All Questions
Tagged with sheaf-cohomology ag.algebraic-geometry
94 questions with no upvoted or accepted answers
10
votes
0
answers
963
views
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
8
votes
0
answers
647
views
Trying to understand "Shtukas"
I'm studying Goss' Basic structures of function field Arithmetic, chapter 6 about Shtukas. I'm trying to understand some details about some concepts. This chapter is based on a Mumford's paper An ...
- 69
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
7
votes
0
answers
574
views
What is the geometric intuition for the sheaf-theoretic terms "soft", "fine", and "flabby"?
The sheaf-theoretic terms "soft", "flabby", and "fine" are of an obviously geometric character, and suggest opposition with "hard", "rigid", and "coarse" sheaves (I'm just inventing these terms here).
...
- 6,025
6
votes
0
answers
1k
views
Intuition behind exceptional inverse image?
The story is probably well-known: given a map $f:X\to Y$ of spaces (say schemes, but there are many other contexts), we have two classical operations between sheaves on $X$ and those on $Y$: the ...
- 28.2k
6
votes
0
answers
637
views
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 ...
- 91
5
votes
0
answers
306
views
Cohomology of coherent sheaves on Deligne Mumford stacks
Suppose that $\cal X$ is tame Deligne Mumford stack with generic trivial inertia. Let $X$ be its muduli space and $f:{\cal X}\to X$ the projection.
Let $\cal F$ be a coherent sheaf on $\cal X$.
Is it ...
- 388
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
5
votes
0
answers
614
views
Does the higher cohomology of a quasi-coherent sheaf on a Stein manifold vanish?
It is a well-known result in algebraic geometry that if $X$ is an affine scheme and $\mathcal{F}$ is a quasi-coherent sheaf on $X$, then the higher cohomologies of $\mathcal{F}$ vanish, i.e.
$$
H^i(X,\...
- 9,019
5
votes
0
answers
374
views
Sheaf Cohomology on Zariski-Riemann Spaces
Can sheaf cohomology on the Zariski-Riemann spaces give some sort of classification for field extensions (even just for function fields)? If not, are there any significant or useful results (e.g. for ...
- 468
4
votes
0
answers
583
views
Etale cohomology of a nodal (cuspidal) curve
Let $k$ be a separably closed field, and $X/k$ be a curve (not necessarily complete) with a single singularity, a simple node $x$. Suppose $\ell$ is a prime number invertible in $k$, how do we compute ...
- 547
4
votes
0
answers
211
views
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
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
4
votes
0
answers
133
views
Theta divisor on compactified jacobian of nodal curve
Let $X$ be a Nodal curve. Let $\bar{J}(X)$ be compactified Jacobian (rank one torsion free sheaf of degree one) and $\Theta$ denote the theta divisor in $J$.
How to compute $H^0(\bar{J}(X);\Theta^k)$, ...
- 137
4
votes
0
answers
343
views
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
4
votes
0
answers
575
views
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
4
votes
0
answers
239
views
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
4
votes
0
answers
137
views
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
4
votes
0
answers
447
views
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
4
votes
0
answers
362
views
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
3
votes
0
answers
250
views
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
3
votes
0
answers
175
views
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
3
votes
0
answers
126
views
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
3
votes
0
answers
641
views
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
3
votes
0
answers
446
views
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
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
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
3
votes
0
answers
152
views
exact sequence of fundamental groups associated to "almost" smooth families of curves
Suppose I have a proper, flat family of curves $X \to S$ that has a section. Fix a basepoint $s \in S$ and let $X_s$ denote the corresponding fiber. Let $\mathbb{L}$ be a set of primes which does not ...
- 1,298
3
votes
0
answers
154
views
$\operatorname{Ext}^2(O,\omega)$ as a higher extension on $\mathbb{P}^1 \times \mathbb{P}^1$
Let $X = \mathbb{P}^1 \times \mathbb{P}^1$ over a field $k$ and consider $Ext^2(\mathcal{O}_X,\omega_X)\cong H^2(\omega_X) = H^2(\mathcal{O}_X(-2,-2)) = k$
Let $C = \mathbb{P}^1$.
By Kunneth $H^2(\...
