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

score 204 · Answer 1 · 2026-05-05 03:34:32

204

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

Published on 05 May 2026 - 3:34

score 199 · Answer 2 · 2026-05-05 13:46:15

199

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

Published on 05 May 2026 - 13:46

score 132 · Answer 3 · 2026-05-07 19:24:24

132

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

Published on 07 May 2026 - 19:24

score 96 · Answer 4 · 2026-04-29 12:50:31

96

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

Published on 29 Apr 2026 - 12:50

score 642 · Answer 5 · 2026-05-05 06:00:35

642

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

Published on 05 May 2026 - 6:00

score 107 · Answer 6 · 2026-05-07 19:06:27

107

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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 07 May 2026 - 19:06

score 54 · Answer 7 · 2026-05-07 09:56:25

54

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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 07 May 2026 - 9:56

score 411 · Answer 8 · 2026-04-29 12:50:31

411

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Cohomology and inverse image of divisors

Published on 29 Apr 2026 - 12:50

score 560 · Answer 9 · 2026-04-29 12:50:21

560

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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 29 Apr 2026 - 12:50

score 372 · Answer 10 · 2026-05-05 15:41:59

372

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Bott vanishing from the Euler sequence

Published on 05 May 2026 - 15:41

score 344 · Answer 11 · 2026-05-07 20:27:39

344

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Calculating Euler Characteristic of Closed Subscheme

Published on 07 May 2026 - 20:27

score 262 · Answer 12 · 2026-04-23 20:35:38

262

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

Published on 23 Apr 2026 - 20:35

score 83 · Answer 13 · 2026-04-29 12:50:50

83

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

Published on 29 Apr 2026 - 12:50

score 194 · Answer 14 · 2026-04-29 12:50:43

194

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Sheaf Cohomology of Injective Sheaves respects Products

Published on 29 Apr 2026 - 12:50

score 122 · Answer 15 · 2026-04-29 12:50:38

122

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Sheaf Cohomology respects Products

Published on 29 Apr 2026 - 12:50

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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