Questions tagged [sheaf-theory]

For questions about sheaves on a topological space.

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Do infinitesimal neighbourhoods help to compute the inverse images of coherent sheaves?

Let $i:Z\to X$ be a closed embedding of (projective) varieties; $S$ is a coherent sheaf on $X$. How could one compute $H^*(Z,i^\ast S)$ (I don't know whether I should write $H^\ast (Z,i^{-1}S)$ ...
Mikhail Bondarko's user avatar
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On inverse images with respect to Zariski-etale topology.

For a variety $X$ I define its Zariski-etale site as follows: the category is the category of etale $X$-schemes, and the coverings are Zariski ones. Note that this topology is more coarse than the ...
Mikhail Bondarko's user avatar
1 vote
1 answer
407 views

Restriction of a sheaf to an infinitely small neighbourhood of a closed submanifold: how to work with this ind-sheaf?

Let $X$ be a manifold, $i: Z\to X$ is a closed embedding. For a sheaf $S$ (of abelian groups) on a manifold $X$ and each $\varepsilon>0$ we denote by $Z_\varepsilon$ the set of points of $X$ ...
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What are the easiest cases of base change (for sheaves on sites)?

I have a closed embedding of schemes $i:X'\to X$, and for each of them I consider three Grothendieck topologies for the category of the corresponding (relatively) \'etale schemes: the \'etale one, the ...
Mikhail Bondarko's user avatar
2 votes
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396 views

Do inverse images respect flabby sheaves?

Let $i:Y\to X$ be a closed embedding of varieties, and let $S$ be a flabby \'etale (or Nisnevich) sheaf of abelian groups on $X$. Is $i^*S$ flabby also? I am mostly interested in the case when $S=i_{x*...
Mikhail Bondarko's user avatar
5 votes
1 answer
2k views

Generalized Beilinson spectral sequences

Assume we are workling on $\mathbb{P}^n$ for some $n\geq 1$ and we have a coherent sheaf $F$ on it. Then there are two (well known?) spectral sequences $E_r^{p,q}$ with $E_1$-term: $E_1^{p,q}=H^q(\...
TonyS's user avatar
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The Mayer-Viertoris exact sequence as a (Zariski) descent spectral sequence.

For certain 'spaces' $U,V$ (they are certain Henselizations of subvarieties) I would like to compute (certain etale) cohomology of $U\cup V$ in terms of the corresponding cohomology of the diagram $U\...
Mikhail Bondarko's user avatar
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1 answer
283 views

sequence of sheaves for studying intersection

I'm studying intersection of curves with a fixed plane cubic, the first case I consider is of course lines, in particular lines intersecting the cubic at only one point. The problem is quite easy and ...
Srks's user avatar
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475 views

Fine and acyclic sheaves on locales

Hey all. Here's the thing, so I have a measure space. According to Johnstone's 'Topos theory' (page 213), let $(X,\Sigma,\mu)$ be a measure space, we can define a Grothendieck pretopology on it (and ...
Mario Carrasco's user avatar
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Can etale $X$-schemes be lifted to $Y$, where $X$ is closed in $Y$?

For a closed embedding (of varieties) $X\to Y$ let $U/X$ be etale. Is is true that there necessarily exists an etale $U'/Y$ such that $U'_X=U$? If this is wrong in general, are there any assumptions ...
Mikhail Bondarko's user avatar
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466 views

Sheaf of sections and local triviality

This is probably not a research level question, I'm sorry if it is inappropriate. I'm reasking here this question on math.se. Suppose that $\xi: E \to B$ is a bundle (by which I mean simply a ...
Alexei Averchenko's user avatar
8 votes
1 answer
1k views

Does one need to sheafify when defining the inverse image of a sheaf with respect to an embedding?

