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?
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|.
≋This is an assistant explanation, not a calculation result. Check the grounds and sources below.
is a finite space with the specialization (Alexandrov) topology: the open subsets are exactly the subsets stable under generalization
x: x
is presented as a subset of (possibly empty), to be tested for openness
u
x
U1
x
U2
x
W
x
X
x
the point belongs to
p
x
a
x
b
x
c
x
0061: — is a specialization of , a generalization of : ∈ closure of
g
p
c
a
c
b
the point lies in the subset
u
p
U1
a
U1
c
U2
b
U2
c
W
c
X
a
X
b
X
c
the covering is aimed at : its members are proposed to cover
c: covu: X
is a member of the covering
c
ui
cov
U1
cov
U2
01FI: , the first member in the total ordering of the covering
c: covu: U1
01FI: , the second member in the total ordering of the covering
c: covu: U2
is the constant sheaf : sections over are the locally constant maps
f: Fx: x
the candidate is a function on the points of
s: fn-U1-00u: U1
the function takes the value at the point
s
p
v
fn-U1-00
a
0
fn-U1-00
c
0
the candidate is a function on the points of
s: fn-U1-01u: U1
the function takes the value at the point
s
p
v
fn-U1-01
a
0
fn-U1-01
c
1
the candidate is a function on the points of
s: fn-U1-10u: U1
the function takes the value at the point
s
p
v
fn-U1-10
a
1
fn-U1-10
c
0
the candidate is a function on the points of
s: fn-U1-11u: U1
the function takes the value at the point
s
p
v
fn-U1-11
a
1
fn-U1-11
c
1
every function is presented as a candidate section of over
f: Fu: U1
the candidate is a function on the points of
s: fn-U2-00u: U2
the function takes the value at the point
s
p
v
fn-U2-00
b
0
fn-U2-00
c
0
the candidate is a function on the points of
s: fn-U2-01u: U2
the function takes the value at the point
s
p
v
fn-U2-01
b
0
fn-U2-01
c
1
the candidate is a function on the points of
s: fn-U2-10u: U2
the function takes the value at the point
s
p
v
fn-U2-10
b
1
fn-U2-10
c
0
the candidate is a function on the points of
s: fn-U2-11u: U2
the function takes the value at the point
s
p
v
fn-U2-11
b
1
fn-U2-11
c
1
every function is presented as a candidate section of over
f: Fu: U2
the candidate is a function on the points of
s: fn-W-0u: W
the function takes the value at the point
s: fn-W-0p: cv: 0
the candidate is a function on the points of
s: fn-W-1u: W
the function takes the value at the point
s: fn-W-1p: cv: 1
every function is presented as a candidate section of over
f: Fu: W
the candidate is a function on the points of
s: fn-X-000u: X
the function takes the value at the point
s
p
v
fn-X-000
a
0
fn-X-000
b
0
fn-X-000
c
0
the candidate is a function on the points of
s: fn-X-001u: X
the function takes the value at the point
s
p
v
fn-X-001
a
0
fn-X-001
b
0
fn-X-001
c
1
the candidate is a function on the points of
s: fn-X-010u: X
the function takes the value at the point
s
p
v
fn-X-010
a
0
fn-X-010
b
1
fn-X-010
c
0
the candidate is a function on the points of
s: fn-X-011u: X
the function takes the value at the point
s
p
v
fn-X-011
a
0
fn-X-011
b
1
fn-X-011
c
1
the candidate is a function on the points of
s: fn-X-100u: X
the function takes the value at the point
s
p
v
fn-X-100
a
1
fn-X-100
b
0
fn-X-100
c
0
the candidate is a function on the points of
s: fn-X-101u: X
the function takes the value at the point
s
p
v
fn-X-101
a
1
fn-X-101
b
0
fn-X-101
c
1
the candidate is a function on the points of
s: fn-X-110u: X
the function takes the value at the point
s
p
v
fn-X-110
a
1
fn-X-110
b
1
fn-X-110
c
0
the candidate is a function on the points of
s: fn-X-111u: X
the function takes the value at the point
s
p
v
fn-X-111
a
1
fn-X-111
b
1
fn-X-111
c
1
every function is presented as a candidate section of over
f: Fu: X
Package: Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
006W with 0061: a function on is locally constant when for every point of and every generalization of in
006W: the function on is locally constant: constant along every generalization inside , hence on connected components: s: urn:case:stacks:cech:fn-U2-00; u: urn:case:stacks:cech:U2
006W with 0061: a function on is locally constant when for every point of and every generalization of in
