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21
votes
0answers
459 views

Why are quasitopological spaces needed in sheaf theoretic approaches to the h-principle?

Recently I have been learning more about the h-principle and in particular the methods of "continuous sheaves". In many treatments of this I see people using "quasi-topological spaces" and I am trying ...
20
votes
0answers
662 views

The topologies for which a presheaf is a sheaf?

Given a set $S$, let $Top(S)$ denote the partially ordered set (poset) of topologies on $S$, ordered by fineness, so the discrete topology, $Disc(S)$, is maximal. Suppose that $Q$ is a presheaf on $...
12
votes
0answers
2k 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 ...
12
votes
0answers
1k views

Idea of presheaf cohomology vs. sheaf cohomology

Let $X$ be a topological space and $U$ an open cover of $X$. In this thread Angelo explained beautifully how presheaf cohomology (Cech cohomology) relates to sheaf cohomology: The zeroth Cech ...
9
votes
0answers
153 views

Is there any notion of “smoothification” from $\mathbb{R}$-schemes to generalized smooth spaces?

I will write $\operatorname{Diff}$ to denote a category of generalized smooth spaces e.g. $Sh(\mathsf{CartSp})$. Is there a version of $\operatorname{Diff}$ for which there exists a functor $\mathcal{...
8
votes
0answers
384 views

Are there simple conditions on a category C which guaranty that Ind(C) is a Grothendieck topos?

The category of finite sets is not a Grothendieck topos, but its Ind category Ind(Finite-Sets) = Sets is a Grothendieck topos. Similarly, given a pro-finite group G, the Grothendieck topos of discrete ...
8
votes
0answers
299 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 ...
8
votes
0answers
371 views

Applications of sheaf theory to the computation of invariants of LS-category type

I would like to know if sheaf theory can be applied to a particular class of questions in topology. The Schwarz genus (also known as sectional category) of a continuous map $p\colon\thinspace E\to ...
7
votes
0answers
165 views

“Flat” and “Affine” morphisms of smooth manifolds

Let $f: X \to Y$ be a qcqs morphism of qcqs schemes. We have the following characterizations: $f$ is flat $\iff$ $f^*:QCoh(Y) \to QCoh(X)$ is exact. $f$ is affine $\iff$ $f_*: QCoh(X) \to QCoh(Y)$ ...
7
votes
0answers
214 views

How do the direct and inverse image sheaf functors interact with homotopy?

This is a crosspost of this MSE question. The direct image sheaf functor $f_\ast$ and inverse image sheaf functor $f^\ast$ (here I mean the usual inverse image sheaf functor often denoted by $f^{-1}$)...
7
votes
0answers
191 views

What's the Hochschild homology of the category of constructible sheaves?

Let $X$ be a manifold. Does the Hochschild homology/cohomology of the category of constructible sheaves on $X$ have a more familiar name?
7
votes
0answers
274 views

Colimits of quasi-coherent sheaves on a ringed space

Recall from the stacks project that a sheaf of modules $F$ on a ringed space $X$ is called quasi-coherent if there is an open covering $\{U_i\}$ such that each $F|_{U_i}$ has a presentation, i.e. is ...
7
votes
0answers
185 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 ...
6
votes
0answers
188 views

Grothendieck - A group as a sheaf over simplicial complexes

In this blog post, Terence Tao gives the following definition of a group. Definition. A group is (identifiable with) a (set-valued) sheaf on the category of simplicial complexes such that the ...
6
votes
0answers
170 views

Making the conceptual leap from locales to Grothendieck topologies?

I find the definition for locales and sheaves on locales to be straightforward, but I'm stumbling over the idea of a Grothendieck topology. Is there a nice way to see roughly how the latter ...
5
votes
0answers
37 views

Expressing the stack of sheaves with 1-limits

Zhen Lin's answer to the MSE question mentions that $\mathsf{Sh}(-)$ is a stack for the canonical topology on the site of open subsets of a space. One of the comments asks whether the stack descent ...
5
votes
0answers
130 views

Push forward of the constant sheaf for a Serre's fibration

Let $f\colon X\to Y$ be a proper continuous map of topological spaces which is a Serre's fibration. $X$ and $Y$ may be assumed to be locally compact, $Y$ is connected topological manifold of finite ...
5
votes
0answers
149 views

Extension of ample vector bundles is ample

As I read Huybrechts-Lehn's book on Moduli of Sheaves, it is making a claim that extensions of several (at least 2) ample vector bundles (on curves) is again ample. Somehow, I am unable to see this ...
5
votes
0answers
317 views

Do 'change of coefficients' functors for sheaves commute with the four functors (formalism)?

