Questions tagged [sheaf-cohomology]
The sheaf-cohomology tag has no usage guidance.
364 questions
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Does hypercohomology of the Koszul complex compute sheaf cohomology?
Let $i:X \to \mathbb{P}^n$ be a smooth projective variety defined by the vanishing locus of polynomials $(\underline{f}) = (f_1,\ldots,f_k)$ which have degrees $>0$ and are pairwise coprime, ...
- 1,716
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575
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Sheaf Cohomology as Glueing of Local Data
For some time I've been trying to find an answer to the question "why do we care about or compute sheaf cohomology". As far as I can tell books like Hartshorne treat this as something we already want ...
- 901
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177
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Reference request: local cohomology in disjoint union
I have a topological space $X$ and two disjoint, closed subspaces $Y$ and $Z$ of $X$. I believe that in this situation, for any abelian sheaf $\mathcal{F}$ on $X$ and any $p \in \mathbb{N}$, there is ...
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Proper base change for non-quasicoherent sheaves
For a proper flat map $f: X \to Y$ (of reasonable schemes) and a closed embedding $i: Y' \to Y$, we know by EGA the base change quasi-isomorphism, where $f'$ and $i'$ are the pullbacks of $f$ and $i$:
...
- 2,040
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137
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Cohomology of $Sym^m Q \otimes Sym^k Q \otimes L^p$
Let $V$ be a complex vector space. Let $L=\mathcal{O}(-1)$ and $Q=V/L$ be the quotient bundle over $\mathbb{P}V$.
I'm trying to compute the cohomologies with coefficients in $Sym^m Q \otimes Sym^k Q \...
- 371
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110
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Convolution of DQ-Modules
On page 92 of Deformation Quantization Modules Kashiwara and Schapira define two different convolution products for DQ-modules that differ by whether one uses $Rp_{13*}$ or $Rp_{13!}$ to push forward....
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Cohomology of a sheaf with only one stalk
Let $X$ be a proper scheme over a henselian discrete valation ring. I have a Nisnevich sheaf $F$ of which has only one stalk at the generic point of $X$ (and all other stalks vanish).
I believe that ...
- 1,553
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362
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Weil Kostant Integrality Result as Stated by Brylisnki
I'm reading through Brylinski's "Loop Spaces, Characteristic Classes and Geometric Quantization" and I am stuck on a piece of Theorem 2.2.15, which asserts that If $K$ is a closed complex-valued 2-...
- 1,611
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orientations on a stratified space
For this question let $k$ be any field of characteristic not equal to $2$ and $X$ a stratified space whose strata are topological manifolds. I'm not sure what the definition of "stratified space" ...
- 3,605
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Cohomology of tangent bundles
Let $X$ be a smooth scheme and $Z\subset X$ a smooth subscheme. Consider the blow-up
$$\pi:\widetilde{X}:=Bl_{Z}X\rightarrow X$$
of $X$ along $Z$.
What is the relation between the cohomology of the ...
- 8,998
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Estimates for certain double-Kloosterman sums
Sorry to disturb. I encounter a double-Kloosterman sum, which needs some help from the experts here.
For any $q\in \mathbb{N}^+$, how can we estimate the type of sum
$$ \sideset{_{}^{}}{^{\ast}_{}}\...
- 563
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2
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371
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Extension between vector bundles inducing non-zero map on cohomology
Let $X$ be a projective variety over a field $k$ equipped with a very ample line bundle $\mathcal{O}_X(1)$. Suppose that $E, F$ are locally free sheaves of finite rank on $X$ and $c\in \mathrm{Ext}^i(...
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Elementary way to compute Hodge numbers of Grassmanian
I know that by using Hodge decomposition and the fact that Schubert cells are Hodge cycles you can compute the Hodge numbers of Grassmanian but is there a more elementary way to compute sheaf ...
- 1,093
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Application of sheaves theory in ring theory
Is there any text that gives some applications of sheaves theory in commutative ring theory? In the other word, is any results in commutative ring theory that be verified by sheaves method?
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Vanishing of sheaf cohomology with compact support
Let $X$ be a smooth manifold. Let $F$ be a sheaf of $\mathbb{R}$-vector spaces on $X$. I have three closely related questions.
1) Under what sufficient conditions on $F$ for any compact subset $K\...
- 21.8k
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What's $H^*(X - \{x_1,\ldots,x_n\},\mathcal{O})$, when $X$ is a projective smooth surface?
Let $X$ be a smooth projective surface over a field $k$. Is there a way to compute $H^1(X - \{x\},\mathcal{O}_{X-\{x\}})$ in terms of similar invariants for $X$? Actually I'd like to remove even ...
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343
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Is any element in $H^2_{et}(X,\mathcal{O}_X^*)$ locally trivial in the Zariski topology?
Let $X$ be an algebraic variety over a field $k$ and we consider the cohomological Brauer group $H^2_{et}(X,\mathcal{O}_X^*)$.
For any element $\alpha \in H^2_{et}(X,\mathcal{O}_X^*)$ and any closed ...
- 9,019
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Help understand a calculation involving RHom of sheaves on manifolds
I am reading a paper and there is some computation of RHom of sheaves that I don't understand. I hope this is the right place to ask.