006W: the function on is locally constant: constant along every generalization inside , hence on connected components: s: urn:case:stacks:cech:fn-U2-11; u: urn:case:stacks:cech:U2
006W with 0061: a function on is locally constant when for every point of and every generalization of in
006W: the function on is locally constant: constant along every generalization inside , hence on connected components: s: urn:case:stacks:cech:fn-W-0; u: urn:case:stacks:cech:W
006W with 0061: a function on is locally constant when for every point of and every generalization of in
006W: the function on is locally constant: constant along every generalization inside , hence on connected components: s: urn:case:stacks:cech:fn-W-1; u: urn:case:stacks:cech:W
006W with 0061: a function on is locally constant when for every point of and every generalization of in
006W: the function on is locally constant: constant along every generalization inside , hence on connected components: s: urn:case:stacks:cech:fn-U1-11; u: urn:case:stacks:cech:U1
006W with 0061: a function on is locally constant when for every point of and every generalization of in
006W: the function on is locally constant: constant along every generalization inside , hence on connected components: s: urn:case:stacks:cech:fn-U1-00; u: urn:case:stacks:cech:U1
01FI, : the pair is a 0-cocycle when both restrict to the same section of
01FI: lies in : and agree on : f: urn:case:stacks:cech:F; c: urn:case:stacks:cech:cov; s1: urn:case:stacks:cech:fn-U1-00; s2: urn:case:stacks:cech:fn-U2-00
tag 01FI, tag 01EF
Identifier
urn:stacks:clir:sheaf-cohomology#CompatiblePair
rule
71
01EF: ; its order is the number of pairs agreeing on
01EF: , the number of compatible pairs: f: urn:case:stacks:cech:F; c: urn:case:stacks:cech:cov; n: 2
tag 01EF, tag 01FI
Identifier
urn:stacks:clir:sheaf-cohomology#CechH0Order
rule
72
, the order of a -dimensional -vector space
k: 0; n: 1
origin not recorded
73
01EF with 01FI: over the field , and , so the order of satisfies
01EF: for a two-member covering: , so and : f: urn:case:stacks:cech:F; c: urn:case:stacks:cech:cov; n: 1
tag 01EF, tag 01FI, tag 01FM
Identifier
urn:stacks:clir:sheaf-cohomology#CechH1Order
rule
74
Query evaluation
query
verified by the engine: 37 · case fact: 36 · origin not recorded: 1 · Full graph: 574 nodes
Steps of the saved proof from the case facts to the answer. Formulas are shown as written in the norm with bound values substituted; the page recomputes nothing.
Basis of this answer
Rules on the saved proof path for this answer.
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
two functions agree at a point when their values there coincide
Identifier
urn:stacks:clir:sheaf-cohomology#AgreeAtPoint
01FI: the order of is the product of the orders of and , all functions presented
Identifier
urn:stacks:clir:sheaf-cohomology#CechC0Order
01FI: the order of is the order of , all functions presented
Identifier
urn:stacks:clir:sheaf-cohomology#CechC1Order
01EF: ; its order is the number of pairs agreeing on
Identifier
urn:stacks:clir:sheaf-cohomology#CechH0Order
01EF with 01FI: over the field , and , so the order of satisfies
Identifier
urn:stacks:clir:sheaf-cohomology#CechH1Order
01FI, : the pair is a 0-cocycle when both restrict to the same section of
Applied in the overall evaluation, but not on the proof path for this answer.
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
01EG on a covering of : with all functions on presented
the members cover when each member lies in and every point of lies in some member
Identifier
urn:stacks:clir:sheaf-cohomology#CoversByPoints
Derived result for this query
01EF: for a two-member covering: , so and
f: Fc: covn: 1
Other derived facts16
, a section of over
s
f
u
fn-W-1
F
W
fn-U2-00
F
U2
fn-W-0
F
W
fn-U2-11
F
U2
fn-X-000
F
X
fn-U1-11
F
U1
fn-U1-00
F
U1
01FI:
f: Fc: covn: 4
, a section of over
s: fn-X-111f: Fu: X
the covering is an open covering of
c: covu: X
01FI:
f: Fc: covn: 2
01FI: lies in : and agree on
f
c
s1
s2
F
cov
fn-U1-11
fn-U2-11
F
cov
fn-U1-00
fn-U2-00
01EF: , the number of compatible pairs
f: Fc: covn: 2
01EG: the natural map is bijective on this covering: the orders coincide
f: Fc: cov
, i.e.
