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.

Results

Condition

Порядок Ȟ¹ стягиваемого пространства

Established

Query parameters · 3

f: urn:case:stacks:cech:F
c: urn:case:stacks:cech:cov
n: 1

Input facts · 88

$X$ is a finite space with the specialization (Alexandrov) topology: the open subsets are exactly the subsets stable under generalization
x: x
$U$ is presented as a subset of $X$ (possibly empty), to be tested for openness
u x
U1 x
U2 x
W x
X x
the point belongs to $X$
p x
a x
b x
c x
0061: $g ⇝ p$ — $p$ is a specialization of $g$ , $g$ a generalization of $p$ : $p$ ∈ closure of $\{g\}$
g p
c a
c b
the point lies in the subset $U$
u p
U1 a
U1 c
U2 b
U2 c
W c
X a
X b
X c
the covering is aimed at $U$ : its members are proposed to cover $U$
c: covu: X
$U_i$ is a member of the covering
c ui
cov U1
cov U2
01FI: $U_1$ , the first member in the total ordering of the covering
c: covu: U1
01FI: $U_2$ , the second member in the total ordering of the covering
c: covu: U2
$F$ is the constant sheaf $(Z/2)_X$ : sections over $U$ are the locally constant maps $U → Z/2$
f: Fx: x
the candidate is a function on the points of $U$
s: fn-U1-00u: U1
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-U1-01u: U1
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-U1-10u: U1
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-U1-11u: U1
the function takes the value $v ∈ \{0, 1\}$ at the point
s p v
fn-U1-11 a 1
fn-U1-11 c 1
every function $U → Z/2$ is presented as a candidate section of $F$ over $U$
f: Fu: U1
the candidate is a function on the points of $U$
s: fn-U2-00u: U2
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-U2-01u: U2
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-U2-10u: U2
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-U2-11u: U2
the function takes the value $v ∈ \{0, 1\}$ at the point
s p v
fn-U2-11 b 1
fn-U2-11 c 1
every function $U → Z/2$ is presented as a candidate section of $F$ over $U$
f: Fu: U2
the candidate is a function on the points of $U$
s: fn-W-0u: W
the function takes the value $v ∈ \{0, 1\}$ at the point
s: fn-W-0p: cv: 0
the candidate is a function on the points of $U$
s: fn-W-1u: W
the function takes the value $v ∈ \{0, 1\}$ at the point
s: fn-W-1p: cv: 1
every function $U → Z/2$ is presented as a candidate section of $F$ over $U$
f: Fu: W
the candidate is a function on the points of $U$
s: fn-X-000u: X
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-X-001u: X
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-X-010u: X
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-X-011u: X
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-X-100u: X
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-X-101u: X
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-X-110u: X
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-X-111u: X
the function takes the value $v ∈ \{0, 1\}$ at the point
s p v
fn-X-111 a 1
fn-X-111 b 1
fn-X-111 c 1
every function $U → Z/2$ is presented as a candidate section of $F$ over $U$
f: Fu: X

u	x
U1	x
U2	x
W	x
X	x

p	x
a	x
b	x
c	x

g	p
c	a
c	b

u	p
U1	a
U1	c
U2	b
U2	c
W	c
X	a
X	b
X	c

c	ui
cov	U1
cov	U2

s	p	v
fn-U1-00	a	0
fn-U1-00	c	0

s	p	v
fn-U1-01	a	0
fn-U1-01	c	1

s	p	v
fn-U1-10	a	1
fn-U1-10	c	0

s	p	v
fn-U1-11	a	1
fn-U1-11	c	1

s	p	v
fn-U2-00	b	0
fn-U2-00	c	0

s	p	v
fn-U2-01	b	0
fn-U2-01	c	1

s	p	v
fn-U2-10	b	1
fn-U2-10	c	0

s	p	v
fn-U2-11	b	1
fn-U2-11	c	1

s	p	v
fn-X-000	a	0
fn-X-000	b	0
fn-X-000	c	0

s	p	v
fn-X-001	a	0
fn-X-001	b	0
fn-X-001	c	1

s	p	v
fn-X-010	a	0
fn-X-010	b	1
fn-X-010	c	0

s	p	v
fn-X-011	a	0
fn-X-011	b	1
fn-X-011	c	1

s	p	v
fn-X-100	a	1
fn-X-100	b	0
fn-X-100	c	0

s	p	v
fn-X-101	a	1
fn-X-101	b	0
fn-X-101	c	1

s	p	v
fn-X-110	a	1
fn-X-110	b	1
fn-X-110	c	0

s	p	v
fn-X-111	a	1
fn-X-111	b	1
fn-X-111	c	1

Why this result? →Sources: 8

На стягиваемом пространстве первая когомология тривиальна: порядок 1.

