All Questions
Tagged with sheaf-theory ag.algebraic-geometry
493 questions
7
votes
2
answers
627
views
Vanishing of higher direct image of finite morphisms relative to the fppf topology
Let $f:X \to Y$ be a finite morphism of schemes.
Let $\mathcal{F}$ be a sheaf of abelian groups on the the etale site of $X$ then we know that $R^{i}f_{*} \mathcal{F} = 0$. Is this statement also true ...
- 345
7
votes
3
answers
2k
views
Are the global sections of a vector bundle a projective module?
Given a scheme $X$ with structure sheaf $\mathcal{O}_X$, we can associate to each $\mathcal{O}_X$-module $\mathcal{F}$ its global sections $\Gamma(\mathcal{F})$, which gets the structure of a $\Gamma(\...
- 12.1k
7
votes
2
answers
796
views
Restriction of Ext sheaves
Let $f \colon X \to Y$ be a map of schemes, $\mathcal{F}, \mathcal{G}$ two coherent sheaves on $Y$. I'm interested in conditions which guarantee an isomorphism
$$f^{*} \mathcal{E}xt^i(\mathcal{F}, \...
- 14.7k
7
votes
3
answers
2k
views
Cohomology with compact support for coherent sheaves on a scheme
Is there a notion (for schemes or just locally ringed spaces) of cohomology with compact support? I guess there is for algebraic schemes over $\mathbf{C}$, but what about schemes in general? Does ...
- 163
7
votes
1
answer
291
views
Direct and inverse image terminology
Let $f\colon X\to Y$ be a continuous map. Then $f$ induces a geometric morphism $f^\ast\dashv f_\ast\colon \mathrm{Sh}(X)\leftrightarrows\mathrm{Sh}(Y)$, whose left adjoint is called inverse image and ...
- 723
7
votes
1
answer
421
views
An example in Mumford's “Picard Groups of Moduli Problems”
I tried asking this at math.stackexchange but I didn't get any responses, so hopefully it's ok to try here.
I'm reading Mumford's paper "Picard Groups of Moduli Problems" and am confused about an ...
- 164
7
votes
1
answer
490
views
Equivariant perverse sheaves and orbit stratification
Let $X$ be a complex algebraic variety with an action of a connected algebraic group $G$.
The forgetful functor from the category of $G$-equivariant perverse sheaves on $X$ to the category of perverse ...
- 3,446
7
votes
1
answer
451
views
Coverage, itself considered as a presheaf
A coverage $J$ on a category $C$ assigns to an object $U$ of $C$ a set of covering families $J(U)$. The covering families are required to be stable under pullback, which amounts to requiring that for ...
- 291
7
votes
1
answer
255
views
Subobject classifier for sheaves on large sites with WISC
Let $\mathsf{C}$ be a possibly large category with a Grothendieck topology satisfying the Weakly Initial Set of Covers condition: there is for each $X$ a set (not a proper class) of covering families ...
- 1,996
7
votes
2
answers
1k
views
Non-zero sheaf cohomology
Let $\mathbb{R}$ denote the real line with its usual topology. Does there exist a sheaf $F$ of abelian groups on $\mathbb{R}$ whose second cohomology group $H^{2}\left(\mathbb{R},F\right)$ is non-zero?...
- 47.5k
7
votes
2
answers
839
views
What is the need for torsion in the definition of lisse sheaves?
I am studying the basics of constructible and lisse sheaves, and am trying to understand SGA 4, IX. As Grothendieck himself observes at the beginning of the chapter, one is forced to work with torsion ...
- 5,670
7
votes
1
answer
537
views
Is the quotient presheaf $\mathbb{G}_m/\mu_p$ an étale sheaf?
I apologize if the question is bit trivial for mathoverflow, but I asked on stack exchange a while ago and haven't got any answer.
Let $k$ be a field of characteristic $p > 0$. Consider the ...
- 536
7
votes
1
answer
787
views
Definition of the cotangent complexes of Artin stacks
I am studying the notion of the cotangent complexes of Artin stacks reading LMB's book and Olsson's paper. According to them, the cotangent complexes are defined as projective systems in their derived ...
- 479
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
7
votes
0
answers
205
views
sheaves on thickened nodal cubics
Suppose F is an algebraically closed field (of any characteristic) and that h in F[x,y,z]
is an irreducible cubic form defining a plane curve C with a node. A lot is known about
sheaves on C; for ...
- 5,422
6
votes
5
answers
4k
views
Why is the rank of a locally free sheaf well-defined?
In Hartshorne p. 109 he defines a sheaf $\mathcal{F}$ of $O_X$-modules to be locally free if there is an open cover of $X$, s.t. on each $U$, $\mathcal{F}|_U$ is a free $O_X|_U$ module of rank $I$. ...
- 1,160
6
votes
2
answers
957
views
An apparent equivalence of the category of affine schemes over $S$ and the category of quasi-coherent $\mathcal{O}_S$-algebras
I had asked something very similar before on math.se (deleted now) but unfortunately it hadn't received a lot of attention. I decided to re-ask here.
