Questions tagged [sheaf-cohomology]
The sheaf-cohomology tag has no usage guidance.
364 questions
1
vote
1
answer
2k
views
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 $\mathcal{L}$ ...
- 39.3k
1
vote
0
answers
191
views
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 ...
- 11
6
votes
1
answer
761
views
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
10
votes
1
answer
719
views
Leray-Hirsch theorem for Dolbeault cohomology
In Bott and Tu's Differential forms in algebraic topology there is a proof of Leray-Hirsch for the De Rham cohomology. The theorem is this:
Theorem (Leray-Hirsch): Let $E$ be a fiber bundle over $M$...
2
votes
1
answer
385
views
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)...
CommunityBot
- 1
6
votes
1
answer
334
views
Naive question on local cohomology
Let $X$ be a smooth, projective variety and $Z_1, Z_2$ two smooth, projective subvarieties in $X$ of the same dimension. Let $E$ be a locally free sheaf on $X$. Recall, there are natural morphims:
$$...
- 63.9k
3
votes
1
answer
460
views
Does local cohomology commute with pullback?
Let $Y$ be a topological spaces and $Z \subset Y$ be locally closed, i.e. $Z=V \cap U^c$ for $U,V \subset Y$ open.
For any abelian sheaf $\mathcal{F}$ on $Y$ let $\Gamma_Z(Y,\mathcal{F}):=\ker(\...
- 473
8
votes
0
answers
257
views
Global functions on a product of schemes over artinian ring
For a morphism of schemes $f:X\to S$ with $S=\text{Spec}(R)$ affine, let's write $A(X)=H^0(X,\mathcal{O}_X)$. I'm interested in the morphism of $R$-algebras
$$
c:A(X)\otimes_R A(Y)\to A(X\times_SY)
$$
...
- 4,492
3
votes
0
answers
188
views
Sheaf cohomology of the complement of a schubert variety
Let $k$ be a field, $d,n \in \mathbb{N}$ and denote by $Gr(d,n)$ the Grassmannian, which parameterizes the $d$-dimensional linear subspaces of $n$-dimensional $k$-vector space, considered as a ...
- 473
12
votes
3
answers
2k
views
How to compute the cohomology of a local system?
Suppose we have a reasonable topological space $X$ (i.e. a complex algebraic variety or a manifold) whose integral singular cohomology and fundamental group we understand well.
Suppose that we are ...
- 5,544
2
votes
1
answer
602
views
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 ...
- 40.3k
1
vote
1
answer
415
views
Multiplicative structure for sheaf cohomology of flag varieties
Let $F,F'$ be two locally free sheaves on a smooth complex algebraic variety. There is a cup-product $H^i(X, F) \otimes H^j(X,F') \to H^{i+j}(X,F \otimes F')$. In particular if $F$ is the sheaf of ...
- 39.3k
8
votes
1
answer
623
views
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 ...
- 35.2k
5
votes
0
answers
268
views
Coherent cohomological dimension and affine morphisms
For simplicity, all varieties in this question are quasiprojective varieties over an algebraically closed field of characteristic $0$.
The coherent cohomological dimension $cd(X)$ of a variety $X$ is ...
- 59
4
votes
1
answer
244
views
Boundness of torsion in $R^i\pi_*\mathcal F$ for a smooth $\pi:X\rightarrow S$
Let $\pi:X\rightarrow S$ be a smooth scheme over $S$ and let's assume for simplicity that $S=\mathrm{Spec} R$ is affine, Noetherian and regular. Let $\mathcal F$ be a coherent sheaf on $X$ and let's ...
- 66
1
vote
0
answers
152
views
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
4
votes
0
answers
240
views
Exact sequence in example in Grothendieck's Tohoku paper resulting from the Cech-to-derived-functor spectral sequence
Grothendieck gives in his Tohoku paper in example 3.8.3 an example for that $\check{\mathrm{H}}^{2}(X,\mathcal{F}) \neq \mathrm{H}^{2}(X,\mathcal{F})$.
In the beginning he states that there exisits ...
- 141
6
votes
1
answer
221
views
When is derived category of ringed space perfectly generated?
Let $(X,\mathcal{O})$ be a ringed space. Also assume that $X$ is nice, e.g. locally compact, Hausdorff, some type of finite dimension, ...
We can then consider $\mathcal{D}(\mathcal{O}\text{-}Mod)$. ...
- 501
1
vote
0
answers
164
views
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
1
vote
0
answers
114
views
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:
...
- 85
1
vote
0
answers
182
views
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
5
votes
2
answers
382
views
Obstructions to abelian sheaf being quasi-coherent
Let $X$ be a Noetherian spectral topological space. A necessary condition for an abelian sheaf on $X$ to be quasi-coherent with respect to some affine scheme structure on $X$ is that its higher ...
- 1,052
1
vote
0
answers
890
views
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 ...
CommunityBot
- 1
1
vote
0
answers
76
views
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
9
votes
1
answer
509
views
Deducing properness from $H^i(X, \mathcal{F})$ finitely generated over $\Gamma(O_X)$
Suppose that $X$ is a quasi-projective variety over a field $k$ and that we further know that for every coherent sheaf $\mathcal{F}$, $H^i(X,\mathcal{F})$ is finitely generated over $\Gamma(O_X)$. Is ...
- 19.4k
1
vote
0
answers
80
views
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
-2
votes
1
answer
203
views
Flat cohomology and finite direct sum
Let $X$ be a scheme (we can assume $X$ is smooth over a field $k$). Let $\mathcal F_1$ and $\mathcal F_2$ be two sheaves of abelian groups on $X$.
Is it always true that $H^i_{\text{flat}}(X, \...
