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8
votes
1answer
281 views

Sheaves on Contractible Analytic Spaces

Let $(X,\mathcal{O}_X)$ be a contractible complex analytic space. Suppose that $\mathcal{F}$ is a coherent sheaf of $\mathcal{O}_X$-modules. Can we invoke the fact that $X$ is contractible to ...
14
votes
1answer
544 views

Why do rigid spaces have “not enough points”?

In Brian Conrad's notes here for the 2007 Arizona winter school, bottom of p18, he says that there is an affinoid rigid-analytic space and a sheaf of abelian groups on it equipped with a non-zero ...
1
vote
2answers
123 views

Terminology: complex of sheaves with cohomology sheaves concentrated in degree zero

What is the proper terminology for a complex of sheaves $\mathcal F^\bullet$ whose homology sheaves $\mathcal H^i\mathcal F^\bullet$ vanish for $i\ne 0$?
5
votes
2answers
305 views

Is the site of (smooth) manifolds hypercomplete?

By site of manifolds Man, I mean the category of manifolds (maybe submanifolds to obtain a small category) with continuous maps between them. A Grothendieck topology is given by open covers. Actually, ...
3
votes
1answer
267 views

Reference request: sheaves on closed sets

I am faced with a context in which the most natural notion of a sheaf $\mathcal F$ is as a functor on the category of compact subsets of a (locally compact Hausdorff) space $X$. Specifically, I say a ...
3
votes
1answer
218 views

Families of local rings coming from a locally ringed space

Let $X$ be a topological space. For every point $x \in X$ let $R_x$ be a local ring. Under what (necessary / sufficient / necessary and sufficient) conditions is there a sheaf ${\cal O}_X$ such that ...
4
votes
0answers
131 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 $\mathscr O_X$ ...
3
votes
0answers
78 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 ...
0
votes
0answers
197 views

Needless axiom for Grothendieck topologies?

Hi, The first axiom for a Grothendieck (pre)topology on a category $C$ says that for every object $X\in C$, the family consisting of just the identity $1_X : X\to X$ should be a covering family. Why ...
8
votes
1answer
508 views

When does the sheaf cohomology of a topological space vanish?

The question is in the title. A more precise formulation is: Let $X$ be a topological space. When does $H^i(X,F) = 0$ for all $i > 0$ and all abelian sheaves $F$ on $X$? The obvious example is a ...
5
votes
1answer
437 views

Hypercohomology of a complex via Cech cohomology

Let $X$ be a reasonable topological space. If $\mathcal{F}$ is a sheaf of abelian groups then Cech cohomology gives us a method to compute the cohomology groups $H^p(X, \mathcal{F})$ - the main input ...
5
votes
2answers
370 views

Are subfunctors of left exact functors also left exact?

Consider the direct image functor $f_*: Sh(X) \rightarrow Sh(Y)$, let $X$ and $Y$ be topological spaces, let $f: X \rightarrow Y$ be a continuous map, let $G \in Sh(X)$ be a sheaf. I was reading this ...
7
votes
4answers
796 views

Coboundaries and Gluing in Cech Cohomology - Intuition?

I'm trying to develop an intuition for Cech cohomology geometrically, but am currently failing. A lot of people seem to say that the groups $H^n$ measure obstructions to gluing local sections to get ...
0
votes
1answer
166 views

Stability of $T_{\mathbb{P}^2}$ and $\Omega_{\mathbb{P}^2}$?

How can one prove that the tangent bundle $T_{\mathbb{P}^2}$ and its dual $\Omega_{\mathbb{P}^2}$ are stable vector bundles with respect to $\mathcal{O}_{\mathbb{P}^2}(1)$? Similarly, is it true that ...
1
vote
0answers
101 views

on geometric Satake and functions

Let $G(F)/G(O)$ the affine grassmanian with $F=k((t))$ where $k$ is a finite field. For $\lambda$ a dominant cocharacter, we have by Cartan decomposition the schubert strata ...
19
votes
0answers
361 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 ...
3
votes
3answers
160 views

Constants sheaves on an open subset

Let $X$ be a topological space and $U\subset X$ an open subset. Let's work in the category of sheaves of abelian groups on $X$. Consider the constant sheaf on $U$, $\mathbb{Z}_U$, given by ...
7
votes
0answers
223 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 ...
1
vote
1answer
318 views

Does the Čech cohomology always yield long exact sequences from short ones?