This seems to be a rather simple (stupid?:)) question; yet I was not able to find an answer quickly. For a morphism $f:X\to Y$ of schemes (or topological spaces) and an (etale or topological) sheaf $...
Mikhail Bondarko's user avatar
2 votes
1 answer
409 views

Nisnevich points

Here is a probably stupid question : If $F$ is a sheaf on the big Nisnevich site, then is the morphism $F(X) \to \amalg F(x)$ injective where the sum is over ALL the points of $X$ (not just the closed ...
name's user avatar
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4 votes
1 answer
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Vector space structure on the tangent bundle of a scheme and relation to the tangent sheaf

First a word of warning: I am not a trained algebraic geometer, so it is possible (likely) that these questions are inappropriate for MO, if so: my apologies. Said this: As far as I understand the ...
Spinorbundle's user avatar
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3 votes
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Sheafification of Arens-Michael algebra-valued presheaves

Let $\mathcal A$ be the category of Arens-Michael algebras, that is, projective limits of Banach algebras. Since $\mathcal A$ is a concrete category, an $\mathcal A$-valued presheaf $A$ admits a set-...
Rodrigo Vargas's user avatar
2 votes
2 answers
248 views

Is restricting the support of an Artinian sheaf a closed condition?

Given a projective surface $S$, and a smooth projective curve $C\subset S$ over $\mathbb{C}$. Furthermore we have a locally free sheaf $E$ of rank $r$ on $S$. Then for any $l\geq 1$, the projective ...
TonyS's user avatar
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5 votes
2 answers
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Are presheaves of constant functions sheaves?

I was reading 'An introduction to homological algebra' by Rotman, and on page 279 in the section about sheaves, example 5.64, Rotman gives an example of a constant presheaf $\mathcal{P}$ that's not ...
Mario Carrasco's user avatar
3 votes
0 answers
851 views

The "pullback presheaf" and the proper base change theorem in topology

Let $f:X\rightarrow Y$ be a continuous map of topological spaces and let $\mathcal{F}$ be a sheaf (say of abelian groups to fix the idea) on $Y$. Define the following rule on open sets of $X$: $$ V\...
Hugo Chapdelaine's user avatar
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Does n-multiplication maps of cohomology groups vanish if it vanishes at the 0th cohomology?

In general, we know that a morphism $f=(f ^ {q})$ between universal (cohomological) $\delta$ functors $S=(S ^ {q}),T=(T ^ {q})\ $vanishes if and only if $f ^ {0} \ \colon \ S^{0} \to T^{0}$ vanishes....
Hiro's user avatar
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4 votes
1 answer
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Question about the definition of a sheaf cohomology group for a sheaf using tensor products of sheaves

In Warner's 'Foundations of differentiable manifolds and Lie groups', in the section about axiomatic sheaf theory (page 178), when establishing the conditions necessary for the existence of a ...
Mario Carrasco's user avatar
2 votes
1 answer
238 views

Behaviour of Morita equivalence in families of sheaves

Given a projective scheme $X$, say over $\mathbb{C}$, another $\mathbb{C}$-scheme $S$ and a coherent and torsion free (as an $O_X$-module) $M_n(O_X)$-module $F$. Now we can use Morita equivalence to ...
TonyS's user avatar
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Does anyone understand the notation in this equation for the sheafification of a presheaf on a site?

Hi there, I'm trying to sheafify a constant presheaf on a site, I went to http://ncatlab.org/nlab/show/sheafification, but can't understand the notation in the equation for W (in the proof for ...
Mario Carrasco's user avatar
5 votes
1 answer
294 views

In what generality is the Verdier biduality map an isomorphism?

Let $X$ be a finite-dimensional, locally compact topological space, and consider the dualizing complex $K_X \in \mathbf{D}^b(X,k)$ (bounded derived category of $k$-sheaves, where $k$ is a noetherian ...
Akhil Mathew's user avatar
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3 votes
1 answer
465 views

Sequences of groups, exact not just in étale but also in the Zariski topology

Let $A=k[[x,y]]$, with say $k=\mathbb{C}$ and $X = \operatorname{Spec}(A)$. Let $B$ denote a free $A$-algebra of rank $e^2$; actually, we have $B=A[u,v]$ with $u^e=x$, $v^e=y$ and $uv = \xi_e vu$, ...
TonyS's user avatar
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8 votes
1 answer
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Sheaves of $\mathbb Z$-modules = sheaves of abelian groups

In his "Algebraic Geometry", Hartshorne proves that for any ringed spaces $(X,\mathcal O_X)$, category $Mod(X)$ of sheaves of $\mathcal O_X$-modules has enough injectives. If we take $\...
Rafael Mrden's user avatar
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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}{\...
Will Chen's user avatar
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3 votes
1 answer
444 views

For an l-adic sheaf (F_n), why is the complex F_n of finite Tor dimension?

