#Point","kind":"type_decl","labels":[{"language":"en","status":"official","text":"point of a topological space"},{"language":"ru","status":"unofficial","text":"точка топологического пространства"}],"name":"Point","package":"urn:stacks:clir:sheaf-cohomology"},{"id":"urn:stacks:clir:sheaf-cohomology#RestrictionByValues","kind":"rule","labels":[{"language":"en","status":"official","text":"006E: for \\(V ⊂ U\\) the restriction of \\(s\\) to \\(V\\) is the function \\(t\\) on \\(V\\) agreeing with \\(s\\) at every point of \\(V\\)"},{"language":"ru","status":"unofficial","text":"006E: при \\(V ⊂ U\\) ограничение \\(s\\) на \\(V\\) — функция \\(t\\) на \\(V\\), совпадающая с \\(s\\) в каждой точке \\(V\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#SameValueAtTwoPoints","kind":"rule","labels":[{"language":"en","status":"official","text":"\\(f(p) = f(g)\\) when the values of \\(f\\) at \\(p\\) and at \\(g\\) coincide"},{"language":"ru","status":"unofficial","text":"\\(f(p) = f(g)\\), когда значения \\(f\\) в \\(p\\) и в \\(g\\) совпадают"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#Section","kind":"type_decl","labels":[{"language":"en","status":"official","text":"section of a presheaf over an open subset"},{"language":"ru","status":"unofficial","text":"сечение предпучка над открытым подмножеством"}],"name":"Section","package":"urn:stacks:clir:sheaf-cohomology"},{"id":"urn:stacks:clir:sheaf-cohomology#SectionsOfConstantSheaf","kind":"rule","labels":[{"language":"en","status":"official","text":"006W: a locally constant function \\(U → Z/2\\) on an open \\(U\\) of \\(X\\) is a section of \\((Z/2)_X\\) over \\(U\\)"},{"language":"ru","status":"unofficial","text":"006W: локально постоянная функция \\(U → Z/2\\) на открытом \\(U ⊂ X\\) — сечение \\((Z/2)_X\\) над \\(U\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#Space","kind":"type_decl","labels":[{"language":"en","status":"official","text":"topological space"},{"language":"ru","status":"unofficial","text":"топологическое пространство"}],"name":"Space","package":"urn:stacks:clir:sheaf-cohomology"},{"id":"urn:stacks:clir:sheaf-cohomology#SubsetByPoints","kind":"rule","labels":[{"language":"en","status":"official","text":"\\(A ⊂ B\\) when every point of \\(A\\) lies in \\(B\\)"},{"language":"ru","status":"unofficial","text":"\\(A ⊂ B\\), когда всякая точка \\(A\\) лежит в \\(B\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#agree_at","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the two functions take the same value at the point"},{"language":"ru","status":"unofficial","text":"две функции принимают одно значение в точке"}],"name":"agree_at","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#agree_at/arg/s","labels":[],"name":"s","type":{"name":"urn:stacks:clir:sheaf-cohomology#Section"}},{"id":"urn:stacks:clir:sheaf-cohomology#agree_at/arg/t","labels":[],"name":"t","type":{"name":"urn:stacks:clir:sheaf-cohomology#Section"}},{"id":"urn:stacks:clir:sheaf-cohomology#agree_at/arg/p","labels":[],"name":"p","type":{"name":"urn:stacks:clir:sheaf-cohomology#Point"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#all_functions_presented","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"every function \\(U → Z/2\\) is presented as a candidate section of \\(F\\) over \\(U\\)"},{"language":"ru","status":"unofficial","text":"всякая функция \\(U → Z/2\\) предъявлена кандидатом в сечения \\(F\\) над \\(U\\)"}],"name":"all_functions_presented","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#all_functions_presented/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#all_functions_presented/arg/u","labels":[],"name":"u","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#candidate_subset","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(U\\) is presented as a subset of \\(X\\) (possibly empty), to be tested for openness"},{"language":"ru","status":"unofficial","text":"\\(U\\) предъявлено как подмножество \\(X\\) (возможно, пустое) для проверки открытости"}],"name":"candidate_subset","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#candidate_subset/arg/u","labels":[],"name":"u","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}},{"id":"urn:stacks:clir:sheaf-cohomology#candidate_subset/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#cech_c0_order","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"01FI: \\(|C^0| = |F(U_1)| · |F(U_2)|\\)"},{"language":"ru","status":"unofficial","text":"01FI: \\(|C^0| = |F(U_1)| · |F(U_2)|\\)"}],"name":"cech_c0_order","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#cech_c0_order/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#cech_c0_order/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:sheaf-cohomology#Covering"}},{"id":"urn:stacks:clir:sheaf-cohomology#cech_c0_order/arg/n","labels":[],"name":"n","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#cech_c1_order","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"01FI: \\(|C^1| = |F(U_12)|\\)"},{"language":"ru","status":"unofficial","text":"01FI: \\(|C^1| = |F(U_12)|\\)"}],"name":"cech_c1_order","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#cech_c1_order/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#cech_c1_order/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:sheaf-cohomology#Covering"}},{"id":"urn:stacks:clir:sheaf-cohomology#cech_c1_order/arg/n","labels":[],"name":"n","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#cech_h0_matches_sections","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"01EG: the natural map \\(F(U) → Ȟ^0(𝒰, F)\\) is bijective on this covering: the orders coincide"},{"language":"ru","status":"unofficial","text":"01EG: естественное отображение \\(F(U) → Ȟ^0(𝒰, F)\\) на этом покрытии биективно: порядки совпадают"}],"name":"cech_h0_matches_sections","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#cech_h0_matches_sections/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#cech_h0_matches_sections/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:sheaf-cohomology#Covering"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#cech_h0_order","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"01EF: \\(|Ȟ^0(𝒰, F)| = |ker d^0|\\), the number of compatible pairs"},{"language":"ru","status":"unofficial","text":"01EF: \\(|Ȟ^0(𝒰, F)| = |ker d^0|\\) — число согласованных пар"}],"name":"cech_h0_order","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#cech_h0_order/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#cech_h0_order/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:sheaf-cohomology#Covering"}},{"id":"urn:stacks:clir:sheaf-cohomology#cech_h0_order/arg/n","labels":[],"name":"n","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#cech_h1_dimension","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(dim_{Z/2} Ȟ^1(𝒰, F) = k\\), i.e. \\(|Ȟ^1| = 2^k\\)"},{"language":"ru","status":"unofficial","text":"\\(dim_{Z/2} Ȟ^1(𝒰, F) = k\\), то есть \\(|Ȟ^1| = 2^k\\)"}],"name":"cech_h1_dimension","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#cech_h1_dimension/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#cech_h1_dimension/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:sheaf-cohomology#Covering"}},{"id":"urn:stacks:clir:sheaf-cohomology#cech_h1_dimension/arg/k","labels":[],"name":"k","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#cech_h1_order","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"01EF: \\(|Ȟ^1(𝒰, F)|\\) for a two-member covering: \\(C^2 = 0\\), so \\(Ȟ^1 = C^1 / im d^0\\) and \\(|Ȟ^1| · |C^0| = |C^1| · |ker d^0|\\)"},{"language":"ru","status":"unofficial","text":"01EF: \\(|Ȟ^1(𝒰, F)|\\) для двучленного покрытия: \\(C^2 = 0\\), потому \\(Ȟ^1 = C^1 / im d^0\\) и \\(|Ȟ^1| · |C^0| = |C^1| · |ker d^0|\\)"}],"name":"cech_h1_order","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#cech_h1_order/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#cech_h1_order/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:sheaf-cohomology#Covering"}},{"id":"urn:stacks:clir:sheaf-cohomology#cech_h1_order/arg/n","labels":[],"name":"n","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#compatible_pair","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"01FI: \\((s_1, s_2) ∈ C^0 = F(U_1) × F(U_2)\\) lies in \\(ker d^0\\): \\(s_1\\) and \\(s_2\\) agree on \\(U_12\\)"},{"language":"ru","status":"unofficial","text":"01FI: \\((s_1, s_2) ∈ C^0 = F(U_1) × F(U_2)\\) лежит в \\(ker d^0\\): \\(s_1\\) и \\(s_2\\) согласованы на \\(U_12\\)"}],"name":"compatible_pair","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#compatible_pair/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#compatible_pair/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:sheaf-cohomology#Covering"}},{"id":"urn:stacks:clir:sheaf-cohomology#compatible_pair/arg/s1","labels":[],"name":"s1","type":{"name":"urn:stacks:clir:sheaf-cohomology#Section"}},{"id":"urn:stacks:clir:sheaf-cohomology#compatible_pair/arg/s2","labels":[],"name":"s2","type":{"name":"urn:stacks:clir:sheaf-cohomology#Section"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#contains","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the point lies in the subset \\(U\\)"},{"language":"ru","status":"unofficial","text":"точка лежит в подмножестве \\(U\\)"}],"name":"contains","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#contains/arg/u","labels":[],"name":"u","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}},{"id":"urn:stacks:clir:sheaf-cohomology#contains/arg/p","labels":[],"name":"p","type":{"name":"urn:stacks:clir:sheaf-cohomology#Point"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#covers","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the covering is an open covering \\(U = ∪ U_i\\) of \\(U\\)"},{"language":"ru","status":"unofficial","text":"покрытие — открытое покрытие \\(U = ∪ U_i\\) множества \\(U\\)"}],"name":"covers","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#covers/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:sheaf-cohomology#Covering"}},{"id":"urn:stacks:clir:sheaf-cohomology#covers/arg/u","labels":[],"name":"u","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#defined_on","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the candidate is a function on the points of \\(U\\)"},{"language":"ru","status":"unofficial","text":"кандидат — функция на точках \\(U\\)"}],"name":"defined_on","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#defined_on/arg/s","labels":[],"name":"s","type":{"name":"urn:stacks:clir:sheaf-cohomology#Section"}},{"id":"urn:stacks:clir:sheaf-cohomology#defined_on/arg/u","labels":[],"name":"u","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#finite_space","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(X\\) is a finite space with the specialization (Alexandrov) topology: the open subsets are exactly the subsets stable under generalization"},{"language":"ru","status":"unofficial","text":"\\(X\\) — конечное пространство с топологией специализации (Александрова): открытые подмножества — в точности устойчивые под генерализацией"}],"name":"finite_space","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#finite_space/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#first_member","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"01FI: \\(U_1\\), the first member in the total ordering of the covering"},{"language":"ru","status":"unofficial","text":"01FI: \\(U_1\\) — первый член в полном упорядочении покрытия"}],"name":"first_member","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#first_member/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:sheaf-cohomology#Covering"}},{"id":"urn:stacks:clir:sheaf-cohomology#first_member/arg/u","labels":[],"name":"u","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#generalizes","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"0061: \\(g ⇝ p\\) — \\(p\\) is a specialization of \\(g\\), \\(g\\) a generalization of \\(p\\): \\(p\\) ∈ closure of \\(\\{g\\}\\)"},{"language":"ru","status":"unofficial","text":"0061: \\(g ⇝ p\\) — \\(p\\) есть специализация \\(g\\), \\(g\\) — генерализация \\(p\\): \\(p\\) ∈ замыканию \\(\\{g\\}\\)"}],"name":"generalizes","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#generalizes/arg/g","labels":[],"name":"g","type":{"name":"urn:stacks:clir:sheaf-cohomology#Point"}},{"id":"urn:stacks:clir:sheaf-cohomology#generalizes/arg/p","labels":[],"name":"p","type":{"name":"urn:stacks:clir:sheaf-cohomology#Point"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#intersection","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(W = U_i ∩ U_j\\)"},{"language":"ru","status":"unofficial","text":"\\(W = U_i ∩ U_j\\)"}],"name":"intersection","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#intersection/arg/w","labels":[],"name":"w","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}},{"id":"urn:stacks:clir:sheaf-cohomology#intersection/arg/u1","labels":[],"name":"u1","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}},{"id":"urn:stacks:clir:sheaf-cohomology#intersection/arg/u2","labels":[],"name":"u2","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#locally_constant","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"006W: the function on \\(U\\) is locally constant: constant along every generalization \\(g ⇝ p\\) inside \\(U\\), hence on connected components"},{"language":"ru","status":"unofficial","text":"006W: функция на \\(U\\) локально постоянна: постоянна вдоль всякой генерализации \\(g ⇝ p\\) внутри \\(U\\), то есть на компонентах связности"}],"name":"locally_constant","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#locally_constant/arg/s","labels":[],"name":"s","type":{"name":"urn:stacks:clir:sheaf-cohomology#Section"}},{"id":"urn:stacks:clir:sheaf-cohomology#locally_constant/arg/u","labels":[],"name":"u","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#member","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(U_i\\) is a member of the covering"},{"language":"ru","status":"unofficial","text":"\\(U_i\\) — член 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\\(X\\)"}],"name":"point_of","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#point_of/arg/p","labels":[],"name":"p","type":{"name":"urn:stacks:clir:sheaf-cohomology#Point"}},{"id":"urn:stacks:clir:sheaf-cohomology#point_of/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#pow2-0","kind":"assertion","labels":[{"language":"en","status":"official","text":"orders of finite \\(Z/2\\)-vector spaces: \\(2^k\\) for \\(k = 0 … 6\\)"},{"language":"ru","status":"unofficial","text":"порядки конечных \\(Z/2\\)-пространств: \\(2^k\\) при \\(k = 0 … 6\\)"}],"package":"urn:stacks:clir:sheaf-cohomology"},{"id":"urn:stacks:clir:sheaf-cohomology#power_of_two","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(n = 2^k\\), the order of a \\(k\\)-dimensional \\(Z/2\\)-vector 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$X$ be a topological space.\n\\begin{enumerate}\n\\item If $x, x' \\in X$ then we say $x$ is a {\\it specialization} of $x'$,\nor $x'$ is a {\\it generalization} of $x$ if $x \\in \\overline{\\{x'\\}}$.\nNotation: $x' \\leadsto x$.\n\\item A subset $T \\subset X$ is {\\it stable under specialization}\nif for all $x' \\in T$ and every specialization $x' \\leadsto x$ we have\n$x \\in T$.