It is this paper, example 3.10 , page 25
arxiv.org/pdf/1005.1517v4....
- 1,893
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1
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Compare global sections of restriction and pullback of sheaves
Let $X$ be a projective scheme and $X \subset \mathbb{P}^n$ for some positive integer $n$. Let $j:Z \hookrightarrow X$ be a closed subscheme. Is it true that $H^0(j^*\mathcal{N}_{X|\mathbb{P}^n}) \...
- 2,323
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2
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390
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Topological information via cohomology of sheaves
On a projective smooth variety $X$ over complex numbers (or rather compact Kahler) we have a specific set of sheaves, namely sheaves of holomorphic forms ${\mathcal \Omega}^p$ of various degrees. The ...
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270
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Čech-like cohomology with the “other nerve”
Let $X$ be a space and $\mathcal U$ a cover of $X$. Instead of Čech cohomology, I would like to take the following construction:
let
$$I= \{ \text{finite nonempty intersections of elements of }\,\...
- 2,368
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1
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Subspace inclusion with non-vanishing higher direct images
I'm looking for concrete topological intuition for the derived pushforward.
Let $f:X\to Y$ be a continuous map. The derived pushforward $\mathbf Rf_\ast$ takes a sheaf $F$ to the sheafification of ...
- 10.5k
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1
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332
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Degeneration of Hodge-de Rham spectral sequence, exactness of a pairing and the trace morphism
If $X\longrightarrow S$, where $S=\operatorname{Spec}(k)$, is a proper smooth morphism of dimension $p$ between $\mathbb{F}_{p}$ schemes with Frobenius $F$, then there is an exact pairing $\wedge:\...
- 1,805
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For what kind of sheaves can we always extend a sheaf map from a closed subset to the whole space?
Let $X$ be a topological space. We know that a sheaf on $X$ is call soft if for any closed subset $Z$ of $X$, a section on $Z$ can be always extend to a section on $X$.
Now we consider a similar ...
- 9,019
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240
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Cohomology of the complement of a subvariety
Let $X$ be a complex manifold, $Y\subset X$ a subvariety, and $U:=X\setminus Y$ of codimension $d$. It is well known that the restriction map
$$
H^i(X,\mathbb Q)\to H^i(U,\mathbb Q)
$$
is an ...
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Does local cohomology commute with pullback?
Let $Y$ be a topological spaces and $Z \subset Y$ be locally closed, i.e. $Z=V \cap U^c$ for $U,V \subset Y$ open.
For any abelian sheaf $\mathcal{F}$ on $Y$ let $\Gamma_Z(Y,\mathcal{F}):=\ker(\...
- 473
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Homology in the sections of an infinite exact sequence of injective sheaves of $\mathcal O_X$-modules?
Let $(X, \mathcal O_X)$ be a scheme and the following an infinite, exact sequence of injective sheaves of $\mathcal O_X$-modules:
$$
\cdots \overset{f_5}\longrightarrow I_5\overset{f_4}\longrightarrow ...
- 355
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Birational Invariants
Let $X$ be a smooth rational variety of dimension $n$. We have $\dim H^0(X,\Omega_X^p) = \dim H^0(\mathbb{P}^n,\Omega_{\mathbb{P}^n}^p)$ for any $p$. These are Hodge numbers. I know that we can not ...
user77919
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Proving the Eichler-Shimura Isomorphism defines a global section
Let $ f \in S_k(\Gamma)$ be a weight k modular cusp form of level $\Gamma$, with modular curve $Y_{\Gamma}$. Let $V^{k-2}$ be the homogenous polynomials in X and Y of degree k-2 with complex ...
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In what sense is the complex $\mathscr{L}^\bullet$ unique?
This is in Section III.12. Algebraic Geometry by Hartshorne. Assume $X\to\mathrm{Spec}(A)$ is a projective morphism of Noetherian schemes. Let $\mathscr{F}$ be coherent over $X$, flat over $A$.
...
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Automorphisms of Frobenius liftings and degeneration of the Hodge-de Rham spectral sequence
I am still studying Deligne and Illusie's paper (https://eudml.org/doc/143480), and I am again stuck, this time on pages 262/263.
Assume $X\longrightarrow S$ is a smooth morphism of $\mathbb{F}_{p}$-...
- 1,805
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1
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177
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is the induced map of an embedding an Iso on Ext-groups?
I am sorry, but I am quite new to Ext groups of sheaves. However, I have a closed embedding of projective $\mathbb{C}$-schemes $\iota : X \hookrightarrow Y$ and was wondering if
$$\iota_*:\mathrm{Ext}^...
- 213
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1
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The sheaf propagation is open in the zero section
Let $X$ a smooth manifold and $F$ a sheaf (let's say of abelian groups) on $X$. We will say that $F$ propagates at $x\in X$ in the (co)-direction $p \in T_x^*X$ if for all $C^1$-function $\phi$ ...
- 1,017
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What is $h^0(\mathcal O_F)$ where $F$ is a fiber of a normal surface over a smooth curve?