f: Fc: covk: 0
, a section of over
s
f
u
urn:case:stacks:cech:fn-W-1
urn:case:stacks:cech:F
urn:case:stacks:cech:W
urn:case:stacks:cech:fn-U2-00
urn:case:stacks:cech:F
urn:case:stacks:cech:U2
urn:case:stacks:cech:fn-W-0
urn:case:stacks:cech:F
urn:case:stacks:cech:W
urn:case:stacks:cech:fn-U2-11
urn:case:stacks:cech:F
urn:case:stacks:cech:U2
urn:case:stacks:cech:fn-X-000
urn:case:stacks:cech:F
urn:case:stacks:cech:X
urn:case:stacks:cech:fn-U1-11
urn:case:stacks:cech:F
urn:case:stacks:cech:U1
urn:case:stacks:cech:fn-U1-00
urn:case:stacks:cech:F
urn:case:stacks:cech:U1
urn:case:stacks:cech:fn-X-111
urn:case:stacks:cech:F
urn:case:stacks:cech:X
01FI:
f
c
n
urn:case:stacks:cech:F
cov
4
the covering is an open covering of
c
u
cov
urn:case:stacks:cech:X
01FI:
f
c
n
urn:case:stacks:cech:F
cov
2
01FI: lies in : and agree on
f
c
s1
s2
urn:case:stacks:cech:F
cov
urn:case:stacks:cech:fn-U1-11
urn:case:stacks:cech:fn-U2-11
urn:case:stacks:cech:F
cov
urn:case:stacks:cech:fn-U1-00
urn:case:stacks:cech:fn-U2-00
01EF: , the number of compatible pairs
f
c
n
urn:case:stacks:cech:F
cov
2
01EG: the natural map is bijective on this covering: the orders coincide
f
c
urn:case:stacks:cech:F
cov
01EF: for a two-member covering: , so and
f
c
n
urn:case:stacks:cech:F
cov
1
, i.e.
f
c
k
urn:case:stacks:cech:F
cov
0
467 further derived facts are not shown: the engine keeps the ones relevant to the question in its compact answer. The full list is in the calculation JSON below.
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
Let be a topological space.
If then we say is a specialization of ,
or is a generalization of if .
Notation: .
A subset is stable under specialization if for all and every specialization we have .
A subset is stable under generalization if for all and every generalization we have .
Original data · JSON
JSONRead only
{"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}"}]}
tag/0062
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
Let be a topological space.
Any closed subset of is stable under specialization.
Any open subset of is stable under generalization.
A subset is stable under specialization
if and only if
the complement is stable under generalization.
Original data · JSON
JSONRead only
{"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}"}]}
tag/006E
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
Let be a topological space.
A presheaf of sets on is a rule which
assigns to each open a set and
to each inclusion a map such that and
whenever we have .
A morphism of presheaves of sets on is a rule which assigns to each
open a map of sets compatible with restriction maps,
i.e., whenever are open the
diagram
The category of presheaves of sets on will be denoted .
Original data · JSON
JSONRead only
{"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}"}]}
tag/006W
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
Let be a topological space. Let be a set.
The constant sheaf with value denoted , or is the sheaf that assigns to an open the set of all locally constant maps with restriction mappings
given by restrictions of functions.
Original data · JSON
JSONRead only
{"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}"}]}
tag/01EF
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
Let be a topological space.
Let be an open covering.
Let be an abelian presheaf on .
The complex is the {\v C}ech complex associated to and the
open covering . Its cohomology groups are
called the {\v C}ech cohomology groups associated to and the covering .
They are denoted .
Original data · JSON
JSONRead only
{"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}"}]}
tag/01EG
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
Let be a topological space.
Let be an abelian presheaf on .
The following are equivalent
is an abelian sheaf and
for every open covering the natural map
is bijective.
Original data · JSON
JSONRead only
{"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}"}]}
tag/01FI
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
Let be a topological space.
Let be an open covering.
Assume given a total ordering on .
Let be an abelian presheaf on .
The complex is the ordered {\v C}ech complex associated to , the
open covering and the given total ordering on .
Original data · JSON
JSONRead only
{"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}"}]}
tag/01FM
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
Let be a topological space.
Let be an open covering.
Assume comes equipped with a total ordering.
The map is homotopic to the identity on .
In particular the inclusion map is a homotopy equivalence.
Original data · JSON
JSONRead only
{"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}"}]}
Packages in the snapshot
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
⌘Technical dataFull response, parameters and checksums⌄
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is a finite space with the specialization (Alexandrov) topology: the open subsets are exactly the subsets stable under generalization
x: x
is presented as a subset of (possibly empty), to be tested for openness
u
x
U1
x
U2
x
W
x
X
x
the point belongs to
p
x
a
x
b
x
c
x
0061: — is a specialization of , a generalization of : ∈ closure of
g
p
c
a
c
b
the point lies in the subset
u
p
U1
a
U1
c
U2
b
U2
c
W
c
X
a
X
b
X
c
the covering is aimed at : its members are proposed to cover
c: covu: X
is a member of the covering
c
ui
cov
U1
cov
U2
01FI: , the first member in the total ordering of the covering
c: covu: U1
01FI: , the second member in the total ordering of the covering
c: covu: U2
is the constant sheaf : sections over are the locally constant maps
f: Fx: x
the candidate is a function on the points of
s: fn-U1-00u: U1
the function takes the value at the point
s
p
v
fn-U1-00
a
0
