The tag has no wiki summary.

learn more… | top users | synonyms

5
votes
0answers
312 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' ...
1
vote
0answers
146 views

Soft sheaves on indiscrete paracompact spaces

Let $X$ be some space, I have basically 2 questions: 1 - Are sheaves on paracompact but not Hausdorff spaces acyclic? I've been doing some reading and some authors say that soft sheaves on ...
11
votes
3answers
682 views

Necessity of hypercovers for sheaf condition for simplicial sheaves

I'm trying to understand where the definition of simplicial sheaf on a space/site comes from. For a presheaf $F$ of sets on a topological space $X$, the sheaf condition can be viewed as saying that ...
2
votes
0answers
481 views

What essential property justifies the name “derivative”?

Most, if not all, of the notions of derivative that I have so far seen have the property that they are locally defined -- meaning that the derivative of a map-type object at a point depends on the map ...
0
votes
1answer
220 views

Direct limit of sheaves over paracompact spaces

Let $\mathcal F_i, i\in I$, be a directed family of sheaves of abelian groups on a paracompact Hausdorff space $X$. Let $\mathcal F=\varinjlim F_i$ denote the direct limit sheaf. Is it true that ...
0
votes
3answers
381 views

Cohomology of complexes

I sometimes see cohomologies of complexes of sheaves. What is the definition of these? Say if $\mathcal F^* $ is a complex of sheaves on $C$, what is $H^i(C,\mathcal F^*) $?
2
votes
0answers
322 views

Quantum sheaves

Are the following definitions known? Consider H a Hilbert space. A "quantum topology" on H is a set Sigma of closed subspaces satisfying the following conditions: (a) {0} and H lie in Sigma (b) If ...
1
vote
1answer
125 views

Is the free R-module on a sheaf of sets still a sheaf?

Let $L$ be a sheaf of sets on some site $S$. Let $F$ be the presheaf obtained by composing $L$ with the free R-module functor, i.e. for any object $U$, we define $F(U)$ to be the free $R$-module on ...
11
votes
1answer
877 views

Affine scheme on spec(A) of a ring A as the sheafification of a pre-sheave on spec(A)?

It is obvious that there is a parallel between the definition of structure sheaf of $\operatorname{Spec}(A)$ versus the sheafification of a pre-sheaf. The definition of the sheaf $\mathscr F^+$ ...
4
votes
1answer
439 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 ...
4
votes
1answer
220 views

When do adjunctions preserve equivalence?

Let $\mathcal{C}'$ and $\mathcal{D}'$ be categories so that $\mathcal{C} \subseteq \mathcal{C}'$ and $\mathcal{D} \subseteq \mathcal{D}'$ are full subcategories. Suppose the forgetful functors ...
2
votes
1answer
902 views

Orientation Sheaf and Double Cover

The orientation sheaf of an $n$-manifold $M$ is $\mathcal{O}_n=Sheaf(U\mapsto H_n(M,M-U;\mathbb{Z}))$, with stalks given by $(\mathcal{O}_n)_x = lim H_n(M,M-U)=H_n(M,M-x)=\mathbb{Z}$ (the limit is ...
4
votes
1answer
329 views

Derived Equivalence of Sheaves and Homotopy

This question loosely elaborates on an earlier question. It is pretty silly, but I'd like to hear some authoritative answers. Recall that if $f:S^{\bullet}\to T^{\bullet}$ is a quasi-isomorphism of ...
4
votes
1answer
377 views

Etale space construction for presheaves on a Grothendieck site

As it is described for example in [Mac Lane-Moerdijk, Sheaves in Geometry and Logic, II.6.], one can construct the sheafification functor very lucidly by associating to a presheaf a certain bundle ...
0
votes
0answers
166 views

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 $H^\ast (Z,i^{-1}S)$ ...
1
vote
0answers
201 views

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 ...
1
vote
1answer
364 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 $X$ ...
2
votes
0answers
308 views

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 ...
2
votes
0answers
255 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 ...
2
votes
1answer
811 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: ...
2
votes
0answers
193 views

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 ...
2
votes
0answers
375 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 ...
1
vote
0answers
224 views

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 ...
2
votes
1answer
506 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 ...
2
votes
1answer
308 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 ...
3
votes
1answer
502 views

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 ...
2
votes
0answers
112 views

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 ...
2
votes
2answers
191 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 ...
2
votes
2answers
676 views

Are presheaves of constant functions sheaves?

Hey there, 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 $\mathcal{P}$ ...
1
vote
0answers
526 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 $X$: $$ ...
1
vote
0answers
295 views

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 $f ^ {0} \ \colon \ S^{0} \to T^{0}$ ...
2
votes
1answer
418 views

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 ...
2
votes
1answer
168 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 ...
2
votes
1answer
335 views

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 ...
5
votes
1answer
225 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 ...
2
votes
1answer
308 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 $uv = \xi_e vu$, ...
6
votes
1answer
755 views

Sheaves of $\mathbb Z$-modules = sheaves of abelian groups

Hi. 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 tako $\mathcal O_X$ ...
2
votes
1answer
266 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 ...
0
votes
2answers
319 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 ...
4
votes
1answer
524 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. ...
0
votes
1answer
249 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 ...
14
votes
4answers
2k 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. ...
12
votes
0answers
1k 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 ...
2
votes
0answers
211 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. ...
8
votes
0answers
288 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 ...
13
votes
4answers
2k 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 ...
13
votes
2answers
2k views

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 ...
3
votes
0answers
312 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 ...
5
votes
1answer
977 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 ...
6
votes
4answers
1k 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