- 3,968
3
votes
0
answers
76
views
$H^{1}(C, N_{C/X}(-m)) = 0$, for $C$ a irreducible curve on $X$ through $m$ general points
I was reading A. de Jong and J. Starr's paper "Low degree complete intersections are rationally simply connected", which can be found at http://www.math.stonybrook.edu/~jstarr/papers/nk1006g.pdf, and ...
- 283
3
votes
0
answers
130
views
Action of automorphisms on cohomology with supports
Let $x$ be the closed point of an $n$-dimensional local scheme $X$, essentially smooth over a field $k$. Let $M$ be a sheaf on the category of smooth $k$-varieties (in either Zariski or Nisnevich ...
- 525
3
votes
0
answers
277
views
Deformation of vector bundle on projective space with same Hilbert polynomial as multiple of structure sheaf
Let $E$ be a vector bundle on projective space ${\bf P}^n$ whose Hilbert polynomial is the same as $\mathcal{O}^{{\rm rank}(E)}$.
Does there exist a vector bundle over ${\bf P}^n \times {\rm Spec}(R)$...
- 10.7k
2
votes
0
answers
62
views
Base change for finding fibers of the pushforward of a line bundle along a proper non-flat morphism
Let $f: X \to Y$ be a proper morphism whose fibers have different dimensions, in particular $f$ is not flat. Let $L$ be a line bundle on $X$. What conditions would be sufficient to be able to conclude ...
- 2,974
2
votes
0
answers
142
views
Computing the coherent cohomology of a quasiprojective variety
I have a quasiprojective variety given by some explicit quations. How do I compute its coherent cohomology (with coefficients in the structure sheaf)? Do I use the Cech complex for an open affine ...
- 2,974
2
votes
0
answers
181
views
Proposition 6.2.3 from Goss "Basic Structures of Function Field Arithmetic"
I'm studying Proposition 6.2.3 from Goss "Basic Structures of Function Field Arithmetic", page 184:
Let $F$ be the sheaf of Data A (a torsion-free coherent $O_{\bar{X}}$-module on $\bar{X}$ ...
- 69
2
votes
0
answers
143
views
Cohomology of equivariant toric vector bundles using Klyachko's filtration
I am trying to understand Klyachko's following description of the cohomology groups of locally free (hopefully more generally of reflexive) sheaves on toric varieties.
Whereas detailed literature ...
- 455
2
votes
0
answers
241
views
Action of algebraic group in cohomology of equivariant algebraic vector bundle
Let $X$ be a projective algebraic variety over an algebraically closed field. Let an algebraic group $G$ act algebraically on $X$. Let $\mathcal{F}$ be a $G$-equivariant vector bundle (or, more ...
- 21.8k
2
votes
0
answers
78
views
How to estimate the locus of non-zero cohomology for a equivariant toric reflexive sheaf, with a Klyachko description
I am trying to analyze Macaulay2 package "ToricVectorBundles". The package deals with equivariant reflexive sheaves on complete toric varieties. Such a sheaf is described by a set of ...
2
votes
0
answers
109
views
Homotopy invariant Bloch-Ogus cohomologies with a vanishing property
I am looking for examples (in any characteristic) of homotopy invariant Bloch-Ogus cohomology theories given by Zariski sheaves $\Gamma(n)$, such that $\Gamma(0) = \mathbb{Z}$ is the constant sheaf. ...
- 5,901
2
votes
0
answers
90
views
$\bigoplus_{k=0}^{\infty}H^n(X,I^k\mathcal{F})$ 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
2
votes
0
answers
124
views
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
votes
0
answers
201
views
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
2
votes
0
answers
405
views
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)...
- 5,986
2
votes
0
answers
136
views
A infinity structure on Yoneda Ext group
I am currently trying to control an $A_\infty$-algebra of the form $\mathrm{Ext}_X(F\oplus F'[2n-2],F\oplus F'[2n-2])$ where $X$ is a nice enough scheme and $F,F'$ are sheaves that are NOT locally ...
- 213
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 $X$. ...
- 9,019
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
2
votes
0
answers
126
views
Local cohomology with supports in a constructible set
Let $X$ be a topological space (I'm interested in the case of $X$ being a complex algebraic variety with the Zariski topology) and $Z$ a constructible subset (i.e. a finite union of locally closed ...
- 3,079
2
votes
0
answers
263
views
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
2
votes
0
answers
265
views
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
2
votes
0
answers
347
views
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