Lately I am studying the bend-and-break, and I follow the proof in the following note written by Olivier Debarre:
http://www.math.ens.fr/~debarre/M2.pdf
There is a detail that I just cannot go ...
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2
answers
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Hartshorne Proposition III 8.1
In the proposition mentioned in the title, Hartshorne states that for each $i\geq 0$ the higher direct image sheaf $R^if_*\mathcal F$ is exactly the sheafification of the presheaf given by
$V \mapsto ...
- 41
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Cohomology and proper base change
Let $\pi:\mathcal{X} \to B$ be a flat, projective surjective morphism over $\mathbb{C}$. Assume that $B$ is a smooth quasi-projective curve. Let $\mathcal{F}$ be a coherent sheaf on $\mathcal{X}$, ...
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Grothendieck spectral sequence (cohomology version) for posets with functor coefficient
In this paper, Quillen mentioned a spectral sequence as follows.
Let $f:X\to Y$ is a poset map and $\mathcal{F}:X\to Ab$, where $Ab$ is the category of abelian groups, a functor which is contravariant ...
- 139
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250
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Is pullback map on sheaf cohomology injective for surjective morphisms?
Consider a surjective map $f\colon X\to Y$ of smooth projective varieties. It is well known (see e.g. Voisin's Hodge theory I, Lemma 7.28) that the map $H^i(Y,\mathbb Q)\to H^i(X,\mathbb Q)$ is ...
- 2,305
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On the exactness of the restriction to small etale site in the case of differentiable stacks
Let $X$ be a manifold considered as a stack on the big etale site of manifolds. Consider the functor $$ F \mapsto F_X$$ which sends a sheaf on the stack $X$ to the sheaf on the manifold $X$.
In Lemma ...
- 313
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227
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Long exact sequence in Borel-Moore homology
The Wikipedia page for Borel-Moore homology states that for a locally compact set $X$ and a closed subset $Z$, if we write $U = X \setminus Z$ we have the following long exact sequence
$$\cdots \to H^{...
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Deligne's integrality theorem in the setting of $ \mathbb{F}_{\ell}((t)) $-adic cohomology
Let $ \mathbb{F}_{q} $ be a finite field of characteristic $ p $ and $ \overline{\mathbb{F}_{q}} $ be an algebraic closure of $ \mathbb{F}_{q} $. Let $ X $ be a smooth projective variety over $ \...
- 863
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Hypercovers with sieves
Consider a covering family $\{Y_i \to X\}$ and the induced sieve $R \subseteq X$, the subpresheaf of all maps to $X$ that factor through some $Y_i$. The family gives me an induced Cech nerve $C_\...
- 1,104
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Is the cohomology $H^1(X, \mathcal{E}^\nabla)$ trivial, for the sheaf of constants of an algebraic connection $\nabla$?
Suppose that $\pi:X\to S$ is a flat morphism between Noetherian, integral schemes (of characteristic zero, if need be).
Let $\mathcal{E}$ be a locally free sheaf on $X$, and
$$\nabla:\mathcal{E} \to \...
- 1,338
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Poincare polynomials for Borel Moore homology and fibrations
For an algebraic variety $X$ over $\mathbb{C}$, we denote $H_k(X)$ as its Borel-Moore homology of degree $k$. Let us define the Poincare polynomial associated it by
$$P(X)=\sum_{k\in \mathbb{N}}dim ...
- 71
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641
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fppf/ etale Cohomology calculate with Cech cohomology
Let $R$ be a commutative ring with one and $S$ commutative faithfully flat $R$-algebra (that is there is a faithfully flat ring map
let $\phi: R \to S$). Then the so called Amitsur complex
$R \to S^{\...
- 5,986
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314
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Sheaf cohomology of Grassmannian G(2,4) with values in twisted tautological bundles over an arbitrary field
Let k be an arbitrary field. Let $G(2,4)_k$ be the Grassmannian of 2-planes in 4-space over that field. Let $\mathcal{E}$ be the tautological quotient bundle on the Grassmannian. I am trying to ...
- 31
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387
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Spectral sequence from resolution of condensed abelian groups
I am watching Scholze's and Clausen's masterclass on Condensed Mathematics and I don't understand or can find any references on something they said.
You have a resolution
$$ \dots \to \mathbb{Z}[\...
3
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0
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446
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Sheaf cohomology on the formal completion of $\mathbb{P}^1_k$ at its north pole
Let $k$ be a field and let $\mathbb{P}^1_k$ be the projective line over $k$. Let $\mathcal{F}$ be a coherent sheaf on $\mathbb{P}^1_k$. The curve $\mathbb{P}^1_k$ is recovered by the two affine ...
- 1,479
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671
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Elementary reference for Borel-Moore/locally finite homology
There is a homology theory called "Borel-Moore" or "locally finite" homology, which can either be constructed by using locally-finite chains or by more advanced sheaf-theoretic ...
- 2,533
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195
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Conceptual definition of derived functors allowing for quick proof of comparison theorems for sheaf cohomology
There are several approaches of increasing sophistication and simplicity to defining derived functors. I know of universal $\delta$-functors and Kan extensions along localizations. More definitions ...
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