Questions tagged [sheaf-cohomology]
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364 questions
3
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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
1
vote
0
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135
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Base change of cohomology when the cohomology is a torsion
Let $(R,m,k)$ be a discrete valuation ring, where $k=R/m$. Let $X\rightarrow \mathrm{Spec}R$ be a projective, integral and flat $R$-scheme. Let $\mathscr F$ be a coherent sheaf such that $H^i(X,\...
- 51
6
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2
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445
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Representability of flat cohomology by a group scheme
In his paper "Supersingular K3 surfaces", Artin states the following theorem (Theorem 3.1) without proof:
Let $\pi:X \to S = \mathrm{Spec}(k)$ be a smooth proper surface with $k$ an ...
- 10.5k
1
vote
0
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98
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Cohomology with coefficient in sheaf of morphisms of an algebraic group
Let $G$ be an affine algebraic group over ${\mathbb C}$. We denote the sheaf of morphisms from ${\mathbb A}^1$ to $G$ by $\bf G$. Then $H^1({\mathbb A}^1,\bf G)=0$ (Cech cohomlogy). Is this fact true? ...
- 1,177
3
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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
6
votes
1
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386
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Elementary proof of the exactness of Čech complex associated to a hypercovering ("Illusie's Conjecture")
Let $\mathcal{E}$ be a sheaf of abelian groups on a topological space (or a site). For an open covering $\mathfrak{U} = (U_i)_i$, it is well known that the augmented Čech complex $0 \to \mathcal{E} \...
- 3,312
2
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1
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503
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Higher direct image with compact support of a constant sheaf
Let $f: X \to Y$ be a locally trivial fibration between locally compact spaces with fiber $F$. It is well known that for a constant sheaf $A_X$ on $X$, the higher direct images $R^n f_* A_X$ are ...
1
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1
answer
634
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Inception of modern view of Sheaf Cohomology in Mathematical Literature
From wikipedia entry on Sheaf Cohomology I have found the intriguing passage: 'The essential point is to fix a topological space X and think of cohomology as a functor from sheaves of abelian groups ...
- 37
3
votes
1
answer
258
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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
2
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0
answers
56
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Conditions for long exact sequence for line bundles on curve to degenerate?
Let $\varphi:X\to Y$ be a morphism of schemes of relative dimension 1, and $\mathcal{L}' \xrightarrow{g} \mathcal{L}$ an injection of line bundles on $X$.
The sequence
$$0\to \mathcal{L}' \xrightarrow{...
- 1,338
6
votes
1
answer
1k
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Poincare duality on the level of complexes
The classical Poincare duality is formulated in terms of cohomology groups. I am wondering if we can also formulate it in terms of complexes.
In particular, suppose $\mathcal{C}^*$ is a complex of $...
- 69
1
vote
1
answer
2k
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Pullback map on global sections surjective
Let $X,Y$ be two irreducible, projective $k$-schemes. $k$ is assumed to be algeraically closed. Consider a dominant morphism $f: X \to Y$ between them which is not an isomorphism!
Let $\mathcal{L}$ ...
- 6,006
13
votes
1
answer
649
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Sheaves in combinatorics and discrete geometry
I am looking for examples for the application of sheaves, sheaf-like constructions or the (co)homology of sheaves to problems in combinatorics and discrete geometry.
For example given a poset $(P,\...
- 430
1
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0
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199
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Künneth formula for local cohomology with support
In "Differential operators on the flag varieties" (http://www.numdam.org/article/AST_1981__87-88__43_0.pdf) by Brylinski, he uses on page 53 a Künneth formula for local cohomology with ...
- 473
1
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1
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454
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Relationship between $H^1(X, \mathbb{T})$ and complex line bundles
Let $X$ be a compact metric space and consider the sheaf cohomology group $H^1(X, \mathbb{T})$. From a class in $H^1(X, \mathbb{T})$, I can get a principal $\mathbb{T}$-bundle over $X$ and from this, ...
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
5
votes
1
answer
209
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Cohomology of doubly pinched torus via spectral sequences
Let $f:T^2\to Y$ be a resolution of singularities where $Y$ is a torus with two "pinched" points (or, if you prefer, two copies of $\mathbb{P}^1$ meeting at two points). I'm interested in ...
- 191
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2
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6k
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When does sheaf cohomology commute with arbitrary direct sums?