Does the Čech cohomology always give rise to a long exact sequences given a short exact sequence of sheaves? Clearly that cannot occur for sheaves on a paracomact (perhaps also Hausdorff, I'm not ...
3
votes
1answer
277 views

Restricting a Soft Sheaf to an Open is again Soft?

Hi everyone! Answered to my satisfaction in the comments - thanks nosr and Jacob Bell! :) Let $X$ be Hausdorff, locally compact, paracompact. Consider $\mathcal{F}$ a soft sheaf on $X$: as there are ...
3
votes
0answers
190 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 $G$. ...
13
votes
6answers
831 views

Understanding Adjointness of Sheaves in Algebraic Geometry

Pushforward and pullback are very basic operations in algebraic geometry, as is the adjointness between them. I worked out a very careful of adjointness of sheaves (below) when I was working out of ...
6
votes
1answer
253 views

Are constructible derived categories invariant up to weak homotopy equivalence?

Let $X$ and $Y$ be two topological spaces and $R$ be commutative ring. Let $D_c^b(X, R)$ and $D_c^b(Y,R)$ be their respective bounded derived categories of constructible sheaves of $R$-modules. I ...
0
votes
1answer
235 views

Minimal Destabilizing Quotient

For a pure $d$-dimensional sheaf $E$ on a projective algebraic variety over a field $k$, one has the Harder-Narasimhan filtration $$0\subset E_1\subset E_2\subset...\subset E_{l-1}\subset E_l:=E,$$ ...
0
votes
1answer
134 views

When is a sheaf of groups (algebras, rings, modules) a group (algebra, ring, module)?

If $\pi:E\to M$ is a vector bundle then the set of sections $\Gamma(E)$ is naturally a vector space under fibrewise addition and scalar multiplication on the bundle $E$. This holds similarily for ...
11
votes
2answers
549 views

Cosheafification

Hello all. I have a pre-cosheaf in the category of vector spaces. How do I cosheafify? I've failed to find literature on this topic. I'll be more specific. Let $\mathbb{X}$ be a topological ...
2
votes
1answer
325 views

On morphisms to projective space arising from a linear system

Context: This question arose as I was reading the proof of Application 6.1 in Mumford's Abelian Varieties. However, I have extracted all of the relevant information below so this question should ...
2
votes
0answers
120 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 ...
1
vote
1answer
281 views

Sheaf Hom and the functor Hom

Let $\varepsilon: 0\to A\to B \to C\to 0$ be an exact sequence of ${\cal O}_X$-modules with $X$ a quasi-compact space. $\varepsilon$ is called pure if the induced sequence $0\rightarrow ...
0
votes
0answers
149 views

restriction and pullback of representable etale sheaf along closed immersion

I find that the restriction and pullback of representable etale sheaf along closed immersion are very confusing. I think they are different in general, I hope some experts can confirm my ...
1
vote
1answer
176 views

quotient of ind scheme

Is I consider an ind scheme such as $G(k((t)))$ for a reductive connected group over $k=\bar{k}$ I have the conjugacy action of $G(k[[t]])$. In what category can I make the quotient ...
1
vote
1answer
289 views

Leray Spectral Sequence

Let $f:X\to Y$ be a smooth map between paracompact differential manifolds $X$ and $Y$. Let $U$ be an open and dense subset of $Y$. For any $y\in U$, let $f^{-1}(y)=F$ be a generic fiber that is a ...
0
votes
0answers
113 views

morphisms in the construction of the moduli space of curves by mumford

Hi fellow mathematicians, I'm just studying mumfords proof of the existence of a coarse modulispace for curves of genus $g$in his great GIT book. On page 102 (in the proof of proposition 5.2) he ...
1
vote
0answers
65 views

How can one compute the cohomology of $i'^*C$, for $i':\mathbb{A}^{N-1}\setminus \{0\}\to \mathbb{A}^{N}\setminus \{0\} $?

