Questions tagged [sheaf-cohomology]
The sheaf-cohomology tag has no usage guidance.
166 questions with no upvoted or accepted answers
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Global section of line bundle on anti-canonical rational surface
Let $X$ be an anti-canonical rational surface(i.e. $-K_X$ is effective) such that $K_X^2\geq 1$. Let $D$ be a $r$-class divisor ($D^2=r, D^2+D.K_X=-2$, the latter condition can be re-interpreted as $\...
- 1,982
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206
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Cohomology of fiber bundles with non constant coefficients
Let $E \to B$ be a topological fiber bundle. We know that the Serre spectral sequence allows you to compute the cohomology of $E$ with constant coefficient in terms of the cohomology of $F$ and the ...
- 121
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265
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Cohomology of intersection of projective hyperplanes
I will change my original question a bit for a bounty:
Let $A$ be a reduced finitely generated $\bar{\mathbb{F}}_p$-algebra (integral, if you want). Let $X$ be a non-empty intersection of ...
- 189
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347
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l-adic cohomology and perverse sheaves
Let consider the map $tr:\mathbb{G}_{m}^{n}\rightarrow\mathbb{A}^{1}_{\mathbb{F}_{q}}$ given by the sum of the coordinates and let $\psi:\mathbb{F}_{q}\rightarrow\mathbb{Q}_{l}^{*}$ a non trivial ...
- 3,472
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287
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$\delta$-functor and commutativity of pull-back with right derivation
Let $f:X \to Y$ be a faithfully flat projective morphism of noetherial $\mathbb{C}$-schemes. Assume that $Y$ is affine, smooth over $\mathbb{C}$. Let $y \in Y$ be a closed point with residue field, ...
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116
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Cohomology and quotients for the canonical topology
Recall that for any category $\mathcal C$, there is a unique finest topology, the canonical topology on $\mathcal C$ for which all representable functors are sheaves. I am interested in the example $\...
- 5,759
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306
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Cech cohomology.
Let us consider the scheme $X$ and the coherent sheaf $\cal F$ on it. We consider finitely many affine open covers $U_{\lambda}$ with $\lambda \in \Lambda$. When I calculate Cech cohomology with $U_{\...
- 181
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151
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Intersections of components of 'simple' ('local") Zariski coverings
I would like to study the ordered Cech cohomology with respect to a Zariski covering of a variety. I can pass to the limit with respect to refinements; the components of the 'limit covering' will be ...
- 16.9k
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238
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Non-realizable CR structures?
Hill, Penrose, and Sparling have an example of a non-realizable CR structure, a 5-manifold $M^5$ that comes equipped with a "twisted version" of the Lewy operator for the quadric $Q^2$,
$v = \frac{1}{...
- 21
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141
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Homeomorphic endomorphism of schemes inducing equivalence of sheaves
Let $F: X \to X$ to be an endomorphism of scheme $X$, which is additionally assumed to induce an universal homeomorphism on the underlying topological space $| X|$. Then it is known that this induces ...
- 5,966
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127
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Local freeness of dualizing sheaf
I am reading the dualizing sheaf and duality theorems from Hartshorne’s algebraic geometry book. I am wondering about the following.
When does the dualizing sheaf of a projective scheme is an locally ...
- 613
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121
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How to increase the second cohomology group of the structure sheaf?
We know that $H^2(\mathcal{O}_{\mathbb{P}^3})=0$. I am looking for blow-ups $$\pi:X \to \mathbb{P}^3$$ such that $X$ is non-singular and $H^2(\mathcal{O}_X)>0$. Of course, if we blow-up along ...
- 2,323
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119
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A question about cohomology with local coefficient
Let's consider the next theorem.
Theorem
[The cohomology Leray-Serre Spectral sequence] Let $R$ be a
commutative ring with unit. Given a fibration $F\hookrightarrow E\overset{p}{%
\rightarrow }B$, ...
- 1,367
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213
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Computing the first sheaf cohomology
I am looking for some examples of computing the dimension of the first sheaf cohomology for smooth projective surfaces. To be more precisely, let $X$ be a smooth, projective surface. Let $D$ be an ...
- 461
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295
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Exterior product of Euler Exact Sequence
Consider the Euler exact sequence:
$ 0\longrightarrow \mathcal{O}_{\mathbb{P}^n} \longrightarrow \mathcal{O}_{\mathbb{P}^n}(1)^{n+1}\longrightarrow \mathcal{T}_{\mathbb{P}^n} \longrightarrow 0 $
This ...
- 131
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153
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Sheaf cohomology definition
I have seen multiple definitions for sheaf cohomology and wanted to ask for the reason.