fn-U1-00
c
0
the candidate is a function on the points of
s: fn-U1-01u: U1
the function takes the value at the point
s
p
v
fn-U1-01
a
0
fn-U1-01
c
1
the candidate is a function on the points of
s: fn-U1-10u: U1
the function takes the value at the point
s
p
v
fn-U1-10
a
1
fn-U1-10
c
0
the candidate is a function on the points of
s: fn-U1-11u: U1
the function takes the value at the point
s
p
v
fn-U1-11
a
1
fn-U1-11
c
1
every function is presented as a candidate section of over
f: Fu: U1
the candidate is a function on the points of
s: fn-U2-00u: U2
the function takes the value at the point
s
p
v
fn-U2-00
b
0
fn-U2-00
c
0
the candidate is a function on the points of
s: fn-U2-01u: U2
the function takes the value at the point
s
p
v
fn-U2-01
b
0
fn-U2-01
c
1
the candidate is a function on the points of
s: fn-U2-10u: U2
the function takes the value at the point
s
p
v
fn-U2-10
b
1
fn-U2-10
c
0
the candidate is a function on the points of
s: fn-U2-11u: U2
the function takes the value at the point
s
p
v
fn-U2-11
b
1
fn-U2-11
c
1
every function is presented as a candidate section of over
f: Fu: U2
the candidate is a function on the points of
s: fn-W-0u: W
the function takes the value at the point
s: fn-W-0p: cv: 0
the candidate is a function on the points of
s: fn-W-1u: W
the function takes the value at the point
s: fn-W-1p: cv: 1
every function is presented as a candidate section of over
f: Fu: W
the candidate is a function on the points of
s: fn-X-000u: X
the function takes the value at the point
s
p
v
fn-X-000
a
0
fn-X-000
b
0
fn-X-000
c
0
the candidate is a function on the points of
s: fn-X-001u: X
the function takes the value at the point
s
p
v
fn-X-001
a
0
fn-X-001
b
0
fn-X-001
c
1
the candidate is a function on the points of
s: fn-X-010u: X
the function takes the value at the point
s
p
v
fn-X-010
a
0
fn-X-010
b
1
fn-X-010
c
0
the candidate is a function on the points of
s: fn-X-011u: X
the function takes the value at the point
s
p
v
fn-X-011
a
0
fn-X-011
b
1
fn-X-011
c
1
the candidate is a function on the points of
s: fn-X-100u: X
the function takes the value at the point
s
p
v
fn-X-100
a
1
fn-X-100
b
0
fn-X-100
c
0
the candidate is a function on the points of
s: fn-X-101u: X
the function takes the value at the point
s
p
v
fn-X-101
a
1
fn-X-101
b
0
fn-X-101
c
1
the candidate is a function on the points of
s: fn-X-110u: X
the function takes the value at the point
s
p
v
fn-X-110
a
1
fn-X-110
b
1
fn-X-110
c
0
the candidate is a function on the points of
s: fn-X-111u: X
the function takes the value at the point
s
p
v
fn-X-111
a
1
fn-X-111
b
1
fn-X-111
c
1
every function is presented as a candidate section of over
f: Fu: X
Package: Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
006W with 0061: a function on is locally constant when for every point of and every generalization of in
006W: the function on is locally constant: constant along every generalization inside , hence on connected components: s: urn:case:stacks:cech:fn-U2-00; u: urn:case:stacks:cech:U2
006W with 0061: a function on is locally constant when for every point of and every generalization of in
006W: the function on is locally constant: constant along every generalization inside , hence on connected components: s: urn:case:stacks:cech:fn-U2-11; u: urn:case:stacks:cech:U2
006W with 0061: a function on is locally constant when for every point of and every generalization of in
006W: the function on is locally constant: constant along every generalization inside , hence on connected components: s: urn:case:stacks:cech:fn-W-0; u: urn:case:stacks:cech:W
006W with 0061: a function on is locally constant when for every point of and every generalization of in
006W: the function on is locally constant: constant along every generalization inside , hence on connected components: s: urn:case:stacks:cech:fn-W-1; u: urn:case:stacks:cech:W
006W with 0061: a function on is locally constant when for every point of and every generalization of in
006W: the function on is locally constant: constant along every generalization inside , hence on connected components: s: urn:case:stacks:cech:fn-U1-11; u: urn:case:stacks:cech:U1
006W with 0061: a function on is locally constant when for every point of and every generalization of in
006W: the function on is locally constant: constant along every generalization inside , hence on connected components: s: urn:case:stacks:cech:fn-U1-00; u: urn:case:stacks:cech:U1
01FI, : the pair is a 0-cocycle when both restrict to the same section of
01FI: lies in : and agree on : f: urn:case:stacks:cech:F; c: urn:case:stacks:cech:cov; s1: urn:case:stacks:cech:fn-U1-00; s2: urn:case:stacks:cech:fn-U2-00
tag 01FI, tag 01EF
Identifier
urn:stacks:clir:sheaf-cohomology#CompatiblePair
rule
71
01EF: ; its order is the number of pairs agreeing on
01EF: , the number of compatible pairs: f: urn:case:stacks:cech:F; c: urn:case:stacks:cech:cov; n: 2
tag 01EF, tag 01FI
Identifier
urn:stacks:clir:sheaf-cohomology#CechH0Order
rule
72
, the order of a -dimensional -vector space
k: 0; n: 1
origin not recorded
73
01EF with 01FI: over the field , and , so the order of satisfies
01EF: for a two-member covering: , so and : f: urn:case:stacks:cech:F; c: urn:case:stacks:cech:cov; n: 1
tag 01EF, tag 01FI, tag 01FM
Identifier
urn:stacks:clir:sheaf-cohomology#CechH1Order
rule
74
the -dimension of is the exponent of its order
, i.e. : f: urn:case:stacks:cech:F; c: urn:case:stacks:cech:cov; k: 0
tag 01EF
Identifier
urn:stacks:clir:sheaf-cohomology#CechH1Dimension
rule
75
Query evaluation
query
verified by the engine: 38 · case fact: 36 · origin not recorded: 1 · Full graph: 574 nodes
Steps of the saved proof from the case facts to the answer. Formulas are shown as written in the norm with bound values substituted; the page recomputes nothing.
