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Question about notation for Čech cohomology and direct image of sheaves in Hartshorne

score 195 · Answer 1 · 2026-03-17 11:38:41

195

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Question about notation for Čech cohomology and direct image of sheaves in Hartshorne

Published on 17 Mar 2026 - 11:38

score 183 · Answer 2 · 2026-03-17 13:38:28

183

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Image of a morphism of chain complexes of sheaves via direct/inverse image functor

Published on 17 Mar 2026 - 13:38

score 124 · Answer 3 · 2026-03-19 22:08:12

124

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Does $H^2(X_{Zar},\mathcal{O}_X^\times)=0$ for $X$ a regular scheme?

Published on 19 Mar 2026 - 22:08

score 90 · Answer 4 · 2026-03-20 16:28:42

90

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Computing the dimension of $H^0(X, \mathcal{O}_X(D))$, where $D \subset X$ is a divisor

Published on 20 Mar 2026 - 16:28

score 632 · Answer 5 · 2026-03-22 03:23:40

632

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Is the cohomology of a stalk the same as the stalk of the cohomology sheaf?

Published on 22 Mar 2026 - 3:23

score 101 · Answer 6 · 2026-03-19 22:11:37

101

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If $H^i(\tilde{X}, \mathcal{F}) = 0$, then is it true that $H^i(X, \mathcal{F}) = 0$?

Published on 19 Mar 2026 - 22:11

score 48 · Answer 7 · 2026-03-19 23:27:55

48

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Conditions on $\mathcal{F}$ such that $\chi(\mathcal{F}) = 0$ for a coherent sheaf on a curve over $k$.

Published on 19 Mar 2026 - 23:27

score 404 · Answer 8 · 2026-03-19 22:49:27

404

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Cohomology and inverse image of divisors

Published on 19 Mar 2026 - 22:49

score 555 · Answer 9 · 2026-03-20 06:51:52

555

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$\dim H^0(X, \mathcal{O}_D) \leq 1 + \deg D$ when $-1 \leq \deg D \leq g - 1$

Published on 20 Mar 2026 - 6:51

score 367 · Answer 10 · 2026-03-22 06:42:51

367

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Bott vanishing from the Euler sequence

Published on 22 Mar 2026 - 6:42

score 337 · Answer 11 · 2026-03-20 01:37:43

337

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Calculating Euler Characteristic of Closed Subscheme

Published on 20 Mar 2026 - 1:37

score 258 · Answer 12 · 2026-03-24 18:12:15

258

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Filtration with ideal sheaf quotients (Hartshorne, Proof of Theorem III.3.7)

Published on 24 Mar 2026 - 18:12

score 77 · Answer 13 · 2026-03-24 08:46:59

77

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Proof that $Pic^0\cong \frac{H^1(X,\mathcal O)}{H^1(X,\mathbb Z)}$

Published on 24 Mar 2026 - 8:46

score 187 · Answer 14 · 2026-03-17 20:28:49

187

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Sheaf Cohomology of Injective Sheaves respects Products

Published on 17 Mar 2026 - 20:28

score 114 · Answer 15 · 2026-03-17 02:33:01

114

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Sheaf Cohomology respects Products

Published on 17 Mar 2026 - 2:33

List Question

Question about notation for Čech cohomology and direct image of sheaves in Hartshorne

Image of a morphism of chain complexes of sheaves via direct/inverse image functor

Does $H^2(X_{Zar},\mathcal{O}_X^\times)=0$ for $X$ a regular scheme?

Computing the dimension of $H^0(X, \mathcal{O}_X(D))$, where $D \subset X$ is a divisor

Is the cohomology of a stalk the same as the stalk of the cohomology sheaf?

If $H^i(\tilde{X}, \mathcal{F}) = 0$, then is it true that $H^i(X, \mathcal{F}) = 0$?

Conditions on $\mathcal{F}$ such that $\chi(\mathcal{F}) = 0$ for a coherent sheaf on a curve over $k$.

Cohomology and inverse image of divisors

$\dim H^0(X, \mathcal{O}_D) \leq 1 + \deg D$ when $-1 \leq \deg D \leq g - 1$

Bott vanishing from the Euler sequence

Calculating Euler Characteristic of Closed Subscheme

Filtration with ideal sheaf quotients (Hartshorne, Proof of Theorem III.3.7)

Proof that $Pic^0\cong \frac{H^1(X,\mathcal O)}{H^1(X,\mathbb Z)}$

Sheaf Cohomology of Injective Sheaves respects Products

Sheaf Cohomology respects Products

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