One goes through injective resolutions and the other through flasque resolutions.
For paracompact bases it holds ...
- 11
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217
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Artin-Winters proof of semi-stable reduction theorem: details
I've been reading through Artin-Winters proof of the semi-stable reduction theorem (Degenerate fibers and stable reduction of curves) and found myself confused about the following detail—
Let $\...
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355
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Global section of pullback of an ideal sheaf
For a local ring $R$ with maximal ideal $\mathfrak{m}\subset R$ and residue field $\kappa$, and a flat morphism $f\colon X\rightarrow \mathrm{Spec} R$ of schemes, we consider the short exact sequence ...
- 119
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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
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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
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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
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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
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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
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469
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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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104
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$L^r_M = i_* \circ \hat{L}^{r-1}_M \circ i^*$ by the projection formula and the Poincare duality
This is a question arising when I am reading
M. A. A. de Cataldo, L. Migliorini - The Hard Lefschetz Theorem and the topology of semismall maps, Ann. sci. École Norm. Sup., Serie 4 35 (2002) 759-772.
...
- 1,168
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163
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Explicit map between $\check{H}^1(M,\underline{\mathbb{R}})$ and $H^1(M,\mathbb{R})$
Is there a way to construct an explicit isomorphism between Cech cohomology and singular cohomology on a smooth manifold for degree 1? If yes can this be extended to higher degee?
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160
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Can morphisms of Mayer-Vietoris triangles be completed into a $3\times 3$ square?
Let $(X,\mathcal{O}_X)$ be a topological ringed space and $(U,V)$ be an open covering of $X$ (i.e. $U$ and $V$ are two open subsets of $X$ such that $U\cup V=X$). Let $\mathcal{F}$ and $\mathcal{F}'$ ...
- 1,479
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117
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Compute Cech cohomology with two open sets
Let $(X,\mathcal{O}_X)$ be a topological ringed space and $(U,V)$ be an open covering of $X$ (i.e. $U$ and $V$ are two open subsets of $X$ such that $U\cup V=X$). Let $\mathcal{F}$ be a sheaf of $\...
- 1,479
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161
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Surjectivity of multiplicative map (in more specific case)
(I have asked the question Surjectivity of multiplicative map. I ask here the more specific case.)
Let $S$ be a smooth complex algebraic surface, and $D$ be a divisor on $S$ such that $D^2>0$ and $...
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local acyclicity when restricting to an hypersurface
Let $X$ be a smooth scheme over $\mathbb{C}$ and a constructible sheaf $K$ of complex vector spaces on $X\times\mathbb{A}^1$ and a function $g:X\rightarrow \mathbb{A}^1$.
Suppose that $K$ is locally ...
- 3,472
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191
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Group cohomology of sheaves under closed immersion
Suppose $X$ is a scheme over Spec $\mathbb{Z}$, and $p$ is a non-zero prime in $\mathbb{Z}$. Then we have a closed immersion from the special fibre $i_p: X_p \rightarrow X$. If $\mathscr{F}$ is a ...
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What is the smallest number $d$ such that $H^1(X,\pi^*\mathcal{O}_{\mathbb{P}_k^1}(d))$ vanishes?
Let $X$ be a reduced projective scheme of pure dimension 1 over the field $k$. Let $\pi: X \to \mathbb{P}_k^1$ be a finite, flat and surjective morphism onto the projective line.
What is the ...
- 435
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114
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Iitaka dimension of a $\mathbb{Q}$-Cartier Prime divisor
Let $X$ be a normal projective variety and $D$ a prime divisor such that $mD$ is Cartier for some integer $m>0$.
Suppose $H^1(X,\mathcal{O}_X)=0$ and $mD|_D\sim 0$.
My questions are the following:
...
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164
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Compute $H^i(S,\underline{\text{Hom}}(A,\mathbb G_m))$ for a semi-abelian scheme $A$
How can I compute $H^i(S,\underline{\text{Hom}}(A,\mathbb G_m))$ (where $A$ denotes a semi-abelian scheme over $S$, $\mathbb G_m$ denotes the multiplicative group over $S$ and $\underline{\text{Hom}}$ ...
- 11
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182
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Cohomological criterion for being projectively normal
Let $X$ be a smooth projective variety over some algebraically closed field $K$ and let $\mathcal{L}$ be a line bundle that is generated by global sections. I want to know whether the ring $\sum_{n\in\...