Basis of this answer
Rules on the saved proof path for this answer.
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
two functions agree at a point when their values there coincide
Identifier
urn:stacks:clir:sheaf-cohomology#AgreeAtPoint
01FI: the order of is the product of the orders of and , all functions presented
Identifier
urn:stacks:clir:sheaf-cohomology#CechC0Order
01FI: the order of is the order of , all functions presented
Identifier
urn:stacks:clir:sheaf-cohomology#CechC1Order
01EF: ; its order is the number of pairs agreeing on
Identifier
urn:stacks:clir:sheaf-cohomology#CechH0Order
the -dimension of is the exponent of its order
Identifier
urn:stacks:clir:sheaf-cohomology#CechH1Dimension
01EF with 01FI: over the field , and , so the order of satisfies
Identifier
urn:stacks:clir:sheaf-cohomology#CechH1Order
01FI, : the pair is a 0-cocycle when both restrict to the same section of
Applied in the overall evaluation, but not on the proof path for this answer.
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
01EG on a covering of : with all functions on presented
the members cover when each member lies in and every point of lies in some member
Identifier
urn:stacks:clir:sheaf-cohomology#CoversByPoints
Derived result for this query
, i.e.
f: Fc: covk: 0
Other derived facts16
, a section of over
s
f
u
fn-W-1
F
W
fn-U2-00
F
U2
fn-W-0
F
W
fn-U2-11
F
U2
fn-X-000
F
X
fn-U1-11
F
U1
fn-U1-00
F
U1
01FI:
f: Fc: covn: 4
, a section of over
s: fn-X-111f: Fu: X
the covering is an open covering of
c: covu: X
01FI:
f: Fc: covn: 2
01FI: lies in : and agree on
f
c
s1
s2
F
cov
fn-U1-11
fn-U2-11
F
cov
fn-U1-00
fn-U2-00
01EF: , the number of compatible pairs
f: Fc: covn: 2
01EG: the natural map is bijective on this covering: the orders coincide
f: Fc: cov
01EF: for a two-member covering: , so and
f: Fc: covn: 1
, a section of over
s
f
u
urn:case:stacks:cech:fn-W-1
urn:case:stacks:cech:F
urn:case:stacks:cech:W
urn:case:stacks:cech:fn-U2-00
urn:case:stacks:cech:F
urn:case:stacks:cech:U2
urn:case:stacks:cech:fn-W-0
urn:case:stacks:cech:F
urn:case:stacks:cech:W
urn:case:stacks:cech:fn-U2-11
urn:case:stacks:cech:F
urn:case:stacks:cech:U2
urn:case:stacks:cech:fn-X-000
urn:case:stacks:cech:F
urn:case:stacks:cech:X
urn:case:stacks:cech:fn-U1-11
urn:case:stacks:cech:F
urn:case:stacks:cech:U1
urn:case:stacks:cech:fn-U1-00
urn:case:stacks:cech:F
urn:case:stacks:cech:U1
urn:case:stacks:cech:fn-X-111
urn:case:stacks:cech:F
urn:case:stacks:cech:X
01FI:
f
c
n
urn:case:stacks:cech:F
cov
4
the covering is an open covering of
c
u
cov
urn:case:stacks:cech:X
01FI:
f
c
n
urn:case:stacks:cech:F
cov
2
01FI: lies in : and agree on
f
c
s1
s2
urn:case:stacks:cech:F
cov
urn:case:stacks:cech:fn-U1-11
urn:case:stacks:cech:fn-U2-11
urn:case:stacks:cech:F
cov
urn:case:stacks:cech:fn-U1-00
urn:case:stacks:cech:fn-U2-00
01EF: , the number of compatible pairs
f
c
n
urn:case:stacks:cech:F
cov
2
01EG: the natural map is bijective on this covering: the orders coincide
f
c
urn:case:stacks:cech:F
cov
01EF: for a two-member covering: , so and
f
c
n
urn:case:stacks:cech:F
cov
1
, i.e.
f
c
k
urn:case:stacks:cech:F
cov
0
467 further derived facts are not shown: the engine keeps the ones relevant to the question in its compact answer. The full list is in the calculation JSON below.