- 345
6
votes
1
answer
1k
views
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 $...
- 16.5k
2
votes
1
answer
1k
views
Cohomology of tangent sheaf of a hypersurface
Let $X\subset\mathbb{P}^n$ be an irreducible and reduced hypersurface of degree $d$. How can one explicitly compute the dimension of the vector spaces $H^0(X,T_X),H^1(X,T_X),H^2(X,T_X)$? Here $T_X$ is ...
- 39.3k
17
votes
2
answers
2k
views
Is there a complex which computes Cech cohomology?
Suppose $X$ is a (paracompact, Hausdorff) topological space and we want to define its Cech cohomology with coefficients in $\mathbb Z$. Here is the way I have seen this constructed. For each open ...
CommunityBot
- 1
6
votes
0
answers
239
views
Sheaves on Rectifiable Sets
Basic question: are there (co)homological or sheaf-based tools which might be useful in geometric measure theory?
Background: The jumping off point here is a simple analogy - geometric measure ...
2
votes
0
answers
251
views
Is there a by-hand prove that $\Gamma(\mathbb{C}P^n,E)$ is finite dimensional for a holomorphic vector bundle $E$?
Please let me know whether this question is suitable for Mathoverflow.
Let $E$ be a finite holomorphic vector bundle (or more generally a coherent analytic sheaf) on a compact complex manifold $X$. ...
- 6,259
1
vote
0
answers
312
views
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
3
votes
1
answer
177
views
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}^...
- 39.3k
1
vote
0
answers
104
views
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
1
vote
1
answer
315
views
What is the hypercohomology of the push-forward of the intersection chain complex of an open cone to its closure?
Let $X = \left(L \times [0, 1]\right) / \left(L \times \{0\}\right)$ be the closed cone over a closed smooth $d$-dimensional manifold $L^{d}$. Let $i \colon Y \hookrightarrow X$ denote the inclusion ...
- 5,544
16
votes
1
answer
1k
views
Does every sheaf embed into a quasicoherent sheaf?
Question. Let $X$ be a scheme. Let $\mathcal{E}$ be a sheaf of $\mathcal{O}_X$-modules. Is there always a quasicoherent sheaf $\mathcal{E}'$ together with a monomorphism $\mathcal{E} \to \mathcal{E}'$?...
- 4,111
4
votes
1
answer
1k
views
Computation of cohomology of ideal sheaves
Let $j: X \to Y$ be a closed embedding. Let $I_{X/Y}$ be the ideal sheaf of this closed embedding. Then there is a exact sequence
$$ I_{X/Y} \to \mathcal{O}_Y \to j_{*}\mathcal{O}_X \to 0$$
One use ...
- 39.3k
0
votes
1
answer
661
views
Sheaf cohomology relative to a closed subspace
Let $i : A \hookrightarrow X$ be a closed subspace of a topological space $X$, and $j : Z := X \setminus A \hookrightarrow X$ denote its open complement. Given a sheaf $F$ of abelian groups on $X$, ...
CommunityBot
- 1
2
votes
0
answers
139
views
Defineing a Sheaf of rings over a topological space
Let $X$ be a topological space and let $R$ be a commutative ring with $1$ such that for each $x\in X$ there exists a multiplicatively closed subset $S_x$ of $R$ such that for each $a\in R$ if $\frac{a}...
- 31
3
votes
0
answers
155
views
Semicontinuity of cohomology of torsion-free sheaves restricted to divisors
Let $X$ be a smooth projective variety, $\mathcal{E}$ a torsion-free coherent sheaf on $X$ and $\mathfrak{d}$ a linear system of divisors in $X$.
I would like to show (at least when $X$ is a surface) ...
- 263
1
vote
1
answer
587
views
Reformulation of Grothendieck vanishing theorem
Let $X$ be a smooth, projective variety, ${F}$ a quasi-coherent $\mathcal{O}_X$-module on $X$ supported on a closed subscheme, say $Z \subset X$. Is it true that $H^i(X,F)=0$ for all $i>\dim Z$?
...
- 36
2
votes
1
answer
202
views
Sheaf cohomology of a complement of finitely many points
Let $X$ be a smooth, projective surface in $\mathbb{P}^3$ and $p \subset X$ a closed point in $X$. How do I compute $H^1(\mathcal{O}_{X\backslash p})$?
Any reference/idea will be most welcome.
- 13.6k
0
votes
1
answer
330
views
Dual of a stable locally free subsheaf is a locally free quotient sheaf
Let $X$ be a compact connected Kähler manifold, of dimension $d\geq3$, with Hermitian metric $\omega$; let $E$ be a vector bundle on $X$ of rank $r\geq2$.
By [1] definition 1.2:
A line bundle $L$ ...
- 828
1
vote
0
answers
97
views
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
5
votes
0
answers
721
views
What is the filtration in Leray's spectral sequence?
Leray's theory of spectral sequences considers a continuous map $f : X \to Y$ between topological spaces. The statement is that there exists a filtration
$$H^n(X,k) = F^0H^n(X,k) \supseteq ... \...
- 1,227
1
vote
0
answers
91
views
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
1
vote
1
answer
283
views
Commutativity between functors on sheaves of abelian groups
I am trying to understand certain properties of sheaf theory, but I'm having trouble finding the notions to answer my questions. I'd be really glad if someone could help me with the following. Let $f :...
- 9,491
1
vote
0
answers
206
views
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
3
votes
0
answers
256
views
Convolution of sheaves on R
I am trying to understand a basic computation of convolution. Throughout, $R$ is the real line as a topological group and $k$ is some base field. I would like to understand the computation of the ...