\n\\item A subset $T \\subset X$ is {\\it stable under generalization}\nif for all $x \\in T$ and every generalization $x' \\leadsto x$ we have\n$x' \\in T$.\n\\end{enumerate}\n\\end{definition}"}]},{"contentHash":"sha256:72a4516ed2a111a21ce235698e7b70959c091ee1e696331bfa395377c0199cd0","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_TOPOLOGY_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:sheaf-cohomology#ST_0062","kind":"fragment","locator":"tag/0062","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:881627a4bc7da511bfc2e3d7cd01f971abb8e2f72ae02f0a295a59b98de782f8","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-open-closed-specialization}\nLet $X$ be a topological space.\n\\begin{enumerate}\n\\item Any closed subset of $X$ is stable under specialization.\n\\item Any open subset of $X$ is stable under generalization.\n\\item A subset $T \\subset X$ is stable under specialization\nif and only if\nthe complement $T^c$ is stable under generalization.\n\\end{enumerate}\n\\end{lemma}"}]},{"contentHash":"sha256:5940ad0150140fcc429bfcb66cf91e0303638f57e88332528ed64e0204a1cc00","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_SHEAVES_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:sheaf-cohomology#ST_006E","kind":"fragment","locator":"tag/006E","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:e728b50fec00b5a51ffd85b7518d87d42e03d9b363884124cfca05a3aa077563","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-presheaf}\nLet $X$ be a topological space.\n\\begin{enumerate}\n\\item A {\\it presheaf $\\mathcal{F}$ of sets on $X$} is a rule which\nassigns to each open $U \\subset X$ a set $\\mathcal{F}(U)$ and\nto each inclusion $V \\subset U$ a map\n$\\rho^U_V : \\mathcal{F}(U) \\to \\mathcal{F}(V)$ such that\n$\\rho^U_U = \\text{id}_{\\mathcal{F}(U)}$ and\nwhenever $W \\subset V \\subset U$ we have\n$\\rho^U_W = \\rho^V_W \\circ \\rho ^U_V$.\n\\item A {\\it morphism $\\varphi : \\mathcal{F} \\to \\mathcal{G}$\nof presheaves of sets on $X$} is a rule which assigns to each\nopen $U \\subset X$ a map of sets $\\varphi : \\mathcal{F}(U)\n\\to \\mathcal{G}(U)$ compatible with restriction maps,\ni.e., whenever $V \\subset U \\subset X$ are open the\ndiagram\n$$\n\\xymatrix{\n\\mathcal{F}(U) \\ar[r]^\\varphi \\ar[d]^{\\rho^U_V} &\n\\mathcal{G}(U) \\ar[d]^{\\rho^U_V} \\\\\n\\mathcal{F}(V) \\ar[r]^\\varphi & \\mathcal{G}(V)\n}\n$$\ncommutes.\n\\item The category of presheaves of sets on $X$ will be denoted\n$\\textit{PSh}(X)$.\n\\end{enumerate}\n\\end{definition}"}]},{"contentHash":"sha256:e15b670cf8eaecb2e9774da8a00edf1ff814ee1bfb132f1eb75f8f353d5ac661","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_SHEAVES_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:sheaf-cohomology#ST_006W","kind":"fragment","locator":"tag/006W","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:dd05660d523049b43c908e00357b644b49aecdc06ba7eb53adb2c325e626e2c3","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-constant-sheaf}\nLet $X$ be a topological space. Let $A$ be a set.\nThe {\\it constant sheaf with value $A$} denoted $\\underline{A}$, or\n$\\underline{A}_X$ is the sheaf that assigns to an open $U \\subset X$\nthe set of all locally constant maps $U \\to A$ with restriction mappings\ngiven by restrictions of functions.\n\\end{definition}"}]},{"contentHash":"sha256:44ca688f1b371dc408a7186efb3a770de31e5cf542f3b48926b9929642de76de","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_COHOMOLOGY_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:sheaf-cohomology#ST_01EF","kind":"fragment","locator":"tag/01EF","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:95b540b6925fdab8fa3a53def4a39d102181a21ef47fcce25d076b6a97bc0b1a","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-cech-complex}\nLet $X$ be a topological space.\nLet $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering.\nLet $\\mathcal{F}$ be an abelian presheaf on $X$.\nThe complex $\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})$\nis the {\\it {\\v C}ech complex} associated to $\\mathcal{F}$ and the\nopen covering $\\mathcal{U}$. Its cohomology groups\n$H^i(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}))$ are\ncalled the {\\it {\\v C}ech cohomology groups} associated to\n$\\mathcal{F}$ and the covering $\\mathcal{U}$.\nThey are denoted $\\check H^i(\\mathcal{U}, \\mathcal{F})$.\n\\end{definition}"}]},{"contentHash":"sha256:32257319184ae0e9c5584919035ad193c88002f0f7ce4eded08cfa8acbb48f20","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_COHOMOLOGY_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:sheaf-cohomology#ST_01EG","kind":"fragment","locator":"tag/01EG","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:4c5b22ab222601aaae95952557fde5950661f40d81749b9bb69b4a8f61ff6108","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-cech-h0}\nLet $X$ be a topological space.\nLet $\\mathcal{F}$ be an abelian presheaf on $X$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is an abelian sheaf and\n\\item for every open covering $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$\nthe natural map\n$$\n\\mathcal{F}(U) \\to \\check{H}^0(\\mathcal{U}, \\mathcal{F})\n$$\nis bijective.\n\\end{enumerate}\n\\end{lemma}"}]},{"contentHash":"sha256:100a796ac383e4193e5e93c36072f00b3cbdec059858ee798aa6bb02c78c3997","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_COHOMOLOGY_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:sheaf-cohomology#ST_01FI","kind":"fragment","locator":"tag/01FI","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:4dbf509c03629779d780d60d6502aae3001874670f84eb6bc4cd9e0dab533882","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-ordered-cech-complex}\nLet $X$ be a topological space.\nLet $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering.\nAssume given a total ordering on $I$.\nLet $\\mathcal{F}$ be an abelian presheaf on $X$.\nThe complex $\\check{\\mathcal{C}}_{ord}^\\bullet(\\mathcal{U}, \\mathcal{F})$\nis the {\\it ordered {\\v C}ech complex} associated to $\\mathcal{F}$, the\nopen covering $\\mathcal{U}$ and the given total ordering on $I$.\n\\end{definition}"}]},{"contentHash":"sha256:13ee1524e1e1e43409a1fb17733206a93162331f710fb544c8218b6036a284e1","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_COHOMOLOGY_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:sheaf-cohomology#ST_01FM","kind":"fragment","locator":"tag/01FM","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:783d414a41461bccfb07b242a527657ee8fda1055c33a6ef41abf92ee925f966","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-alternating-usual}\nLet $X$ be a topological space.\nLet $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering.\nAssume $I$ comes equipped with a total ordering.\nThe map $c \\circ \\pi$ is homotopic to the identity on\n$\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})$.\nIn particular the inclusion map\n$\\check{\\mathcal{C}}_{alt}^\\bullet(\\mathcal{U}, \\mathcal{F}) \\to\n\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})$\nis a homotopy equivalence.