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
Let be a topological space.
If then we say is a specialization of ,
or is a generalization of if .
Notation: .
A subset is stable under specialization if for all and every specialization we have .
A subset is stable under generalization if for all and every generalization we have .
Original data · JSON
JSONRead only
{"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}"}]}
tag/0062
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
Let be a topological space.
Any closed subset of is stable under specialization.
Any open subset of is stable under generalization.
A subset is stable under specialization
if and only if
the complement is stable under generalization.
Original data · JSON
JSONRead only
{"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}"}]}
tag/006E
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
Let be a topological space.
A presheaf of sets on is a rule which
assigns to each open a set and
to each inclusion a map such that and
whenever we have .
A morphism of presheaves of sets on is a rule which assigns to each
open a map of sets compatible with restriction maps,
i.e., whenever are open the
diagram
The category of presheaves of sets on will be denoted .
Original data · JSON
JSONRead only
{"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}"}]}
tag/006W
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
Let be a topological space. Let be a set.
The constant sheaf with value denoted , or is the sheaf that assigns to an open the set of all locally constant maps with restriction mappings
given by restrictions of functions.
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{"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}"}]}
tag/01EF
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
Let be a topological space.
Let be an open covering.
Let be an abelian presheaf on .
The complex is the {\v C}ech complex associated to and the
open covering . Its cohomology groups are
called the {\v C}ech cohomology groups associated to and the covering .
They are denoted .
Original data · JSON
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{"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}"}]}
tag/01EG
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
Let be a topological space.
Let be an abelian presheaf on .
The following are equivalent
is an abelian sheaf and
for every open covering the natural map
is bijective.
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{"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}"}]}
tag/01FI
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
Let be a topological space.
Let be an open covering.
Assume given a total ordering on .
Let be an abelian presheaf on .
The complex is the ordered {\v C}ech complex associated to , the
open covering and the given total ordering on .
Original data · JSON
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{"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}"}]}
tag/01FM
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
Let be a topological space.
Let be an open covering.
Assume comes equipped with a total ordering.
The map is homotopic to the identity on .
In particular the inclusion map is a homotopy equivalence.
Original data · JSON
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{"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}"}]}
Packages in the snapshot
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
⌘Technical dataFull response, parameters and checksums⌄
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Rule: 01EF with 01FI: over the field , and , so the order of satisfies
·v6 × v2 = v3 × v4DEPENDS
✓ , the order of a -dimensional -vector space1, 2established
·01FI: F, cov, v3DEPENDS
·01FI: F, cov, v2DEPENDS
·01EF: , the number of compatible pairsF, cov, v4DEPENDS
Input parameters
What we are finding
01EF: for a two-member covering: , so and
Fcov2
Input facts
is a finite space with the specialization (Alexandrov) topology: the open subsets are exactly the subsets stable under generalization
x: x
is presented as a subset of (possibly empty), to be tested for openness
u
x
U1
x
U2
x
W
x
X
x
the point belongs to
p
x
a
x
b
x
c
x
0061: — is a specialization of , a generalization of : ∈ closure of
g
p
c
a
c
b
the point lies in the subset
u
p
U1
a
U1
c
U2
b
U2
c
W
c
X
a
X
b
X
c
the covering is aimed at : its members are proposed to cover
c: covu: X
is a member of the covering
c
ui
cov
U1
cov
U2
01FI: , the first member in the total ordering of the covering
c: covu: U1
01FI: , the second member in the total ordering of the covering
c: covu: U2
is the constant sheaf : sections over are the locally constant maps
f: Fx: x
the candidate is a function on the points of
s: fn-U1-00u: U1
the function takes the value at the point
s
p
v
fn-U1-00
a
0
fn-U1-00
c
0
the candidate is a function on the points of
s: fn-U1-01u: U1
the function takes the value at the point