For an (etale or 'topological', constructible bounded) complex of sheaves $C$ on $X'=\mathbb{A}^{N}\setminus \{0\} $, $i'$ being the embedding $\mathbb{A}^{N-1}\setminus \{0\}\to ...
3
votes
0answers
300 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 ...
7
votes
2answers
487 views

Interpreting $f^*f_*$

For a morphism of schemes $f: X\rightarrow Y$, one often considers the function $f^*f_*$ on sheaves. For example, this appears in adjunction for sheaves of $\mathcal{O}_X$-modules, the projection ...
8
votes
1answer
355 views

Injective objects in Mor(Ab)

Consider the abelian (Grothendieck) category $\mathcal{C} := \mathrm{Fun}(\{0<1\},\mathrm{Ab}) = \mathrm{Mor}(\mathrm{Ab})$. Objects are morphisms $(A \to B)$ of abelian groups, morphisms are ...
2
votes
0answers
330 views

large cardinal tree properties as properties of sheaves

As follows from this talk Large Properties for Small Cardinals, p.7,p.4 http://www2.dm.unito.it/paginepersonali/viale/SEMINARS-TORINO/Fontanella-Torino-19.1.2012.pdf, the definitions of weakly compact ...
4
votes
0answers
196 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
291 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 ...
2
votes
1answer
285 views

Showing morphism of sheaves is zero

I work in derived category $D^b(X)$ of constructible sheaves on a reasonable space $X$. Let $j\colon U\to X$ be an open inclusion and $i\colon Y\to X$ the closed complement. Let $M,N\in D^b(X)$ and ...
1
vote
1answer
228 views

Inclusion of logarithmic de-Rham complex into differentials

Let $X$ be a complex manifold and $D$ a normal crossing divisor. Let $U = X - D$ and $j: U \rightarrow X$ the natural map. Voisen observes that there is a natural inclusion $$\Omega^k_X(\log D) ...
2
votes
1answer
275 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 ...
5
votes
2answers
246 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 ...
7
votes
3answers
479 views

Representable Presheaf

I have a very quick question. Is there an easy example of a representable presheaf on a site that is not a sheaf? This certainly can't happen on a small FPPF site so I would expect a counterexample to ...
2
votes
1answer
270 views

smooth manifold vs. exceptional inverse image

A well-known theorem in topology says that for a smooth manifold $M$ of dimension $n$ the map $f: M \rightarrow point$ satisfies $$f^! \mathbf R = \mathbf R[n]$$ Here $\mathbf R$ is the constant ...
9
votes
1answer
379 views

When do sheaves which are constant along the fibers come from the base?

Assume we are given a map $f: X \rightarrow Y$ between topological spaces which is open, surjective and has (pathwise) connected fibers. Consider categories $\text{Sh}(X)$, $\text{Sh}(Y)$ of sheaves ...
3
votes
1answer
185 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 ...
6
votes
2answers
455 views

Can one characterize those sheaves which have Hausdorff etale spaces?

Given a sheaf of sets $F$ on a space $X,$ under the equivalence of categories between etale spaces over $X$ and sheaves over $X,$ $F$ is associated to a local homeomorphism $$E\left(F\right) \to X$$ ...
2
votes
1answer
342 views

system of local coefficients on X, locally constant sheaves and orientation sheaves

Hi, I try to understand the orientation sheaves. When searching it in the google, i meet new areas such as local coefficient system and locally constant sheaves. I realize that any system of local ...