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10
votes
1answer
826 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
408 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
218 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 ...
1
vote
1answer
811 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
327 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
373 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
198 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
362 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
306 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
251 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
768 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
192 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
368 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
501 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
306 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
479 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
110 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
187 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
639 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
508 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
289 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 $f ^ {0} \ \colon \ S^{0} \to T^{0}$ ...
1
vote
1answer
409 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
166 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
332 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
221 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
302 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
731 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
263 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
309 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
509 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
243 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
204 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
287 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 ...
2
votes
0answers
307 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
904 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
5
votes
2answers
950 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 $X$, ...
3
votes
0answers
129 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
1answer
533 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 ...
4
votes
1answer
329 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 ...
0
votes
1answer
424 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 ...
3
votes
4answers
1k 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 ...
3
votes
2answers
2k 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. ...
4
votes
2answers
457 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 ...