s
p
v
fn-U1-01
a
0
fn-U1-01
c
1
the candidate is a function on the points of
s: fn-U1-10u: U1
the function takes the value at the point
s
p
v
fn-U1-10
a
1
fn-U1-10
c
0
the candidate is a function on the points of
s: fn-U1-11u: U1
the function takes the value at the point
s
p
v
fn-U1-11
a
1
fn-U1-11
c
1
every function is presented as a candidate section of over
f: Fu: U1
the candidate is a function on the points of
s: fn-U2-00u: U2
the function takes the value at the point
s
p
v
fn-U2-00
b
0
fn-U2-00
c
0
the candidate is a function on the points of
s: fn-U2-01u: U2
the function takes the value at the point
s
p
v
fn-U2-01
b
0
fn-U2-01
c
1
the candidate is a function on the points of
s: fn-U2-10u: U2
the function takes the value at the point
s
p
v
fn-U2-10
b
1
fn-U2-10
c
0
the candidate is a function on the points of
s: fn-U2-11u: U2
the function takes the value at the point
s
p
v
fn-U2-11
b
1
fn-U2-11
c
1
every function is presented as a candidate section of over
f: Fu: U2
the candidate is a function on the points of
s: fn-W-0u: W
the function takes the value at the point
s: fn-W-0p: cv: 0
the candidate is a function on the points of
s: fn-W-1u: W
the function takes the value at the point
s: fn-W-1p: cv: 1
every function is presented as a candidate section of over
f: Fu: W
the candidate is a function on the points of
s: fn-X-000u: X
the function takes the value at the point
s
p
v
fn-X-000
a
0
fn-X-000
b
0
fn-X-000
c
0
the candidate is a function on the points of
s: fn-X-001u: X
the function takes the value at the point
s
p
v
fn-X-001
a
0
fn-X-001
b
0
fn-X-001
c
1
the candidate is a function on the points of
s: fn-X-010u: X
the function takes the value at the point
s
p
v
fn-X-010
a
0
fn-X-010
b
1
fn-X-010
c
0
the candidate is a function on the points of
s: fn-X-011u: X
the function takes the value at the point
s
p
v
fn-X-011
a
0
fn-X-011
b
1
fn-X-011
c
1
the candidate is a function on the points of
s: fn-X-100u: X
the function takes the value at the point
s
p
v
fn-X-100
a
1
fn-X-100
b
0
fn-X-100
c
0
the candidate is a function on the points of
s: fn-X-101u: X
the function takes the value at the point
s
p
v
fn-X-101
a
1
fn-X-101
b
0
fn-X-101
c
1
the candidate is a function on the points of
s: fn-X-110u: X
the function takes the value at the point
s
p
v
fn-X-110
a
1
fn-X-110
b
1
fn-X-110
c
0
the candidate is a function on the points of
s: fn-X-111u: X
the function takes the value at the point
s
p
v
fn-X-111
a
1
fn-X-111
b
1
fn-X-111
c
1
every function is presented as a candidate section of over
f: Fu: X
Package: Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
verified by the engine: 1 · case fact: 0 · Full graph: 574 nodes
Steps of the saved proof from the case facts to the answer. Formulas are shown as written in the norm with bound values substituted; the page recomputes nothing.
Applied rules16
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
two functions agree at a point when their values there coincide
Identifier
urn:stacks:clir:sheaf-cohomology#AgreeAtPoint
01FI: the order of is the product of the orders of and , all functions presented
Identifier
urn:stacks:clir:sheaf-cohomology#CechC0Order
01FI: the order of is the order of , all functions presented
Identifier
urn:stacks:clir:sheaf-cohomology#CechC1Order
01EG on a covering of : with all functions on presented
01EG: the natural map is bijective on this covering: the orders coincide
f: Fc: cov
01EF: for a two-member covering: , so and
f: Fc: covn: 1
, i.e.
f: Fc: covk: 0
, a section of over
s
f
u
urn:case:stacks:cech:fn-W-1
urn:case:stacks:cech:F
urn:case:stacks:cech:W
urn:case:stacks:cech:fn-U2-00
urn:case:stacks:cech:F
urn:case:stacks:cech:U2
urn:case:stacks:cech:fn-W-0
urn:case:stacks:cech:F
urn:case:stacks:cech:W
urn:case:stacks:cech:fn-U2-11
urn:case:stacks:cech:F
urn:case:stacks:cech:U2
urn:case:stacks:cech:fn-X-000
urn:case:stacks:cech:F
urn:case:stacks:cech:X
urn:case:stacks:cech:fn-U1-11
urn:case:stacks:cech:F
urn:case:stacks:cech:U1
urn:case:stacks:cech:fn-U1-00
urn:case:stacks:cech:F
urn:case:stacks:cech:U1
urn:case:stacks:cech:fn-X-111
urn:case:stacks:cech:F
urn:case:stacks:cech:X
01FI:
f
c
n
urn:case:stacks:cech:F
cov
4
the covering is an open covering of
c
u
cov
urn:case:stacks:cech:X
01FI:
f
c
n
urn:case:stacks:cech:F
cov
2
01FI: lies in : and agree on
f
c
s1
s2
urn:case:stacks:cech:F
cov
urn:case:stacks:cech:fn-U1-11
urn:case:stacks:cech:fn-U2-11
urn:case:stacks:cech:F
cov
urn:case:stacks:cech:fn-U1-00
urn:case:stacks:cech:fn-U2-00
01EF: , the number of compatible pairs
f
c
n
urn:case:stacks:cech:F
cov
2
01EG: the natural map is bijective on this covering: the orders coincide
f
c
urn:case:stacks:cech:F
cov
01EF: for a two-member covering: , so and
f
c
n
urn:case:stacks:cech:F
cov
1
, i.e.
f
c
k
urn:case:stacks:cech:F
cov
0
467 further derived facts are not shown: the engine keeps the ones relevant to the question in its compact answer. The full list is in the calculation JSON below.