\n\\end{lemma}"}]}],"text":"cech_h1_dimension: **TRUE_ONLY** — установлено\nВыведено правом: section_over(urn:case:stacks:cech:fn-W-1, urn:case:stacks:cech:F, urn:case:stacks:cech:W); section_over(urn:case:stacks:cech:fn-U2-00, urn:case:stacks:cech:F, urn:case:stacks:cech:U2); section_over(urn:case:stacks:cech:fn-W-0, urn:case:stacks:cech:F, urn:case:stacks:cech:W); section_over(urn:case:stacks:cech:fn-U2-11, urn:case:stacks:cech:F, urn:case:stacks:cech:U2); section_over(urn:case:stacks:cech:fn-X-000, urn:case:stacks:cech:F, urn:case:stacks:cech:X); section_over(urn:case:stacks:cech:fn-U1-11, urn:case:stacks:cech:F, urn:case:stacks:cech:U1); section_over(urn:case:stacks:cech:fn-U1-00, urn:case:stacks:cech:F, urn:case:stacks:cech:U1); cech_c0_order(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, 4); section_over(urn:case:stacks:cech:fn-X-111, urn:case:stacks:cech:F, urn:case:stacks:cech:X); covers(urn:case:stacks:cech:cov, urn:case:stacks:cech:X); cech_c1_order(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, 2); compatible_pair(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, urn:case:stacks:cech:fn-U1-11, urn:case:stacks:cech:fn-U2-11); compatible_pair(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, urn:case:stacks:cech:fn-U1-00, urn:case:stacks:cech:fn-U2-00); cech_h0_order(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, 2); cech_h0_matches_sections(urn:case:stacks:cech:F, urn:case:stacks:cech:cov); cech_h1_order(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, 1); cech_h1_dimension(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, 0)\n…и ещё 467 выведенных фактов вне предмета вопроса (полный вывод — law_explain)\nПрименены правила: AgreeAtPoint, CechC0Order, CechC1Order, CechH0MatchesSections, CechH0Order, CechH1Dimension, CechH1Order, CompatiblePair, CoversByPoints, IntersectionByPoints, LocallyConstantAlongSpecialization, OpenByGeneralizationStability, RestrictionByValues, SameValueAtTwoPoints, SectionsOfConstantSheaf, SubsetByPoints\nПраво (вне юрисдикции государства): Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — доктрина (programHash sha256:6b62eb59e903…)\nproof-граф: 574 узлов — поле evaluation готово для law_explain"},{"answer":{"answer":{"evaluationStatus":"COMPUTED","kind":"TRUTH","meaning":"НЕ УСТАНОВЛЕНО: в формализованном праве нет ни подтверждения, ни опровержения (открытый мир §69) — это не «нет»","missingInputs":[],"truthStatus":"NEITHER"},"closedEditionRules":[],"derived":["section_over(urn:case:stacks:cech:fn-W-1, urn:case:stacks:cech:F, urn:case:stacks:cech:W)","section_over(urn:case:stacks:cech:fn-U2-00, urn:case:stacks:cech:F, urn:case:stacks:cech:U2)","section_over(urn:case:stacks:cech:fn-W-0, urn:case:stacks:cech:F, urn:case:stacks:cech:W)","section_over(urn:case:stacks:cech:fn-U2-11, urn:case:stacks:cech:F, urn:case:stacks:cech:U2)","section_over(urn:case:stacks:cech:fn-X-000, urn:case:stacks:cech:F, urn:case:stacks:cech:X)","section_over(urn:case:stacks:cech:fn-U1-11, urn:case:stacks:cech:F, urn:case:stacks:cech:U1)","section_over(urn:case:stacks:cech:fn-U1-00, urn:case:stacks:cech:F, urn:case:stacks:cech:U1)","cech_c0_order(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, 4)","section_over(urn:case:stacks:cech:fn-X-111, urn:case:stacks:cech:F, urn:case:stacks:cech:X)","covers(urn:case:stacks:cech:cov, urn:case:stacks:cech:X)","cech_c1_order(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, 2)","compatible_pair(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, urn:case:stacks:cech:fn-U1-11, urn:case:stacks:cech:fn-U2-11)","compatible_pair(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, urn:case:stacks:cech:fn-U1-00, urn:case:stacks:cech:fn-U2-00)","cech_h0_order(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, 2)","cech_h0_matches_sections(urn:case:stacks:cech:F, urn:case:stacks:cech:cov)","cech_h1_order(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, 1)","cech_h1_dimension(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, 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совпадают"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#CechC0Order","kind":"rule","labels":[{"language":"en","status":"official","text":"01FI: the order of \\(C^0\\) is the product of the orders of \\(F(U_1)\\) and \\(F(U_2)\\), all functions presented"},{"language":"ru","status":"unofficial","text":"01FI: порядок \\(C^0\\) — произведение порядков \\(F(U_1)\\) и \\(F(U_2)\\), все функции предъявлены"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#CechC1Order","kind":"rule","labels":[{"language":"en","status":"official","text":"01FI: the order of \\(C^1\\) is the order of \\(F(U_12)\\), all functions presented"},{"language":"ru","status":"unofficial","text":"01FI: порядок \\(C^1\\) — порядок \\(F(U_12)\\), все функции 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\\(U_12\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#CechH1Dimension","kind":"rule","labels":[{"language":"en","status":"official","text":"the \\(Z/2\\)-dimension of \\(Ȟ^1\\) is the exponent of its order"},{"language":"ru","status":"unofficial","text":"\\(Z/2\\)-размерность \\(Ȟ^1\\) — показатель её порядка"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#CechH1Order","kind":"rule","labels":[{"language":"en","status":"official","text":"01EF with 01FI: over the field \\(Z/2\\), \\(|im d^0| = |C^0| / |ker d^0|\\) and \\(Ȟ^1 = C^1 / im d^0\\), so the order \\(n\\) of \\(Ȟ^1\\) satisfies \\(n · |C^0| = |C^1| · |ker d^0|\\)"},{"language":"ru","status":"unofficial","text":"01EF с 01FI: над полем \\(Z/2\\) \\(|im d^0| = |C^0| / |ker d^0|\\) и \\(Ȟ^1 = C^1 / im d^0\\), потому порядок \\(n\\) группы \\(Ȟ^1\\) удовлетворяет \\(n · |C^0| = |C^1| · |ker d^0|\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#CompatiblePair","kind":"rule","labels":[{"language":"en","status":"official","text":"01FI, \\(d^0(s_1, s_2) = s_2|U_12 − s_1|U_12\\): the pair is a 0-cocycle when both restrict to the same section of \\(U_12\\)"},{"language":"ru","status":"unofficial","text":"01FI, \\(d^0(s_1, s_2) = s_2|U_12 − s_1|U_12\\): пара — 0-коцикл, когда обе ограничиваются в одно сечение над \\(U_12\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#Covering","kind":"type_decl","labels":[{"language":"en","status":"official","text":"open covering \\(U = ∪ U_i\\)"},{"language":"ru","status":"unofficial","text":"открытое покрытие \\(U = ∪ U_i\\)"}],"name":"Covering","package":"urn:stacks:clir:sheaf-cohomology"},{"id":"urn:stacks:clir:sheaf-cohomology#CoversByPoints","kind":"rule","labels":[{"language":"en","status":"official","text":"the members cover \\(U\\) when each member lies in \\(U\\) and every point of \\(U\\) lies in some member"},{"language":"ru","status":"unofficial","text":"члены покрывают \\(U\\), когда каждый член лежит в \\(U\\) и всякая точка \\(U\\) лежит в некотором члене"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#IntersectionByPoints","kind":"rule","labels":[{"language":"en","status":"official","text":"\\(W = U ∩ V\\) when \\(W ⊂ U, W ⊂ V\\) and every point common to \\(U\\) and \\(V\\) lies in \\(W\\)"},{"language":"ru","status":"unofficial","text":"\\(W = U ∩ V\\), когда \\(W ⊂ U, W ⊂ V\\) и всякая общая точка \\(U\\) и \\(V\\) лежит в \\(W\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#LocallyConstantAlongSpecialization","kind":"rule","labels":[{"language":"en","status":"official","text":"006W with 0061: a function on \\(U\\) is locally constant when \\(f(p) = f(g)\\) for every point \\(p\\) of \\(U\\) and every generalization \\(g\\) of \\(p\\) in \\(U\\)"},{"language":"ru","status":"unofficial","text":"006W с 0061: функция на \\(U\\) локально постоянна, когда \\(f(p) = f(g)\\) для всякой точки \\(p\\) из \\(U\\) и всякой генерализации \\(g\\) точки \\(p\\) в \\(U\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#Open","kind":"type_decl","labels":[{"language":"en","status":"official","text":"open subset of a topological space"},{"language":"ru","status":"unofficial","text":"открытое подмножество топологического пространства"}],"name":"Open","package":"urn:stacks:clir:sheaf-cohomology"},{"id":"urn:stacks:clir:sheaf-cohomology#OpenByGeneralizationStability","kind":"rule","labels":[{"language":"en","status":"official","text":"0062 (2) on a finite space: a subset of \\(X\\) stable under generalization is open"},{"language":"ru","status":"unofficial","text":"0062 (2) на конечном пространстве: подмножество \\(X\\), устойчивое под генерализацией, открыто"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#Point","kind":"type_decl","labels":[{"language":"en","status":"official","text":"point of a topological space"},{"language":"ru","status":"unofficial","text":"точка топологического пространства"}],"name":"Point","package":"urn:stacks:clir:sheaf-cohomology"},{"id":"urn:stacks:clir:sheaf-cohomology#RestrictionByValues","kind":"rule","labels":[{"language":"en","status":"official","text":"006E: for \\(V ⊂ U\\) the restriction of \\(s\\) to \\(V\\) is the function \\(t\\) on \\(V\\) agreeing with \\(s\\) at every point of \\(V\\)"},{"language":"ru","status":"unofficial","text":"006E: при \\(V ⊂ U\\) ограничение \\(s\\) на \\(V\\) — функция \\(t\\) на \\(V\\), совпадающая с \\(s\\) в каждой точке \\(V\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#SameValueAtTwoPoints","kind":"rule","labels":[{"language":"en","status":"official","text":"\\(f(p) = f(g)\\) when the values of \\(f\\) at \\(p\\) and at \\(g\\) coincide"},{"language":"ru","status":"unofficial","text":"\\(f(p) = f(g)\\), когда значения \\(f\\) в \\(p\\) и в \\(g\\) совпадают"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#Section","kind":"type_decl","labels":[{"language":"en","status":"official","text":"section of a presheaf over an open subset"},{"language":"ru","status":"unofficial","text":"сечение предпучка над открытым подмножеством"}],"name":"Section","package":"urn:stacks:clir:sheaf-cohomology"},{"id":"urn:stacks:clir:sheaf-cohomology#SectionsOfConstantSheaf","kind":"rule","labels":[{"language":"en","status":"official","text":"006W: a locally constant function \\(U → Z/2\\) on an open \\(U\\) of \\(X\\) is a section of \\((Z/2)_X\\) over \\(U\\)"},{"language":"ru","status":"unofficial","text":"006W: локально постоянная функция \\(U → Z/2\\) на открытом \\(U ⊂ X\\) — сечение \\((Z/2)_X\\) над \\(U\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#Space","kind":"type_decl","labels":[{"language":"en","status":"official","text":"topological space"},{"language":"ru","status":"unofficial","text":"топологическое пространство"}],"name":"Space","package":"urn:stacks:clir:sheaf-cohomology"},{"id":"urn:stacks:clir:sheaf-cohomology#SubsetByPoints","kind":"rule","labels":[{"language":"en","status":"official","text":"\\(A ⊂ B\\) when every point of \\(A\\) lies in \\(B\\)"},{"language":"ru","status":"unofficial","text":"\\(A ⊂ B\\), когда всякая точка \\(A\\) лежит в \\(B\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#agree_at","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the two functions take the same value at the point"},{"language":"ru","status":"unofficial","text":"две функции принимают одно значение в точке"}],"name":"agree_at","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#agree_at/arg/s","labels":[],"name":"s","type":{"name":"urn:stacks:clir:sheaf-cohomology#Section"}},{"id":"urn:stacks:clir:sheaf-cohomology#agree_at/arg/t","labels":[],"name":"t","type":{"name":"urn:stacks:clir:sheaf-cohomology#Section"}},{"id":"urn:stacks:clir:sheaf-cohomology#agree_at/arg/p","labels":[],"name":"p","type":{"name":"urn:stacks:clir:sheaf-cohomology#Point"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#all_functions_presented","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"every function \\(U → Z/2\\) is presented as a candidate section of \\(F\\) over \\(U\\)"},{"language":"ru","status":"unofficial","text":"всякая функция \\(U → Z/2\\) предъявлена кандидатом в сечения \\(F\\) над \\(U\\)"}],"name":"all_functions_presented","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#all_functions_presented/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#all_functions_presented/arg/u","labels":[],"name":"u","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#candidate_subset","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(U\\) is presented as a subset of \\(X\\) (possibly empty), to be tested for openness"},{"language":"ru","status":"unofficial","text":"\\(U\\) предъявлено как подмножество \\(X\\) (возможно, пустое) для проверки открытости"}],"name":"candidate_subset","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#candidate_subset/arg/u","labels":[],"name":"u","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}},{"id":"urn:stacks:clir:sheaf-cohomology#candidate_subset/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#cech_c0_order","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"01FI: \\(|C^0| = |F(U_1)| · |F(U_2)|\\)"},{"language":"ru","status":"unofficial","text":"01FI: \\(|C^0| = |F(U_1)| · |F(U_2)|\\)"}],"name":"cech_c0_order","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#cech_c0_order/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#cech_c0_order/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:sheaf-cohomology#Covering"}},{"id":"urn:stacks:clir:sheaf-cohomology#cech_c0_order/arg/n","labels":[],"name":"n","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#cech_c1_order","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"01FI: \\(|C^1| = |F(U_12)|\\)"},{"language":"ru","status":"unofficial","text":"01FI: \\(|C^1| = |F(U_12)|\\)"}],"name":"cech_c1_order","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#cech_c1_order/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#cech_c1_order/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:sheaf-cohomology#Covering"}},{"id":"urn:stacks:clir:sheaf-cohomology#cech_c1_order/arg/n","labels":[],"name":"n","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#cech_h0_matches_sections","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"01EG: the natural map \\(F(U) → Ȟ^0(𝒰, F)\\) is bijective on this covering: the orders coincide"},{"language":"ru","status":"unofficial","text":"01EG: естественное отображение \\(F(U) → Ȟ^0(𝒰, F)\\) на этом покрытии биективно: порядки совпадают"}],"name":"cech_h0_matches_sections","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#cech_h0_matches_sections/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#cech_h0_matches_sections/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:sheaf-cohomology#Covering"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#cech_h0_order","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"01EF: \\(|Ȟ^0(𝒰, F)| = |ker d^0|\\), the number of compatible pairs"},{"language":"ru","status":"unofficial","text":"01EF: \\(|Ȟ^0(𝒰, F)| = |ker d^0|\\) — число согласованных пар"}],"name":"cech_h0_order","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#cech_h0_order/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#cech_h0_order/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:sheaf-cohomology#Covering"}},{"id":"urn:stacks:clir:sheaf-cohomology#cech_h0_order/arg/n","labels":[],"name":"n","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#cech_h1_dimension","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(dim_{Z/2} Ȟ^1(𝒰, F) = k\\), i.e. \\(|Ȟ^1| = 2^k\\)"},{"language":"ru","status":"unofficial","text":"\\(dim_{Z/2} Ȟ^1(𝒰, F) = k\\), то есть \\(|Ȟ^1| = 2^k\\)"}],"name":"cech_h1_dimension","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#cech_h1_dimension/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#cech_h1_dimension/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:sheaf-cohomology#Covering"}},{"id":"urn:stacks:clir:sheaf-cohomology#cech_h1_dimension/arg/k","labels":[],"name":"k","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#cech_h1_order","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"01EF: \\(|Ȟ^1(𝒰, F)|\\) for a two-member covering: \\(C^2 = 0\\), so \\(Ȟ^1 = C^1 / im d^0\\) and \\(|Ȟ^1| · |C^0| = |C^1| · |ker d^0|\\)"},{"language":"ru","status":"unofficial","text":"01EF: \\(|Ȟ^1(𝒰, F)|\\) для двучленного покрытия: \\(C^2 = 0\\), потому \\(Ȟ^1 = C^1 / im d^0\\) и \\(|Ȟ^1| · |C^0| = |C^1| · |ker d^0|\\)"}],"name":"cech_h1_order","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#cech_h1_order/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#cech_h1_order/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:sheaf-cohomology#Covering"}},{"id":"urn:stacks:clir:sheaf-cohomology#cech_h1_order/arg/n","labels":[],"name":"n","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#compatible_pair","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"01FI: \\((s_1, s_2) ∈ C^0 = F(U_1) × F(U_2)\\) lies in \\(ker d^0\\): \\(s_1\\) and \\(s_2\\) agree on \\(U_12\\)"},{"language":"ru","status":"unofficial","text":"01FI: \\((s_1, s_2) ∈ C^0 = F(U_1) × F(U_2)\\) лежит в \\(ker d^0\\): \\(s_1\\) и \\(s_2\\) согласованы на \\(U_12\\)"}],"name":"compatible_pair","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#compatible_pair/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#compatible_pair/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:sheaf-cohomology#Covering"}},{"id":"urn:stacks:clir:sheaf-cohomology#compatible_pair/arg/s1","labels":[],"name":"s1","type":{"name":"urn:stacks:clir:sheaf-cohomology#Section"}},{"id":"urn:stacks:clir:sheaf-cohomology#compatible_pair/arg/s2","labels":[],"name":"s2","type":{"name":"urn:stacks:clir:sheaf-cohomology#Section"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#contains","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the point lies in the subset \\(U\\)"},{"language":"ru","status":"unofficial","text":"точка лежит в подмножестве \\(U\\)"}],"name":"contains","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#contains/arg/u","labels":[],"name":"u","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}},{"id":"urn:stacks:clir:sheaf-cohomology#contains/arg/p","labels":[],"name":"p","type":{"name":"urn:stacks:clir:sheaf-cohomology#Point"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#covers","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the covering is an open covering \\(U = ∪ U_i\\) of \\(U\\)"},{"language":"ru","status":"unofficial","text":"покрытие — открытое покрытие \\(U = ∪ U_i\\) множества \\(U\\)"}],"name":"covers","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#covers/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:sheaf-cohomology#Covering"}},{"id":"urn:stacks:clir:sheaf-cohomology#covers/arg/u","labels":[],"name":"u","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#defined_on","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the candidate is a function on the points of \\(U\\)"},{"language":"ru","status":"unofficial","text":"кандидат — функция на точках \\(U\\)"}],"name":"defined_on","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#defined_on/arg/s","labels":[],"name":"s","type":{"name":"urn:stacks:clir:sheaf-cohomology#Section"}},{"id":"urn:stacks:clir:sheaf-cohomology#defined_on/arg/u","labels":[],"name":"u","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#finite_space","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(X\\) is a finite space with the specialization (Alexandrov) topology: the open subsets are exactly the subsets stable under generalization"},{"language":"ru","status":"unofficial","text":"\\(X\\) — конечное пространство с топологией специализации (Александрова): открытые подмножества — в точности устойчивые под генерализацией"}],"name":"finite_space","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#finite_space/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#first_member","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"01FI: \\(U_1\\), the first member in the total ordering of the covering"},{"language":"ru","status":"unofficial","text":"01FI: \\(U_1\\) — первый член в полном упорядочении покрытия"}],"name":"first_member","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#first_member/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:sheaf-cohomology#Covering"}},{"id":"urn:stacks:clir:sheaf-cohomology#first_member/arg/u","labels":[],"name":"u","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#generalizes","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"0061: \\(g ⇝ p\\) — \\(p\\) is a specialization of \\(g\\), \\(g\\) a generalization of \\(p\\): \\(p\\) ∈ closure of \\(\\{g\\}\\)"},{"language":"ru","status":"unofficial","text":"0061: \\(g ⇝ p\\) — \\(p\\) есть специализация \\(g\\), \\(g\\) — генерализация \\(p\\): \\(p\\) ∈ замыканию \\(\\{g\\}\\)"}],"name":"generalizes","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#generalizes/arg/g","labels":[],"name":"g","type":{"name":"urn:stacks:clir:sheaf-cohomology#Point"}},{"id":"urn:stacks:clir:sheaf-cohomology#generalizes/arg/p","labels":[],"name":"p","type":{"name":"urn:stacks:clir:sheaf-cohomology#Point"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#intersection","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(W = U_i ∩ U_j\\)"},{"language":"ru","status":"unofficial","text":"\\(W = U_i ∩ U_j\\)"}],"name":"intersection","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#intersection/arg/w","labels":[],"name":"w","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}},{"id":"urn:stacks:clir:sheaf-cohomology#intersection/arg/u1","labels":[],"name":"u1","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}},{"id":"urn:stacks:clir:sheaf-cohomology#intersection/arg/u2","labels":[],"name":"u2","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#locally_constant","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"006W: the function on \\(U\\) is locally constant: constant along every generalization \\(g ⇝ p\\) inside \\(U\\), hence on connected components"},{"language":"ru","status":"unofficial","text":"006W: функция на \\(U\\) локально постоянна: постоянна вдоль всякой генерализации \\(g ⇝ p\\) внутри \\(U\\), то есть на компонентах связности"}],"name":"locally_constant","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#locally_constant/arg/s","labels":[],"name":"s","type":{"name":"urn:stacks:clir:sheaf-cohomology#Section"}},{"id":"urn:stacks:clir:sheaf-cohomology#locally_constant/arg/u","labels":[],"name":"u","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#member","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(U_i\\) is a member of the covering"},{"language":"ru","status":"unofficial","text":"\\(U_i\\) — член покрытия"}],"name":"member","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#member/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:sheaf-cohomology#Covering"}},{"id":"urn:stacks:clir:sheaf-cohomology#member/arg/ui","labels":[],"name":"ui","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#open_of","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(U\\) is an open subset of \\(X\\)"},{"language":"ru","status":"unofficial","text":"\\(U\\) — открытое подмножество \\(X\\)"}],"name":"open_of","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#open_of/arg/u","labels":[],"name":"u","type":{"name":"urn:stacks:clir:sheaf-cohomology#Open"}},{"id":"urn:stacks:clir:sheaf-cohomology#open_of/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#point_of","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the point belongs to \\(X\\)"},{"language":"ru","status":"unofficial","text":"точка принадлежит \\(X\\)"}],"name":"point_of","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#point_of/arg/p","labels":[],"name":"p","type":{"name":"urn:stacks:clir:sheaf-cohomology#Point"}},{"id":"urn:stacks:clir:sheaf-cohomology#point_of/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#pow2-0","kind":"assertion","labels":[{"language":"en","status":"official","text":"orders