Condition

Размерность Ȟ¹ стягиваемого пространства

Established

Query parameters · 3

f: urn:case:stacks:cech:F
c: urn:case:stacks:cech:cov
k: 0

Input facts · 88

$X$ is a finite space with the specialization (Alexandrov) topology: the open subsets are exactly the subsets stable under generalization
x: x
$U$ is presented as a subset of $X$ (possibly empty), to be tested for openness
u x
U1 x
U2 x
W x
X x
the point belongs to $X$
p x
a x
b x
c x
0061: $g ⇝ p$ — $p$ is a specialization of $g$ , $g$ a generalization of $p$ : $p$ ∈ closure of $\{g\}$
g p
c a
c b
the point lies in the subset $U$
u p
U1 a
U1 c
U2 b
U2 c
W c
X a
X b
X c
the covering is aimed at $U$ : its members are proposed to cover $U$
c: covu: X
$U_i$ is a member of the covering
c ui
cov U1
cov U2
01FI: $U_1$ , the first member in the total ordering of the covering
c: covu: U1
01FI: $U_2$ , the second member in the total ordering of the covering
c: covu: U2
$F$ is the constant sheaf $(Z/2)_X$ : sections over $U$ are the locally constant maps $U → Z/2$
f: Fx: x
the candidate is a function on the points of $U$
s: fn-U1-00u: U1
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-U1-01u: U1
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-U1-10u: U1
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-U1-11u: U1
the function takes the value $v ∈ \{0, 1\}$ at the point
s p v
fn-U1-11 a 1
fn-U1-11 c 1
every function $U → Z/2$ is presented as a candidate section of $F$ over $U$
f: Fu: U1
the candidate is a function on the points of $U$
s: fn-U2-00u: U2
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-U2-01u: U2
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-U2-10u: U2
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-U2-11u: U2
the function takes the value $v ∈ \{0, 1\}$ at the point
s p v
fn-U2-11 b 1
fn-U2-11 c 1
every function $U → Z/2$ is presented as a candidate section of $F$ over $U$
f: Fu: U2
the candidate is a function on the points of $U$
s: fn-W-0u: W
the function takes the value $v ∈ \{0, 1\}$ at the point
s: fn-W-0p: cv: 0
the candidate is a function on the points of $U$
s: fn-W-1u: W
the function takes the value $v ∈ \{0, 1\}$ at the point
s: fn-W-1p: cv: 1
every function $U → Z/2$ is presented as a candidate section of $F$ over $U$
f: Fu: W
the candidate is a function on the points of $U$
s: fn-X-000u: X
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-X-001u: X
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-X-010u: X
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-X-011u: X
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-X-100u: X
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-X-101u: X
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-X-110u: X
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-X-111u: X
the function takes the value $v ∈ \{0, 1\}$ at the point
s p v
fn-X-111 a 1
fn-X-111 b 1
fn-X-111 c 1
every function $U → Z/2$ is presented as a candidate section of $F$ over $U$
f: Fu: X