Let $S$ be a fixed scheme. Is the following true?
...
- 7,789
6
votes
1
answer
494
views
Prove category of constructible sheaves is abelian
Let $X$ be a nice enough topological space, perhaps a complex algebraic variety with its analytic topology. I'm hoping someone could help me prove that the category $\text{Constr}(X)$ of ...
- 1,701
6
votes
1
answer
1k
views
Tensor product of sheaves separated?
Let $(X, O_X)$ be an arbitrary scheme and $\mathcal{F}$ a presheaf of $O_X$-modules. The presheaf $\mathcal{F}$ is said to be separated if the natural map to it's sheafification $\mathcal{F}^+$ is an ...
- 3,555
6
votes
1
answer
911
views
Putting sheaves to work for algebraic topology?
This is cross-posted from math.se after receiving points and no answers. I apologise if this question is too basic for MathOverflow.
I'm refreshing my memory of ...
- 1,221
6
votes
2
answers
385
views
cohomology and $j_!$
I have a projective variety $X$ and an open immersion $j : U \to X$.
Say I have a sheaf, locally free in my case of interest, $\mathcal{S}$ on $U$. Is there any reasonable relationship between $H^i(X,...
- 758
6
votes
2
answers
631
views
Geometric meaning of coherent sheaves $\mathcal{F} \otimes \mathcal{O}_{\mathbb{P}^n}(d)$ over $\mathbb{P}^n$
Maybe it sounds like a silly question but I'm not able to figure out in my head the geometric meaning of "twisting" vector bundles (or more generaly coherent sheaf) over $\mathbb{P}^n$.
I ...
- 1,343
6
votes
2
answers
1k
views
Explicit Direct Summands in the Decomposition Theorem
Let f:X→Y be a semismall resolution of singularities. Then the pushforward of the constant sheaf on X is a semisimple perverse sheaf on Y. Under these conditions, I know how to calculate the ...
- 8,866
6
votes
1
answer
798
views
Example of non-holonomic D-module and explicit computation of characteristic variety
I'm currently trying to have a better understanding of the concepts of characteristic variety and holonomic $D$-modules (let us assume that they are coherent) on a holomorphic manifold $X$. I know ...
- 1,017
6
votes
1
answer
2k
views
Is the stalk of the (co)limit of sheaves equal to the (co)limit of the stalks?
More precisely, if $\mathcal F_i$ is a system of sheaves, is it the case that
$$
(\lim \mathcal F_i)_p = \lim ((\mathcal F_i)_p)
$$
and similarly for colimits? I can see how to get a map
$$
(\lim ...
- 1,509
6
votes
1
answer
455
views
Subsheaves of Spec K, K a field
$\DeclareMathOperator\Spec{Spec}\newcommand\Ring{\mathrm{Ring}}\newcommand\op{^\text{op}}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Sh{Sh}$In the category of schemes the objects of the form $\...
- 775
6
votes
2
answers
790
views
Do quotients of representable sheaves represent quotients?
Here's the context for the question: Proposition 4.6 of Freitag and Kiehl's book on etale cohomology shows that a sheaf (of sets) $\mathcal{F}$ (on the site Et(X)) is constructible if and only if it ...
- 1,988
6
votes
1
answer
441
views
Where can I find a definition of $\underline{H}^p(X, \mathscr{F})$?
Let $X$ be a topological space and $\mathscr{F}$ a sheaf on $X$. In the paper Tropical cycle classes for non-archimedean spaces and weight decomposition of de Rham cohomology sheaves by Yifeng Liu, ...
- 1,153
6
votes
1
answer
281
views
Cokernel of map of étale sheaves
Let $p:\mathbb{G}_m\to \operatorname{Spec} k$ be the structure map, and let $T$ be an algebraic $k$-torus viewed as an étale sheaf over $k$. Why is the cokernel of the canonical map $T\to p_*p^*T$ ...
- 115
6
votes
1
answer
2k
views
Information and intuition packed in the Chern character for coherent sheaves
even after quite some time learning it, I still get somehow puzzled by the Chern character. Let me recall some stuff to get notation and setting.
Let us consider a smooth projective algebraic variety ...
- 155
6
votes
1
answer
752
views
Sections of the conormal bundle
Let $X\subset\mathbb{P}^N$ be a quadratic manifold. That is $I(X)$ is generated by quadratic polynomials $Q_1,...,Q_m$.
Let $\mathcal{I}_X$ be the ideal sheaf of $X$ and $\mathcal{I}_X/\mathcal{I}_X^...
user82886
6
votes
1
answer
1k
views
When are non-quasi-coherent sheaves used?
Non-quasi-coherent sheaves of $\mathcal O_X$ modules on a scheme seem like a wild concept to me; are they actually used for something?
- 459
6
votes
1
answer
727
views
What do nearby/vanishing cycles have to do with Fourier transforms?
Let $E$ be a vector bundle on some smooth algebraic variety and $E^*$ its dual. Suppose $A$ is a sheaf (constructible or a $D$-module) on $E$. Given a linear function $f$ on $E$, we may compute the ...