- 3,031
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890
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Cohomology of the pullback of a coherent sheaf (considered as an abelian sheaf)
Let $X$ be an affine Noetherian scheme, $Y$ a separated Noetherian scheme, $f:X\rightarrow Y$ a morphism of schemes inducing a homeomorphism on the underlying topological spaces.
Let $F$ be a ...
user137767
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Identities for beta functions and twisted cohomology
This is a question about notation, I apologize if it is too basic. In the paper
Cho, Koji; Matsumoto, Keiji, Intersection theory for twisted cohomologies and twisted Riemann’s period relations. I, ...
- 8,992
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80
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Sections of nodal curves
We work over an algebraically closed field. Suppose $X\subset \mathbf{P}^n$ is an integral projective curve and $\pi:X\to Y$ is a linear projection that identifies two distinct points $p,q\in X$ to a ...
- 141
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312
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Cohomology of constant sheaves
Let $X= spec(k)$ where $k$ is an algebraically closed field. Consider the constant sheaf $\mathbb{Z}$ on the fppf site of $X$. I'm interested in computing $H^1_{fppf}(X, \mathbb{Z})$. I know that $H^...
- 11
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A sheaf for factorization
Let $R$ be a commutative ring with $1$ and let $X$ be the space of connected componens of $Spec (R) $ with Zariski topology ( The boolean spectrum of $R $ )and let for each $x\in X$ there exists a ...
- 11
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97
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Thom-type isomorphism on sheaf cohomology
Let $X$ be a smooth, projective surface and $T$ a finite set of points in $X$ i.e., of codimension $2$ in $X$. Is it true that $H^i(\mathcal{O}_X)=H^i(\mathcal{O}_{X\backslash T})$ for $i \ge 1$?
- 2,032
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91
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Continuous maps vs filtrations construction of the Leray spectral sequence
The Leray spectral sequence of a continuous map $f : X \to Y$ between two topological spaces can be constructed, as far as I understand, in two ways, one very general, and the other a bit more ...
- 1,227
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How to calculate : $ \mathrm{Hdg}^{ 2 \bullet } ( \mathcal{H}\mathrm{ilb} ( \mathbb{P}^n ),\mathbb{Q} ) $?
I try to calculate the rational cohomology algebra $ \mathrm{Hdg}^{ 2 \bullet } ( \mathcal{H}\mathrm{ilb} ( \mathbb{P}^n ),\mathbb{Q} ) = \displaystyle \bigoplus_{k=0}^{+ \infty} \mathrm{Hdg}^{ 2 k } (...
- 325
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122
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maps between two Leray spectral sequences based on maps on cochain complexes
Let $f_1:X_1 \to Y_1$ and $f_2:X_2 \to Y_2$ be two continuous maps between topological spaces. I am trying to understand under what condition there could be a map relating the Leray spectral sequences ...
- 1,227
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328
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Cohomology of a structure sheaf of a normal affine variety
I can't find the reference for the following fact:
Let $X$ be an affine variety and let $Y$ be its smooth resolution. $H^0(X,\mathcal{O}_x)=H^0(Y,\mathcal{O}_Y)$ if and only if $X$ is normal.
- 371
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205
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Sheaves and isomorphisms with chain complex of singular chains (Sheaf Theory, Bredon)
Let $\Delta_{\ast}(X,A)$ (resp. $\Delta_{\ast}^c(X,A)$) be the chain complex of locally finite (resp. finite) singular chains of $X$ modulo those chains in $A$.
How to show that the homomorphism of ...
- 11
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236
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Canonicity of Čech cohomology
For a topological space $X$, consider the Leray covering $U_\lambda$ (i.e. $\cap U_\lambda$ is sufficiently fine, e.g. affine for Zariski topology) of $X$.
For a sheaf $F$ on $X,$ the cohomology $H^...
- 781
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132
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Picard sequence for sujective morphisms
Given $\phi:X\rightarrow Y$ a surjective morphism of $k$-algebraic varieties ($k$ separably closed), I wanted to find how the write an exact sequence involving Pic(X) and Pic(Y). We can use the long ...
- 101
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508
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Equivariant Sheaves, Local system
Let $(G,m)$ be a group scheme unipotent, and $L$ a local system of rank 1 on $G$ such that:
$m^*(L) \simeq L \boxtimes L $.
Then why is $L$ an equivariant sheaf on $G$ with the action the ...
- 335
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445
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Pull-back of globally generated sheaves
Let $X$ be a smooth projective surface in $\mathbb{P}^3$, $D=\sum_i n_iD_i$ an effective Cartier divisor. Let $C$ be a smooth irreducible curve on $X$. Denote by $i:C \hookrightarrow X$ is the closed ...
- 1,981