For a morphism $f$ of varieties over a field of characteristic $\neq l$ I can consider the functors $Rf_*$, $f^\ast$, $f_!$, and $f^!$ both for the corresponding derived categories of 'all' (...
5
votes
0answers
238 views

Topologies (and sheaves) on Cat and CAT

I've been wondering lately what sort of Grothendieck (pre)topologies there are on $Cat$ (the category of small categories) and $CAT$ (the v. large category of large categories - to forestall criticism ...
4
votes
0answers
70 views

Applications and main properties of hyperfunctions

I am trying to get familiar with hyperfunctions, and I do have some familiarity with the classical theory of distributions. I am wondering whether hyperfunctions have any advantages over ...
4
votes
0answers
152 views

Homotopy-theoretic measure of operations on sheaves failing to be sheaves

Here's something I've been wondering about for a few weeks: Consider a topological space $X$ and a sheaf of rings $\mathscr O_X$ on $X$. Suppose $\mathscr{F}$ and $\mathscr{G}$ are $\mathscr O_X$ ...
4
votes
0answers
210 views

Mixed structures on Hom spaces induced by mixed sheaves

Let $D^b_m(X)$ (resp $D^b(X)$) denote the derived category of mixed Hodge modules (resp. constructible sheaves) on a complex variety $X$. Let $rat\colon D^b_m(X)\to D^b(X)$ be the `forgetful' ...
4
votes
0answers
384 views

About an argument in `Koszul duality patterns in representation theory' by Beilinson-Ginzburg-Soergel.

I am trying to understand Proposition 3.4.2 in `Koszul duality patterns in representation theory' by Beilinson-Ginzburg-Soergel [BGS]. A copy of the paper can be found at http://home.mathematik.uni-...
4
votes
0answers
328 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 ...
4
votes
0answers
492 views

Monomorphisms of sheaves

The following is surely pretty standard, but I have been unable to prove it or find a proof in the literature (many sources assert it without proof). Let $\phi : \mathcal{F} \rightarrow \mathcal{G}$ ...
4
votes
0answers
371 views

A question on a proof that fine sheaves are soft

Let's open R.O.Wells "Differential Analysis on Complex Manifolds" p. 53 and have a look at the Proposition 3.5 stating that all fine sheaves are soft (over a paracompact Hausdorff $X$). In the proof ...
3
votes
0answers
58 views

When is a functorial coverage a sheaf, and what universal property does it have?

In The Elephant (A.2.1.9), Johnstone defines the notion of a coverage on a category $\mathcal{C}$. Quoting verbatim, a coverage on $\mathcal{C}$ is a function assigning to each object $A$ of $\...
3
votes
0answers
125 views

First sheaf cohomology $H^1(\mathscr{O}_D, \mathbb{D})=0$

Given a finite divisor$$D=p_1+\dots +p_m -q_1 -\dots -q_n$$on the unit disk $\mathbb{D}$, does it necessarily follow that the first sheaf cohomology group equals zero, i.e.$$H^1(\mathscr{O}_D, \mathbb{...
3
votes
0answers
134 views

A question about the adjunction between pushforward and pullback of sheaves

I am reading this article: http://arxiv.org/pdf/1310.5978.pdf. In definition 2.6 on page 4 there is claim that is made and I don't see why it is true. I will recall it here: Let $X$ be an integral ...
3
votes
0answers
89 views

Are there Coherent Cosheaves?

Is there a well-defined notion of coherent cosheaves in a similar sense to coherent sheaves? If so, what properties do they hold?
3
votes
0answers
206 views

Defining Inertia Stack

Let $X$ be a topos and $F: \zeta \rightarrow X$ a stack on $X$. Now in the paper http://arxiv.org/pdf/math/0411337v2.pdf ( Definitions 2.1.1.5 and 2.1.1.1) of Lieblich, he describes the inertia stack ...
3
votes
0answers
112 views

Which sheaves on a projective bundle are flat over the base scheme?

Assume $X$ is a noetherian scheme over $\mathbb{C}$ and $E$ a locally free sheaf of finite rank on $X$. Denote the the associated projective bundle by $f: \mathbb{P}(E)\rightarrow X$. Are there any ...
3
votes
0answers
86 views

Can we extend a homotopy invertible chain morphisms between complexes of sheaves from a closed subspace to the whole space?