- 3,019
6
votes
1
answer
728
views
Sheaf cohomology with support vanishes
I am trying to solve the exercise 2.4 chapter III in Hartshorne's "Algebraic Geometry". For this I would like to prove for a sheaf $F$ of Abelian groups on a topological space $X$ and $U$ open subset ...
- 233
6
votes
1
answer
1k
views
Base change for Borel-Moore homology
For a seperated scheme of finite type $X$ over $\mathbf{C}$, let $H_*(X)$ denote its Borel-Moore homology, which is defined by
$$
H_k(X) = R^{-k}\Gamma(X, \omega_X)
$$
where $\omega\in D_c(X, \mathbf{...
- 1,121
6
votes
1
answer
800
views
What kind of colimits are preserved by a certain Yoneda embedding?
(This question is related to this one)
Let $k$ be a field and consider the category $Sch/k$ of schemes over $k$, say also separable and of finite type. The Yoneda embedding
$$
Y:Sch/k \to Pre(Sch/k)
$...
- 2,782
6
votes
1
answer
417
views
Etale sheaves on algebraic spaces vs. Etale sheaves on affines
Let's fix a field $k$. First, consider $Aff_k$ to be the category of affine finite type $k$ schemes. On this category, one can define the etale topology and thus consider the site $Aff_k^{et}$, then ...
- 595
6
votes
1
answer
239
views
What does an ideal correspond to in the internal language of sheaves?
Suppose I have a sheaf $\mathcal F$ in some topos $\mathrm{Sh}(\mathcal C)$. Then this becomes the sheaf of rings from algebraic geometry when described as a ring in the internal language of the topos....
- 61
6
votes
1
answer
760
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
6
votes
1
answer
2k
views
How to understand the Fourier-Sato transform and microlocalization functors?
Given a smooth real vector bundle $\pi: E \to M$ I can look at the (bounded from below) derived category of sheaves on $E$. Since $E$ admits a very natural action of $\mathbb{R}^{\geq 0}$ by scaling, ...
- 272
6
votes
0
answers
187
views
Algebraic model for the abelian category of descent data for modules in the non-affine case
Let $f: X \to Y$ be a morphism of schemes. I'd like to have a completely algebraic description of the belian category of descent data for modules along $f$. Here's my attempt:
The category of quasi-...
- 7,789
5
votes
2
answers
985
views
Chern classes in flat families
Given a smooth projective variety $X$ over an algebraically closed field $k$. Now given a another projective variety $T$ and a coherent $O_{X\times T}$-module $F$, which is flat over $T$.
Given $r,s \...
- 1,391
5
votes
1
answer
1k
views
Euler characteristic and inclusion-exclusion
Define the Euler characteristic of a scheme to be the Euler characteristic of its structure sheaf. I remember being told that for curves, this invariant satisfies inclusion-exclusion. That is, if $C_1,...
- 303
5
votes
3
answers
680
views
Deequivariantisation of indecomposable sheaves
Let G be a connected group acting on a space X. All spaces should be reasonable, so e.g. G is a complex affine algebraic group acting on an algebraic variety X, with everything done using the usual ...
- 8,866
5
votes
2
answers
3k
views
morphisms of affine schemes question
So, in chapter 2, section 2 of Hartshorne, (prop 2.3), he describes how if $\varphi : A\rightarrow B$ is a homomorphism of rings, then you get a morphism of (affine schemes):
$\newcommand{\Spec}{\...
- 10.7k
5
votes
1
answer
442
views
Is the determinant line bundle of a coherent sheaf functorial (between sheaves of the same rank)?
The determinant line bundle of a coherent sheaf $\mathcal{F}$ on an $n$-dimensional (smooth) analytic space is defined as
\begin{equation}
\det \mathcal{F} := \bigotimes_i^n (\det \mathcal{E}_i)^{⊗...
- 1,141
5
votes
2
answers
358
views
Canonical conics pulling back to polynomials on rational normal curve
(In following all schemes are formed over $\Bbb C$)
Let $C:=\nu_d(\Bbb P^1)$ the rational normal curve obtained via $d$-folded Veronese map $\nu_d: \Bbb P^1 \to \Bbb P^d$. The quadrics on $\Bbb P^d$ ...
- 6,028
5
votes
1
answer
367
views
Check that a Sheaf is Invertible Etale Locally
A question about following statement from Martin Olsson's book on Stacks. In the proof of Proposition 13.2.9. (p 269) is claimed that certain sheaf $K$ on a nodal curve $C$ is invertible it suffice to ...
- 6,028
5
votes
1
answer
1k
views
"Role" of cohomology of coherent sheaves in SGA 4.5, étale cohomology
As the question title suggests, what is the role cohomology of coherent sheaves plays for SGA 4.5, étale cohomology? Why are they so important for the construction and establishing properties of étale ...
user97565
5
votes
1
answer
723
views
Sheaf condition and representability in the category Top
This is a rather nice question I got from this user via private communication.
Let $\mathcal{C} = Top$ the category of topological spaces. Let $\mathcal{C}^\prime$ be the category $Funct(\mathcal{C}^{...
- 7,442