of finite \\(Z/2\\)-vector spaces: \\(2^k\\) for \\(k = 0 … 6\\)"},{"language":"ru","status":"unofficial","text":"порядки конечных \\(Z/2\\)-пространств: \\(2^k\\) при \\(k = 0 … 6\\)"}],"package":"urn:stacks:clir:sheaf-cohomology"},{"id":"urn:stacks:clir:sheaf-cohomology#power_of_two","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(n = 2^k\\), the order of a \\(k\\)-dimensional \\(Z/2\\)-vector 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predicate":"defined_on"},{"args":["urn:case:stacks:cech:fn-W-0","urn:case:stacks:cech:c",0],"predicate":"value_at"},{"args":["urn:case:stacks:cech:fn-W-1","urn:case:stacks:cech:W"],"predicate":"defined_on"},{"args":["urn:case:stacks:cech:fn-W-1","urn:case:stacks:cech:c",1],"predicate":"value_at"},{"args":["urn:case:stacks:cech:F","urn:case:stacks:cech:W"],"predicate":"all_functions_presented"},{"args":["urn:case:stacks:cech:fn-X-000","urn:case:stacks:cech:X"],"predicate":"defined_on"},{"args":["urn:case:stacks:cech:fn-X-000","urn:case:stacks:cech:a",0],"predicate":"value_at"},{"args":["urn:case:stacks:cech:fn-X-000","urn:case:stacks:cech:b",0],"predicate":"value_at"},{"args":["urn:case:stacks:cech:fn-X-000","urn:case:stacks:cech:c",0],"predicate":"value_at"},{"args":["urn:case:stacks:cech:fn-X-001","urn:case:stacks:cech:X"],"predicate":"defined_on"},{"args":["urn:case:stacks:cech:fn-X-001","urn:case:stacks:cech:a",0],"predicate":"value_at"},{"args":["urn:case:stacks:cech:fn-X-001","urn:case:stacks:cech:b",0],"predicate":"value_at"},{"args":["urn:case:stacks:cech:fn-X-001","urn:case:stacks:cech:c",1],"predicate":"value_at"},{"args":["urn:case:stacks:cech:fn-X-010","urn:case:stacks:cech:X"],"predicate":"defined_on"},{"args":["urn:case:stacks:cech:fn-X-010","urn:case:stacks:cech:a",0],"predicate":"value_at"},{"args":["urn:case:stacks:cech:fn-X-010","urn:case:stacks:cech:b",1],"predicate":"value_at"},{"args":["urn:case:stacks:cech:fn-X-010","urn:case:stacks:cech:c",0],"predicate":"value_at"},{"args":["urn:case:stacks:cech:fn-X-011","urn:case:stacks:cech:X"],"predicate":"defined_on"},{"args":["urn:case:stacks:cech:fn-X-011","urn:case:stacks:cech:a",0],"predicate":"value_at"},{"args":["urn:case:stacks:cech:fn-X-011","urn:case:stacks:cech:b",1],"predicate":"value_at"},{"args":["urn:case:stacks:cech:fn-X-011","urn:case:stacks:cech:c",1],"predicate":"value_at"},{"args":["urn:case:stacks:cech:fn-X-100","urn:case:stacks:cech:X"],"predicate":"defined_on"},{"args":["urn:case:stacks:cech:fn-X-100","urn:case:stacks:cech:a",1],"predicate":"value_at"},{"args":["urn:case:stacks:cech:fn-X-100","urn:case:stacks:cech:b",0],"predicate":"value_at"},{"args":["urn:case:stacks:cech:fn-X-100","urn:case:stacks:cech:c",0],"predicate":"value_at"},{"args":["urn:case:stacks:cech:fn-X-101","urn:case:stacks:cech:X"],"predicate":"defined_on"},{"args":["urn:case:stacks:cech:fn-X-101","urn:case:stacks:cech:a",1],"predicate":"value_at"},{"args":["urn:case:stacks:cech:fn-X-101","urn:case:stacks:cech:b",0],"predicate":"value_at"},{"args":["urn:case:stacks:cech:fn-X-101","urn:case:stacks:cech:c",1],"predicate":"value_at"},{"args":["urn:case:stacks:cech:fn-X-110","urn:case:stacks:cech:X"],"predicate":"defined_on"},{"args":["urn:case:stacks:cech:fn-X-110","urn:case:stacks:cech:a",1],"predicate":"value_at"},{"args":["urn:case:stacks:cech:fn-X-110","urn:case:stacks:cech:b",1],"predicate":"value_at"},{"args":["urn:case:stacks:cech:fn-X-110","urn:case:stacks:cech:c",0],"predicate":"value_at"},{"args":["urn:case:stacks:cech:fn-X-111","urn:case:stacks:cech:X"],"predicate":"defined_on"},{"args":["urn:case:stacks:cech:fn-X-111","urn:case:stacks:cech:a",1],"predicate":"value_at"},{"args":["urn:case:stacks:cech:fn-X-111","urn:case:stacks:cech:b",1],"predicate":"value_at"},{"args":["urn:case:stacks:cech:fn-X-111","urn:case:stacks:cech:c",1],"predicate":"value_at"},{"args":["urn:case:stacks:cech:F","urn:case:stacks:cech:X"],"predicate":"all_functions_presented"}],"kind":"truth","legalTime":"2026-09-06","package":"stacks-sheaf-cohomology","predicate":"cech_h1_order","proof":true},"sources":[{"contentHash":"sha256:9bf62fa9de5f46cfcbefb534e889ce088952a428b4991bf8fb33c2d3d212d9a3","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_TOPOLOGY_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:sheaf-cohomology#ST_0061","kind":"fragment","locator":"tag/0061","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:1d6ee8a1b98bc09677428df262ad2123d411e5bb168e3ac20a1c67cb0dd9d0a5","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-specialization}\nLet $X$ be a topological space.\n\\begin{enumerate}\n\\item If $x, x' \\in X$ then we say $x$ is a {\\it specialization} of $x'$,\nor $x'$ is a {\\it generalization} of $x$ if $x \\in \\overline{\\{x'\\}}$.\nNotation: $x' \\leadsto x$.\n\\item A subset $T \\subset X$ is {\\it stable under specialization}\nif for all $x' \\in T$ and every specialization $x' \\leadsto x$ we have\n$x \\in T$.\n\\item A subset $T \\subset X$ is {\\it stable under generalization}\nif for all $x \\in T$ and every generalization $x' \\leadsto x$ we have\n$x' \\in T$.\n\\end{enumerate}\n\\end{definition}"}]},{"contentHash":"sha256:72a4516ed2a111a21ce235698e7b70959c091ee1e696331bfa395377c0199cd0","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_TOPOLOGY_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:sheaf-cohomology#ST_0062","kind":"fragment","locator":"tag/0062","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:881627a4bc7da511bfc2e3d7cd01f971abb8e2f72ae02f0a295a59b98de782f8","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-open-closed-specialization}\nLet $X$ be a topological space.\n\\begin{enumerate}\n\\item Any closed subset of $X$ is stable under specialization.\n\\item Any open subset of $X$ is stable under generalization.\n\\item A subset $T \\subset X$ is stable under specialization\nif and only if\nthe complement $T^c$ is stable under generalization.\n\\end{enumerate}\n\\end{lemma}"}]},{"contentHash":"sha256:5940ad0150140fcc429bfcb66cf91e0303638f57e88332528ed64e0204a1cc00","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_SHEAVES_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:sheaf-cohomology#ST_006E","kind":"fragment","locator":"tag/006E","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:e728b50fec00b5a51ffd85b7518d87d42e03d9b363884124cfca05a3aa077563","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-presheaf}\nLet $X$ be a topological space.\n\\begin{enumerate}\n\\item A {\\it presheaf $\\mathcal{F}$ of sets on $X$} is a rule which\nassigns to each open $U \\subset X$ a set $\\mathcal{F}(U)$ and\nto each inclusion $V \\subset U$ a map\n$\\rho^U_V : \\mathcal{F}(U) \\to \\mathcal{F}(V)$ such that\n$\\rho^U_U = \\text{id}_{\\mathcal{F}(U)}$ and\nwhenever $W \\subset V \\subset U$ we have\n$\\rho^U_W = \\rho^V_W \\circ \\rho ^U_V$.\n\\item A {\\it morphism $\\varphi : \\mathcal{F} \\to \\mathcal{G}$\nof presheaves of sets on $X$} is a rule which assigns to each\nopen $U \\subset X$ a map of sets $\\varphi : \\mathcal{F}(U)\n\\to \\mathcal{G}(U)$ compatible with restriction maps,\ni.e., whenever $V \\subset U \\subset X$ are open the\ndiagram\n$$\n\\xymatrix{\n\\mathcal{F}(U) \\ar[r]^\\varphi \\ar[d]^{\\rho^U_V} &\n\\mathcal{G}(U) \\ar[d]^{\\rho^U_V} \\\\\n\\mathcal{F}(V) \\ar[r]^\\varphi & \\mathcal{G}(V)\n}\n$$\ncommutes.\n\\item The category of presheaves of sets on $X$ will be denoted\n$\\textit{PSh}(X)$.