It is well known and more or less proven in Hartshorne's 'Algebraic Geometry' (p. 209) that for every noetherian scheme $X$ and every collection of abelian sheaves $\mathcal{F}_i$ the canonical map
$$...
- 12.1k
1
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1
answer
521
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Connecting homomorphism in Cech cohomology
Let $M$ be a smooth manifold and $\mathcal{U}$ be a good open cover of $M$. If I have an exact sequence of sheaves
$$0 \longrightarrow A \stackrel{f}\longrightarrow B \stackrel{g}\longrightarrow C \...
- 19
2
votes
0
answers
124
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The cohomology groups corresponding to a modified global sections functor
Let $\mathcal{F}$ be a sheaf on the big etale site of $Sm_k$. I am looking for a way to calculate a modified version of sheaf cohomology. Let $X$ be a smooth scheme and $Z$ a closed sub-scheme. After ...
- 5,901
2
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1
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279
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Help about "Varieties with small Dual Varieties" by L.Ein
I'm studying the paper "Varieties with small Dual Varieties" by L.Ein and in the construction he gives about the $10-$dimensional spinor variety $S_4 \subset \mathbb{P}^{15}$ I'm finding ...
- 1,343
1
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0
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128
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understanding higher direct images of $\mathbb{G}_m$ for a finite Galois map
Let $X$ be a smooth quasi-projective variety over $\mathbb{C}$, and let $\mu_r$ denote the group of $r$-th roots of unity, and moreover suppose $\mu_r$ (algebraically) acts on $X$ freely. Then $Y:= X/\...
- 737
3
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0
answers
126
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Poincare polynomials for Borel Moore homology and fibrations
For an algebraic variety $X$ over $\mathbb{C}$, we denote $H_k(X)$ as its Borel-Moore homology of degree $k$. Let us define the Poincare polynomial associated it by
$$P(X)=\sum_{k\in \mathbb{N}}dim ...
- 71
2
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0
answers
201
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Higher cohomology of projective bundles
Let $C$ be a curve and $L$ be a line bundle with sufficiently large degree. Let $C_p$ denote the $p$-th symmetric product of $C$, which consists of all the effective divisors of degree $p$ on $C$. Let ...
- 439
3
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0
answers
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
10
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0
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964
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intuition about perverse sheaves
firstly, I would know if my very basic intuition on perverse sheaves is correct .
secondly, I would have some clarification in what perverse sheaves behaves better than regular sheaves .
my intuition ...
- 546
3
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0
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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}[\...
2
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0
answers
90
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$\bigoplus_{k=0}^{\infty}H^n(X,I^k\mathcal{F})$ is a finitely-generated $\bigoplus_{k=0}^nI^k-$graded module
Does anyone know where I can find a proof of the following result ?
Given a Noetherian ring $A$, a proper morphism of schemes $X\rightarrow \operatorname{Spec}A$, a coherent $O_X-$module $\mathcal{F}$ ...
- 121
5
votes
0
answers
370
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Continuity property for Čech cohomology
Suppose we have an inverse system of compact Hausdorff spaces $\lbrace X_i , \varphi_{ij} \rbrace_{i\in I}$ and that each space has a presheaf $\Gamma_i$ assigned to it in such a way that $\Gamma_i(\...
- 275
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1
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Can one determine the trace map for a nonsingular projective variety explicitly?
I've never understood how one would actually go about computing a trace map associated with the canonical sheaf on a smooth projective variety, if it's even possible. Hartshorne proves that the ...
- 615
6
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1
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761
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The Yoneda pairing, hypercohomology, and cup product
Let $\mathcal{F}$ and $\mathcal{G}$ be coherent analytic sheaves on $\mathbb{P}^n$. Let $\mathcal{F}_\bullet$ be a locally free resolution of $\mathcal{F}$. In Principles of Algebraic Geometry by ...
- 294
1
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0
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Dimension of global holomorphic sections of a line bundle
Let $K$ be the canonical line bundle of a compact Riemann surface $M$ of genus $g$. Consider the pull back of $K$ on $M \times M$ via projection on the first factor. What is the dimension of the space ...
- 35
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0
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91
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Continuity of motivic cohomology under direct limit
Given the motivic complexes $\mathbb{Z}(n)$ on the big Zariski site of finite type smooth $k$-schemes denoted by $FinSm_k$, we pullback it to the smooth $k$-schemes i.e. $Sm_k$. For example for a ...
- 5,901
8
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1
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506
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Does the sheaf $\mathcal{O}^*$ on a complex manifold have an acyclic cover?