Let $X$ be a variety and let $\mathcal{F}=(\mathcal{F}_n)_{n\geq 0}$ be a (constructible) $\ell$-adic sheaf on $X$. Let $K_n$ be the object in the derived category $D(X,\mathbf{Z}/\ell^{n+1})$ of ...
Ariyan Javanpeykar's user avatar
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2 answers
674 views

Subsheaf of quotient of quasi coherent sheaves

We know that any submodule of a quotient module $\frac{M}{N}$ is of the form $\frac{K}{N}$, where $K$ is a submodule of $M$ containing $N$. Now here is a question: Let $\cal F$ and ${\cal G}$ be quasi ...
Gholam's user avatar
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6 votes
1 answer
924 views

Sheaves with isomorphic cohomology, but not quasi-isomorphic

Suppose I have two (constructible) sheaves of vector spaces $F$ and $G$ over the same base space that have isomorphic cohomology (degree by degree), but no sheaf map inducing this isomorphism (i.e. ...
Justin Curry's user avatar
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1 answer
381 views

The behavior of pure sheaves under functor Hom( F, -)

We know that a submodule A of B is pure if and only if the functor $Hom(M, -)$ is exact on the sequence $ 0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ for every finitely presented module ...
Gholam's user avatar
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20 votes
4 answers
4k views

The tangent bundle to an infinite-dimensional manifold

Suppose that $A,B$ are smooth ($\mathrm C^\infty$) manifolds, and denote by $\hom(A,B)$ the set of $\mathrm C^\infty$-maps $A \to B$. It is a perfectly well-defined set, but often one wants more. ...
Theo Johnson-Freyd's user avatar
3 votes
0 answers
398 views

Two definitions of Čech cohomology

Hello, I have found different definitions of Čech complex for sheaf $F$ od abelian groups on topological space $X$ with respect to the cover $\mathcal U$. One in Gelfand-Manin says to take product of ...
Rafael Mrden's user avatar
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16 votes
0 answers
4k views

Sheaf cohomology and inverse limits

In proving the formal function theorem, Grothendieck uses a rather technical lemma in EGA 0-III.13: Lemma: Let $\mathcal{F}_n$ be an inverse system of sheaves on a space $X$ with surjective ...
Akhil Mathew's user avatar
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3 votes
0 answers
353 views

Exercise concerning locally constant presheaves [closed]

Let $\mathscr{F}$ be a presheaf of abelian groups on some topological space $X$. We say that $\mathscr{F}$ is locally constant if there exists an open cover $\mathcal{U}$ of $X$ (i.e. $X=\bigcup_{U\in\...
Jesko Hüttenhain's user avatar
8 votes
0 answers
368 views

Dualizing complex of the product of two locally compact spaces

Hello! In the setting of locally compact Hausdorff spaces, I would like to understand the relation between the exterior product ${\mathbb D}_X\boxtimes{\mathbb D}_Y$ of the dualizing complexes of two ...
Hanno's user avatar
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31 votes
4 answers
13k views

When will the pushforward of a structure sheaf still be a structure sheaf?

Let $f:X\rightarrow Y$ be a morphism of schemes. When $PicY\rightarrow PicX$ is an embedding and $f_{*}\mathscr{O}_{X}$ is invertible, it is the structure sheaf of $Y$. In the proof of Zariski's Main ...
YOURS's user avatar
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21 votes
2 answers
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Elementary short exact sequence of sheaves

This question arised when I was trying to use this answer to understand Reid's "Young Person's guide to Canonical Singularities". In particular page 352 when computing the blow-up $Y\rightarrow A^2/\...
Jesus Martinez Garcia's user avatar
5 votes
0 answers
369 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 ...
Jizhan Hong's user avatar
9 votes
2 answers
4k views

Inverse Image as the left adjoint to pushforward

This is a repost of a question on Math stackexchange. No one is biting at it there, so I guess it is harder than I thought. Assume $X$ and $Y$ are topological spaces, $f : X \to Y$ is a continuous ...
Daniel Barter's user avatar
7 votes
4 answers
4k views

Is the direct image of a constant sheaf a constant sheaf?