u	x
U1	x
U2	x
W	x
X	x

p	x
a	x
b	x
c	x

g	p
c	a
c	b

u	p
U1	a
U1	c
U2	b
U2	c
W	c
X	a
X	b
X	c

c	ui
cov	U1
cov	U2

s	p	v
fn-U1-00	a	0
fn-U1-00	c	0

s	p	v
fn-U1-01	a	0
fn-U1-01	c	1

s	p	v
fn-U1-10	a	1
fn-U1-10	c	0

s	p	v
fn-U1-11	a	1
fn-U1-11	c	1

s	p	v
fn-U2-00	b	0
fn-U2-00	c	0

s	p	v
fn-U2-01	b	0
fn-U2-01	c	1

s	p	v
fn-U2-10	b	1
fn-U2-10	c	0

s	p	v
fn-U2-11	b	1
fn-U2-11	c	1

s	p	v
fn-X-000	a	0
fn-X-000	b	0
fn-X-000	c	0

s	p	v
fn-X-001	a	0
fn-X-001	b	0
fn-X-001	c	1

s	p	v
fn-X-010	a	0
fn-X-010	b	1
fn-X-010	c	0

s	p	v
fn-X-011	a	0
fn-X-011	b	1
fn-X-011	c	1

s	p	v
fn-X-100	a	1
fn-X-100	b	0
fn-X-100	c	0

s	p	v
fn-X-101	a	1
fn-X-101	b	0
fn-X-101	c	1

s	p	v
fn-X-110	a	1
fn-X-110	b	1
fn-X-110	c	0

s	p	v
fn-X-111	a	1
fn-X-111	b	1
fn-X-111	c	1

Why this result? →Sources: 8

Тривиальной группе отвечает размерность 0.

Condition

Порядок 2 на стягиваемом пространстве

Not established

Query parameters · 3

f: urn:case:stacks:cech:F
c: urn:case:stacks:cech:cov
n: 2

Input facts · 88

$X$ is a finite space with the specialization (Alexandrov) topology: the open subsets are exactly the subsets stable under generalization
x: x
$U$ is presented as a subset of $X$ (possibly empty), to be tested for openness
u x
U1 x
U2 x
W x
X x
the point belongs to $X$
p x
a x
b x
c x
0061: $g ⇝ p$ — $p$ is a specialization of $g$ , $g$ a generalization of $p$ : $p$ ∈ closure of $\{g\}$
g p
c a
c b
the point lies in the subset $U$
u p
U1 a
U1 c
U2 b
U2 c
W c
X a
X b
X c
the covering is aimed at $U$ : its members are proposed to cover $U$
c: covu: X
$U_i$ is a member of the covering
c ui
cov U1
cov U2
01FI: $U_1$ , the first member in the total ordering of the covering
c: covu: U1
01FI: $U_2$ , the second member in the total ordering of the covering
c: covu: U2
$F$ is the constant sheaf $(Z/2)_X$ : sections over $U$ are the locally constant maps $U → Z/2$
f: Fx: x
the candidate is a function on the points of $U$
s: fn-U1-00u: U1
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-U1-01u: U1
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-U1-10u: U1
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-U1-11u: U1
the function takes the value $v ∈ \{0, 1\}$ at the point
s p v
fn-U1-11 a 1
fn-U1-11 c 1
every function $U → Z/2$ is presented as a candidate section of $F$ over $U$
f: Fu: U1
the candidate is a function on the points of $U$
s: fn-U2-00u: U2
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-U2-01u: U2
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-U2-10u: U2
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-U2-11u: U2
the function takes the value $v ∈ \{0, 1\}$ at the point
s p v
fn-U2-11 b 1
fn-U2-11 c 1
every function $U → Z/2$ is presented as a candidate section of $F$ over $U$
f: Fu: U2
the candidate is a function on the points of $U$
s: fn-W-0u: W
the function takes the value $v ∈ \{0, 1\}$ at the point
s: fn-W-0p: cv: 0
the candidate is a function on the points of $U$
s: fn-W-1u: W
the function takes the value $v ∈ \{0, 1\}$ at the point
s: fn-W-1p: cv: 1
every function $U → Z/2$ is presented as a candidate section of $F$ over $U$
f: Fu: W
the candidate is a function on the points of $U$
s: fn-X-000u: X
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-X-001u: X
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-X-010u: X
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-X-011u: X
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-X-100u: X
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-X-101u: X
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-X-110u: X
the function takes the value $v ∈ \{0, 1\}$ 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 $U$
s: fn-X-111u: X
the function takes the value $v ∈ \{0, 1\}$ at the point
s p v
fn-X-111 a 1
fn-X-111 b 1
fn-X-111 c 1
every function $U → Z/2$ is presented as a candidate section of $F$ over $U$
f: Fu: X

u	x
U1	x
U2	x
W	x
X	x

p	x
a	x
b	x
c	x

g	p
c	a
c	b

u	p
U1	a
U1	c
U2	b
U2	c
W	c
X	a
X	b
X	c

c	ui
cov	U1
cov	U2

s	p	v
fn-U1-00	a	0
fn-U1-00	c	0

s	p	v
fn-U1-01	a	0
fn-U1-01	c	1

s	p	v
fn-U1-10	a	1
fn-U1-10	c	0

Why this result? →Sources: 8

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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