The sheaf-theory tag has no usage guidance.

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### How does torsion behave under the direct image functor?

Assume we have a finite morphism $f: X\rightarrow Y$ of smooth projective varieties of degree $d$ over $k=\mathbb{C}$. Then $f_{*}$ induces an equivalence between the categoy of coherent $O_X$-modules ...

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273 views

### Cohomology of a cochain complex of acyclic sheaves

Ok, sort of as a follow up to my previous question, let's recall the de Rham-Weil theorem:
Let $F$ be a sheaf on a topological space $X$ and let $\mathcal{L}^{\bullet}$ be an acyclic resolution of ...

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2k views

### Where can I find a proof of the de Rham-Weil theorem?

Where can I find a proof of the de Rham-Weil theorem?
Does anyone know?

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**5**answers

818 views

### Sheafification via hypercovers

The sheafification of a presheaf on a site is often constructed in a two-step process $X^{++}$, where $X^+$ consists of matching families in $X$, is always separated, and is a sheaf if $X$ is ...

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1k views

### Is Sheafification Functor Exact?

I know that sheafification functor from the category of abelian presheaves on $C$ to the category of abelian sheaves on $C$. Here, $C$ is a category with Grothendieck pretopology.
My question is:
...

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**1**answer

237 views

### Do Categorical Quotients Preserve Covering Maps?

Before asking a question, please let me write down settings.
SETTINGS:
Let $C$ be a category with fiber products and $B$ be a closed subcategory of $C$ (i.e. $B$ contains any isomorphism of $C$, and ...

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453 views

### Is $H^i(X,F)$ finitely generated over $\Gamma(O_X)$ if $F$ is coherent?

Suppose $\mathcal{X}$ is a smooth quasi-projective variety over $\mathbb{C}$ (I apologize if these hypotheses have little to do with the question at hand). Let $\mathcal{F}$ be a coherent sheaf on ...

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4k views

### Is there a sheaf theoretical characterization of a differentiable manifold?

I'm going through the crisis of being unhappy with the textbook definition of a differentiable manifold. I'm wondering whether there is a sheaf-theoretic approach which will make me happier. In a ...

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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' ...

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147 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 ...

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**3**answers

710 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 ...

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**0**answers

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 ...

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224 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 ...

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383 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^*) $?

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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 ...

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**1**answer

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 ...

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**1**answer

905 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 $\mathscr F^+$ ...

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**1**answer

449 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 ...

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votes

**1**answer

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 ...

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**1**answer

931 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 ...

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**1**answer

331 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 ...

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**1**answer

380 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 ...

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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 $H^\ast (Z,i^{-1}S)$ ...

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203 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 ...

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**1**answer

365 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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310 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 ...

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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 ...

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833 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:
...

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196 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 ...

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376 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 ...

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225 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 ...

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507 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 ...

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312 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 ...

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**1**answer

522 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 ...

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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 ...

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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 ...

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697 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 $\mathcal{P}$ ...

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530 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 $X$:
$$
...

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296 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 $f ^ {0} \ \colon \ S^{0} \to T^{0}$ ...

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424 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 ...

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169 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 ...

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**1**answer

339 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 ...

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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 ...

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316 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 $uv = \xi_e vu$, ...

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763 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 $\mathcal O_X$ ...

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269 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 ...

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323 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 ...

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**1**answer

537 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. ...

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**1**answer

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 ...

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**4**answers

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. ...