Is the direct image of a constant sheaf a constant sheaf? I'm not an expert on sheaf theory and can't find this anywhere
Mario Carrasco's user avatar
8 votes
2 answers
2k views

Can we relate Cech cohomology and derived functor cohomology even when the cover we choose isn't nice?

In my algebraic geometry class this semester, we've learned about Leray's Theorem, which states that for a sheaf $\mathcal{F}$ on a topological space $X$, and $\mathcal{U}$ a countable cover of $X$, ...
Zev Chonoles's user avatar
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3 votes
0 answers
145 views

Classification of Sheaves of Q-modules over R

Every constructible sheaf of $\mathbb{Q}-$modules over $\mathbb{R}$ is the direct sum of indecomposable sheaves, which are either sheaves with stalk $\mathbb{Q}$ at a point or constant sheaves with ...
Ben's user avatar
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6 votes
1 answer
1k views

Question on the interpretation of a presheaf category as a co-completion

The category of presheaves $Pre(C)$ on a small category $C$ is the category of functors $C^{op}\to Sets$. Since the category of sets is co-complete and every presheaf is a colimit of representable ...
roger123's user avatar
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4 votes
1 answer
576 views

Extension of a first order deformation of a sheaf

Given a coherent and torsion free sheaf $F$ on a smooth projective scheme $S$. Then we have a bijection between $Ext^1_S(F,F)$ and deformations of $F$ over $k[\epsilon]$, $\epsilon^2=0$. Assume all ...
TonyS's user avatar
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3 votes
1 answer
1k views

Chern classes of the direct image of an ideal sheaf resp. skyscraper sheaf

Given a double cover $\pi: X\rightarrow \mathbb{P}^2$ of the projective plane by choosing a square root $S$ of $O_{\mathbb{P}^2}(Q)$, where $Q$ is a quartic in the plane. Choose a closed point $p\...
TonyS's user avatar
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5 votes
4 answers
3k views

Interesting examples of flasque sheaves?

Does anyone know any interesting examples of flasque sheaves? Ideally, I would like to see one that both arises naturally and is geometric in some sense. On the other hand, I know so few examples ...
user332's user avatar
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5 votes
2 answers
6k views

On a proof of the existence of tubular neighborhoods.

Studying analysis on manifolds, I have found, in the proof of the existence of tubular neighborhoods, a reference to theorem 3.1.2 in "Topologie algebrique et theorie des faisceaux" of Godement. ...
agt's user avatar
  • 4,246
1 vote
1 answer
652 views

when a section descends?

Let $C$ be a (reduced, possibly reducible, complex) projective singular curve. Let $\nu: C'\to C$ a finite surjective birational morphism. (For example the normalization, but could be some ...
Dmitry Kerner's user avatar
4 votes
2 answers
551 views

If one wants to work with $Q_l$-adic sheaves, should the scheme be of finite type over a 1-dimensional one?

In section 6 of his 'Adic Formalism' T. Ekedahl states that $l$-adic sheaves 'behave nicely' for finite type separated schemes over $S$ that is regular of dimension $\le 1$. Is the dimension ...
Mikhail Bondarko's user avatar
5 votes
1 answer
1k views

How to compute cohomology groups of a closed subscheme Z of projective space, defined by a homogeneous polynomial of degree d?

Let $Z = \mathrm{Proj}\,k[x_{0},x_{1},\ldots,x_{r}]/f$ be a closed subscheme of degree $d$, i.e., $f$ is a homogeneous polynomial of degree $d$, and $\mathcal{O}_{Z}(1)=i^{*}\mathcal{O}_\mathbb{P}(1)$....
jaz's user avatar
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