\n\\end{enumerate}\n\\end{definition}"}]},{"contentHash":"sha256:e15b670cf8eaecb2e9774da8a00edf1ff814ee1bfb132f1eb75f8f353d5ac661","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_SHEAVES_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:sheaf-cohomology#ST_006W","kind":"fragment","locator":"tag/006W","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:dd05660d523049b43c908e00357b644b49aecdc06ba7eb53adb2c325e626e2c3","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-constant-sheaf}\nLet $X$ be a topological space. Let $A$ be a set.\nThe {\\it constant sheaf with value $A$} denoted $\\underline{A}$, or\n$\\underline{A}_X$ is the sheaf that assigns to an open $U \\subset X$\nthe set of all locally constant maps $U \\to A$ with restriction mappings\ngiven by restrictions of functions.\n\\end{definition}"}]},{"contentHash":"sha256:44ca688f1b371dc408a7186efb3a770de31e5cf542f3b48926b9929642de76de","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_COHOMOLOGY_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:sheaf-cohomology#ST_01EF","kind":"fragment","locator":"tag/01EF","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:95b540b6925fdab8fa3a53def4a39d102181a21ef47fcce25d076b6a97bc0b1a","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-cech-complex}\nLet $X$ be a topological space.\nLet $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering.\nLet $\\mathcal{F}$ be an abelian presheaf on $X$.\nThe complex $\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})$\nis the {\\it {\\v C}ech complex} associated to $\\mathcal{F}$ and the\nopen covering $\\mathcal{U}$. Its cohomology groups\n$H^i(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}))$ are\ncalled the {\\it {\\v C}ech cohomology groups} associated to\n$\\mathcal{F}$ and the covering $\\mathcal{U}$.\nThey are denoted $\\check H^i(\\mathcal{U}, \\mathcal{F})$.\n\\end{definition}"}]},{"contentHash":"sha256:32257319184ae0e9c5584919035ad193c88002f0f7ce4eded08cfa8acbb48f20","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_COHOMOLOGY_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:sheaf-cohomology#ST_01EG","kind":"fragment","locator":"tag/01EG","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:4c5b22ab222601aaae95952557fde5950661f40d81749b9bb69b4a8f61ff6108","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-cech-h0}\nLet $X$ be a topological space.\nLet $\\mathcal{F}$ be an abelian presheaf on $X$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is an abelian sheaf and\n\\item for every open covering $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$\nthe natural map\n$$\n\\mathcal{F}(U) \\to \\check{H}^0(\\mathcal{U}, \\mathcal{F})\n$$\nis bijective.\n\\end{enumerate}\n\\end{lemma}"}]},{"contentHash":"sha256:100a796ac383e4193e5e93c36072f00b3cbdec059858ee798aa6bb02c78c3997","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_COHOMOLOGY_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:sheaf-cohomology#ST_01FI","kind":"fragment","locator":"tag/01FI","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:4dbf509c03629779d780d60d6502aae3001874670f84eb6bc4cd9e0dab533882","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-ordered-cech-complex}\nLet $X$ be a topological space.\nLet $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering.\nAssume given a total ordering on $I$.\nLet $\\mathcal{F}$ be an abelian presheaf on $X$.\nThe complex $\\check{\\mathcal{C}}_{ord}^\\bullet(\\mathcal{U}, \\mathcal{F})$\nis the {\\it ordered {\\v C}ech complex} associated to $\\mathcal{F}$, the\nopen covering $\\mathcal{U}$ and the given total ordering on $I$.\n\\end{definition}"}]},{"contentHash":"sha256:13ee1524e1e1e43409a1fb17733206a93162331f710fb544c8218b6036a284e1","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_COHOMOLOGY_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:sheaf-cohomology#ST_01FM","kind":"fragment","locator":"tag/01FM","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:783d414a41461bccfb07b242a527657ee8fda1055c33a6ef41abf92ee925f966","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-alternating-usual}\nLet $X$ be a topological space.\nLet $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering.\nAssume $I$ comes equipped with a total ordering.\nThe map $c \\circ \\pi$ is homotopic to the identity on\n$\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})$.\nIn particular the inclusion map\n$\\check{\\mathcal{C}}_{alt}^\\bullet(\\mathcal{U}, \\mathcal{F}) \\to\n\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})$\nis a homotopy equivalence.\n\\end{lemma}"}]}],"text":"cech_h1_order: **NEITHER** — НЕ УСТАНОВЛЕНО: в формализованном праве нет ни подтверждения, ни опровержения (открытый мир §69) — это не «нет»\nВыведено правом: section_over(urn:case:stacks:cech:fn-W-1, urn:case:stacks:cech:F, urn:case:stacks:cech:W); section_over(urn:case:stacks:cech:fn-U2-00, urn:case:stacks:cech:F, urn:case:stacks:cech:U2); section_over(urn:case:stacks:cech:fn-W-0, urn:case:stacks:cech:F, urn:case:stacks:cech:W); section_over(urn:case:stacks:cech:fn-U2-11, urn:case:stacks:cech:F, urn:case:stacks:cech:U2); section_over(urn:case:stacks:cech:fn-X-000, urn:case:stacks:cech:F, urn:case:stacks:cech:X); section_over(urn:case:stacks:cech:fn-U1-11, urn:case:stacks:cech:F, urn:case:stacks:cech:U1); section_over(urn:case:stacks:cech:fn-U1-00, urn:case:stacks:cech:F, urn:case:stacks:cech:U1); cech_c0_order(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, 4); section_over(urn:case:stacks:cech:fn-X-111, urn:case:stacks:cech:F, urn:case:stacks:cech:X); covers(urn:case:stacks:cech:cov, urn:case:stacks:cech:X); cech_c1_order(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, 2); compatible_pair(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, urn:case:stacks:cech:fn-U1-11, urn:case:stacks:cech:fn-U2-11); compatible_pair(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, urn:case:stacks:cech:fn-U1-00, urn:case:stacks:cech:fn-U2-00); cech_h0_order(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, 2); cech_h0_matches_sections(urn:case:stacks:cech:F, urn:case:stacks:cech:cov); cech_h1_order(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, 1); cech_h1_dimension(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, 0)\n…и ещё 467 выведенных фактов вне предмета вопроса (полный вывод — law_explain)\nПрименены правила: AgreeAtPoint, CechC0Order, CechC1Order, CechH0MatchesSections, CechH0Order, CechH1Dimension, CechH1Order, CompatiblePair, CoversByPoints, IntersectionByPoints, LocallyConstantAlongSpecialization, OpenByGeneralizationStability, RestrictionByValues, SameValueAtTwoPoints, SectionsOfConstantSheaf, SubsetByPoints\nПравило «01EF with 01FI: over the field \\(Z/2\\), \\(|im d^0| = |C^0| / |ker d^0|\\) and \\(Ȟ^1 = C^1 / im d^0\\), so the order \\(n\\) of \\(Ȟ^1\\) satisfies \\(n · |C^0| = |C^1| · |ker d^0|\\)» вывело бы это при посылках:\n    · v6 × v2 = v3 × v4 — DEPENDS\n    ✓ power_of_two(1, 2) — TRUE_ONLY\n    · cech_c1_order(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, v3) — DEPENDS\n    · cech_c0_order(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, v2) — DEPENDS\n    · cech_h0_order(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, v4) — DEPENDS\n(? — факт не подан и не выведен; ✗ — установлено обратное)\nПолные правила с посылками: law_rules({\"predicate\": \"cech_h1_order\"})\nПраво (вне юрисдикции государства): Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — доктрина (programHash sha256:6b62eb59e903…)\nproof-граф: 574 узлов — поле evaluation готово для law_explain"}],"language":"ru","question":{"origin":"user","text":"Стягиваемое конечное пространство с покрытием из двух открытых множеств и постоянный пучок Z/2. Чему равна первая когомология Чеха и что модель отвечает на предъявленное ей предположение о порядке 2?"},"schemaVersion":"law.answers.document/0.1"}