Let $X$ be a complex manifold and let $\mathcal{O}^*$ be the sheaf nonvanishing holomorphic functions on it. Does it have an acyclic cover? That is, a cover for which all open sets and all ...
- 1,198
1
vote
1
answer
1k
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Injectivity of the cohomology map associated to the pullback of line bundles
Let $f:X\to Y$ be a flat, surjective, smooth morphism between smooth algebraic varieties (over $\mathbb C$). We assume that $f$ has relative dimension $n$ and we assume also that $\dim Y\ge 2$ (just ...
- 321
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0
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637
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Calculation in prismatic cohomology
In the standard references for prismatic cohomology, most theorems are proved in a local context (i.e. with completeness assumptions), and the devissage to the global case (i.e. smooth proper ...
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4
votes
1
answer
289
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Yoga on coherent flat sheaves $\mathcal{F}$ over projective space $\mathbb{P}^n$
I'm reading Mumfords's Lectures on Curves on an Algebraic Surface (jstor-link: https://www.jstor.org/stable/j.ctt1b9x2g3)
and I found in Lecture 7 (RESUME OF THE COHOMOLOGY OF COHERENT SHEAVES ON
$\...
- 6,006
2
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0
answers
405
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Cohomology of a family of twisted cubic curves (Hartshorne III, 12.9.2)
I'm trying to understand following Example in Hartshorne (Chapter III, Example 9.8.2
& Example 12.9.2):
Let $X_1 \subset \mathbb{P}^{3}$ be a twisted cubic curve not containing the point
$(0:...:1)...
- 6,006
3
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1
answer
460
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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
3
votes
1
answer
225
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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
2
votes
0
answers
670
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Čech-Alexander complex in computing (crystalline/prismatic) cohomology
I have a naive question about Čech-Alexander complexes in prismatic cohomology (although I suspect that the situation is similar for crystalline cohomology).
They seemed to be introduced as a method ...
- 21
9
votes
2
answers
4k
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Top cohomology detecting compactness
I am looking for a reference for the fact that the top cohomology $H^n(X;A)$ of an $n$-dimensional manifold $X$ is non-trivial precisely when $X$ is compact.
I tried to ask this question on Math....
- 1,211
15
votes
5
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Non-cohomological proof that the pullback of an ample bundle by a finite morphism is ample
It is a standard fact that for any finite morphism of proper Noetherian $A$-schemes ($A$ being Noetherian), the pullback of an ample line bundle is ample. The usual proof of this fact is via Serre's ...
- 7,338
3
votes
0
answers
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
28
votes
1
answer
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Two points of view about Borel-moore homology
They are several ways to define the Borel-Moore homology on a locally compact space $X$.
The first one is by analogy with the singular homology but instead of using finite chains, we use locally ...
- 1,017
2
votes
1
answer
385
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Very weak Riemann-Roch on curves (by J. Kollar)
I have a question on an unequality used in the proof of the Very weak Riemann-Roch on curves in Janos Kollar's Lecture on Resolution of Singularities (page 14):
1.13 (Very weak Riemann-Roch on curves)...
- 6,006
4
votes
0
answers
211
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Does cohomology and base change hold if supported at a point?
I have a flat, quasicompact, and separated map $p : X \to \mathbb{A}^1$ and I know that $R^i p_* \mathcal{O}_X$ vanishes everywhere except possibly $0 \in \mathbb{A}^1$.
Q1: Does "cohomology and ...
- 1,104
1
vote
1
answer
278
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Relation between characteristic cycle and singular support of constructible sheaf
Let $M$ be a real analytic manifold. Let $F$ be an object of the bounded derived category of sheaves on $M$ with real constructible cohomology sheaves. Let $CC(F)$ denote the characteristic cycle of $...
- 21.8k
2
votes
1
answer
602
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Pushforward in Compactly Supported Cohomology
Suppose $X,Y$ are locally compact Hausdorff spaces and $f:X\to Y$ is a topological submersion of relative dimension $n$. By this we mean that for all points $x\in X$, there exists an open neighborhood ...
- 1,761
6
votes
1
answer
284
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Formal character of local cohomology groups with support in Schubert cells
Let $k$ be a field of characteristic zero, $G$ a connected semi-simple algebraic group over $k$ and $B$ a fixed Borel subgroup of $G$ with maximal torus $T$. Also denote by $W$ the Weyl-group of $G$. ...
- 473