Why the conclusion was not reached1 rules
1
01EF with 01FI: over the field , and , so the order of satisfies
What is missing
v6 × v2 = v3 × v4DEPENDS
, the order of a -dimensional -vector space1, 2Established
01FI: F, cov, v3DEPENDS
01FI: F, cov, v2DEPENDS
01EF: , the number of compatible pairsF, cov, v4DEPENDS
Source: tag 01EF, tag 01FI, tag 01FM
Identifier
urn:stacks:clir:sheaf-cohomology#CechH1Order
rule
These are the rules whose head answers the question, with their unmet premises. A missing fact is not a refuted one.
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
Let be a topological space.
If then we say is a specialization of ,
or is a generalization of if .
Notation: .
A subset is stable under specialization if for all and every specialization we have .
A subset is stable under generalization if for all and every generalization we have .
Original data · JSON
JSONRead only
{"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}"}]}
tag/0062
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
Let be a topological space.
Any closed subset of is stable under specialization.
Any open subset of is stable under generalization.
A subset is stable under specialization
if and only if
the complement is stable under generalization.
Original data · JSON
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{"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}"}]}
tag/006E
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
Let be a topological space.
A presheaf of sets on is a rule which
assigns to each open a set and
to each inclusion a map such that and
whenever we have .
A morphism of presheaves of sets on is a rule which assigns to each
open a map of sets compatible with restriction maps,
i.e., whenever are open the
diagram
The category of presheaves of sets on will be denoted .
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{"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}"}]}
tag/006W
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
Let be a topological space. Let be a set.
The constant sheaf with value denoted , or is the sheaf that assigns to an open the set of all locally constant maps with restriction mappings
given by restrictions of functions.
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{"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}"}]}
tag/01EF
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
Let be a topological space.
Let be an open covering.
Let be an abelian presheaf on .
The complex is the {\v C}ech complex associated to and the
open covering . Its cohomology groups are
called the {\v C}ech cohomology groups associated to and the covering .
They are denoted .
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{"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}"}]}
tag/01EG
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
Let be a topological space.
Let be an abelian presheaf on .
The following are equivalent
is an abelian sheaf and
for every open covering the natural map
is bijective.
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{"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}"}]}
tag/01FI
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
Let be a topological space.
Let be an open covering.
Assume given a total ordering on .
Let be an abelian presheaf on .
The complex is the ordered {\v C}ech complex associated to , the
open covering and the given total ordering on .
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{"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}"}]}
tag/01FM
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
Let be a topological space.
Let be an open covering.
Assume comes equipped with a total ordering.
The map is homotopic to the identity on .
In particular the inclusion map is a homotopy equivalence.
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{"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}"}]}
Packages in the snapshot
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
⌘Technical dataFull response, parameters and checksums⌄
Calculation status
COMPUTED
Full engine response
cech_h1_order: NEITHER — НЕ УСТАНОВЛЕНО: в формализованном праве нет ни подтверждения, ни опровержения (открытый мир §69) — это не «нет»
Выведено 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)
…и ещё 467 выведенных фактов вне предмета вопроса (полный вывод — law_explain)
Применены правила: AgreeAtPoint, CechC0Order, CechC1Order, CechH0MatchesSections, CechH0Order, CechH1Dimension, CechH1Order, CompatiblePair, CoversByPoints, IntersectionByPoints, LocallyConstantAlongSpecialization, OpenByGeneralizationStability, RestrictionByValues, SameValueAtTwoPoints, SectionsOfConstantSheaf, SubsetByPoints
Правило «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:
· v6 × v2 = v3 × v4 — DEPENDS
✓ power_of_two(1, 2) — TRUE_ONLY
· cech_c1_order(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, v3) — DEPENDS
· cech_c0_order(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, v2) — DEPENDS
· cech_h0_order(urn:case:stacks:cech:F, urn:case:stacks:cech:cov, v4) — DEPENDS
(? — факт не подан и не выведен; ✗ — установлено обратное)
Полные правила с посылками: law_rules({"predicate": "cech_h1_order"})
Право (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…)
proof-граф: 574 узлов — поле evaluation готово для law_explain
Complete machine result · JSON
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Утверждение о порядке 2 здесь не подтверждено и не опровергнуто: модель считает порядок, а не отрицает чужие значения.
How to cite
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Citation
“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?”. Arxo Lens, as of 2026-09-06. https://lens.arxo.io/a/a_D8izPKYnhehPBQQFpqUuGlhN. Snapshot SHA-256: 05dc05bab7d971bd7b4152d586de3205a754f97cfc755d7e9fc726eb5cad9a1d.
BibTeX
@misc{arxo-lens-a_D8izPKYnhehP,
title = {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?},
howpublished = {Arxo Lens},
url = {https://lens.arxo.io/a/a_D8izPKYnhehPBQQFpqUuGlhN},
note = {as of 2026-09-06; SHA-256 05dc05bab7d971bd7b4152d586de3205a754f97cfc755d7e9fc726eb5cad9a1d}
}
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