Let $X$ be a (say, topological) space and $i: Z\hookrightarrow X$ be a closed subspace. Let $Sh(X)$ and $Sh(Z)$ denote the categories of sheaves of abelian groups on $X$ and $Z$ respectively. ...
3
votes
0answers
111 views

Elementary examples on sheaf extension

Let $V\subset\mathbb{P}^n$ be a projective variety and $C_V$ its conormal subvariety in $T^\ast\mathbb{P}^n$. Denote by $\mathscr{O}_{C_V}$ its structure sheaf, then when will the condition $\mathit{...
3
votes
0answers
89 views

Characterization of global sections (which are not products) of a sheaf which is locally a product

In order to compute certain group cohomology sets I have come upon a construction which seems rather general concerning sheaves which are locally products. So I will state the problem here in a ...
3
votes
0answers
66 views

Convolution of DQ-Modules

On page 92 of Deformation Quantization Modules Kashiwara and Schapira define two different convolution products for DQ-modules that differ by whether one uses $Rp_{13*}$ or $Rp_{13!}$ to push forward....
3
votes
0answers
84 views

Monodromy along strata of a pushforward

Work with complex varieties and constructible sheaves on the complex analytic site. All functors will be tacitly derived. Let $X$ be a variety acted upon by a connected linear algebraic group. Let $X ...
3
votes
0answers
187 views

Can one construct Freyd-Mitchell's embeddings that respect sheafifications?

For a certain presheaf $P$ with values in an abelian category $A$ satisfying AB5 and its sheafification $S$ (with respect to a small Grothendieck site) I would like to prove: $S(f):S(X)\to S(Y)$ is ...
3
votes
0answers
240 views

serre duality for sheaves of logarithmic differentials

This question is motivated by a comment of Donu Arapura here dimension of compact support cohomology Let $D$ be a divisor with normal crossings on some smooth projective (complex algebraic) variety $...
3
votes
0answers
101 views

How does the machinery of left-exact comonads generalize from sheaves to stacks?

Suppose that we have two Grothendieck sites, their associated sheaves $\mathcal{E}=\rm{Sh}(\bf{C},J)$ and $\mathcal{F}=\rm{Sh}(\bf{D},K)$ and a geometric surjection $f:\mathcal{E}\to\mathcal{F}$. This ...
3
votes
0answers
248 views

Definition of derived category of a stack

In their book, Bernstein an Lunts define the equivariant derived category in several ways. One can be expressed as follows: Let $X$ be a say complex variety with an action by an algebraic group $G$. ...
3
votes
0answers
154 views

Examples of Sheafification via Hypercovers

For a presheaf $F$ on a category equipped with a pretopology, one has the sheafification $F^{\sharp}$ of $F$. I know well the plus-construction of sheafification, which is presented in Artin's paper "...
3
votes
0answers
303 views

Does this property of subgroups (or sheaves of ideals) already have a name?

I'm constructing an example of a group which has a particular property on its subgroups, and the property looks like something that might have been considered before. Fix a group $G$ and a pair of ...
3
votes
0answers
134 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 ...
3
votes
0answers
216 views

Descent of singular cohomology

When proving that singular cohomology of an appropriate space $X$ equals sheaf cohomology of $X$ with "values" (does one say that?) in the sheaf $\mathbb{Z}_X$ of locally constant functions, the ...
2
votes
0answers
107 views

Double dual of ample sheaf

Let $X$ be a projective manifold. Then we can define ample sheaves on $X$, and many results of ample vector bundles still hold in this more general case (See K. Kubota, Ample sheaves). Now I was ...
2
votes
0answers
107 views

Derived Categories provide a good Framework for Sheaf Cohomology?

I'm a bit new to this sheaf cohomology business. Can someone explain how derived categories provide a good setting for Sheaf Cohomology? I understand that sheaf coho arises as right derived functors, ...
2
votes
0answers
85 views

Sheaf on a filtered topological space?

Is there any nice way of defining a sheaf of abelian groups on a filtered topological space? Let $X$ equipped with filtration $X_0\subset X_1\subset X_2\subset ... \subset X_k=X$ be an object in the ...
2
votes
0answers
63 views

Pre-cosheaf of connected components

Consider a continous map $f:Y \to X$ between topological spaces. The pre-cosheaf $\mathcal{F}: Open(X) \to Set$ of connected components of the inverse image is defined as $\mathcal{F}(U):= \pi_0(f^{-1}...