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0
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1answer
201 views

A functorial isomorphism in derived category

This question is a direct continuation of Question 1 in this post: Two basic questions on derived categories Let $f\colon \mathcal{A}\to\mathcal{B}$ be a left exact functor between two abelian ...
3
votes
2answers
504 views

Two basic questions on derived categories

Let $\mathcal{A}, \mathcal{B}$ be two abelian categories with sufficiently many injective objects (in my case these are categories of sheaves of vector spaces on a manifold). Let $f_*\colon ...
0
votes
1answer
138 views

Can a smooth function on a cross be extended to the whole plane?

Consider a real function on the union of two lines R×0 and 0×R in R² whose restrictions to R×0 and 0×R are smooth functions R→R. Is it possible to extend this function to a smooth function on R²? ...
2
votes
1answer
239 views

Stalks of étale sheaves

I want to prove that $0 \to F\to G\to H \to 0$ is an exact sequence of étale sheaves. I understand that it is enough to show that $0\to F_{\bar{x}}\to G_{\bar{x}}\to H_{\bar{x}}\to 0$ is exact at ...
10
votes
1answer
645 views

Reference request: Book of topology from “Topos” point of view

Question: Is there any book of topology in the modern language of topos theory? Motivation: In "Sheaves in Geometry and Logic" Mac Lane and Moerdijk say: "For Grothendieck, topology became the ...
0
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0answers
151 views

Hermitian metric on $f_*(\Omega_{X/X_{can}}^{n-k})$

Let $X$ be an n- dimensional algebraic manifold . Suppose that its canonical line bundle $K_X$ is semi-positive and $0<k=Kod(X)<n $ . Let $f: X\to X_{can}\subset \mathbb CP^N$ Here $X_{can}$ ...
0
votes
1answer
116 views

Open subset of the moduli space of stable sheaves on a noetherian scheme

This is my question: Given a projective noetherian scheme $X$, the structural sheaf $\mathcal{O}_X$ is a coherent sheaf, so every locally free sheaf is coherent. This means that the family of stable ...
1
vote
1answer
102 views

Are the pullback functors of adjoint functors also adjoint?

Given adjoint functors $F: A \to B$, $G: B \to A$, if you then take their pullback functors $F^* : Set^B \to Set^A$ and $G^* : Set^A \to Set^B$ given by pre-composition, are these two also adjoint ...
4
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0answers
155 views

Atlas of a manifold as a Sheaf

--Hopefully this question does not dublicate another-- In this question Tom Goodwillie pointed out, that the 'atlas part' of the definition of a smooth manifold can be redefined in terms of sheaves. ...
4
votes
1answer
164 views

Hypercovers of sheaves in classical and quasi-categories

I am interested in relating the definition of hypercovers in the $\infty$-topos of sheaves on an $\infty$-Grothendieck site to the classical definition of hypercovers of presheaves on a Grothendieck ...
0
votes
0answers
119 views

Cohomology group vs sheaf of cohomology group

Suppose $F$ is a coherent sheaf on a smooth (algebraic or complex) variety $X$. Then we can consider the cohomology groups $$H^p(X,F)$$ for all $i$. Now, let we consider the sheaf ...
0
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0answers
61 views

the push forward of the differential idea of sheaf

This is about the differential idea of a sheaf as a vector bundle $E$ (not necessarily locally trivial) on a real manifold $M$ with a zero curvature connection $\nabla$. The zeroth cohomology is then ...
2
votes
0answers
46 views

Sheaves, colimits and closure

I am considering the sheaf topos $\mathbf{Sh}(\mathfrak{X})$ on a topological space $\mathfrak{X}$. Limits in this category are constructed pointwise, as in presheaves. Colimits, however, are not ...
4
votes
1answer
147 views

About the construction of the Universal Enveloping Lie Algebroid

Let $X$ be a reasonable smooth scheme over some base $S$. The tangent sheaf $T_X$ is a Lie algebroid, locally free as a $\mathcal{O}_X$ module, and its Universal Enveloping Lie Algebroid ...
6
votes
1answer
329 views

What information is lost in $X \to \mathrm{Sh}(X)$?

Given a topological space or site $X$. Construct $\mathrm{Sh}(X)$ - the sheaves on $X$ with values in $\mathrm{Set}$. Is it known what information is lost in this procedure? Thanks, Adrian.
0
votes
0answers
68 views

Equivariant Sheaves, Local system

Let $(G,m)$ be a group scheme unipotent, and $L$ a local system of rank 1 on $G$ such that: $m^*(L) \simeq L \boxtimes L $. Then why is $L$ an equivariant sheaf on $G$ with the action the ...
2
votes
1answer
119 views

Example for pullback of stable sheaf not stable

Suppose $C$ is a complete algebraic curve. Define a coherent locally free sheaf $\mathcal{F}$ over $C$ to be stable if $\mu(\mathcal{E})<\mu(\mathcal{F})$ for any subsheaf $\mathcal{E}$, where ...
1
vote
1answer
132 views

Unique decomposition of locally free sheaf

Below let's work over coherent sheaves on a smooth projective algebraic curve. We call a subsheaf $\mathcal{F'}$ of $\mathcal{F}$ saturated if it $\mathcal{F/F'}$ is locally free. We call a locally ...
4
votes
1answer
128 views

Associating a principal bundle to a torsor

in arXiv:math/0212266, Moerdijk defines a torsor to be a sheaf $\mathcal{S}$ on $X$ with a freely transitive left-action of a sheaf of groups $\mathcal{G}$, such that $X =\bigcup \{ U \in ...
6
votes
3answers
312 views

Is there a name for a “rigid” sheaf?

Is there a name for the property of a sheaf $\mathcal F$ such that the restriction maps $\mathcal F(V) \to \mathcal F(U)$ are injective when $V$ is connected and $U$ is nonempty? In other words, this ...
7
votes
0answers
156 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?
1
vote
1answer
125 views

what's the cohomological dimension of a Stein space?

I want to know the "cohomological dimension" of a Stein manifold. I know that: for $X$ differential manifold and for every sheaf $F$ of abelian groups, I have $H_c^j(X,F)=H^j(X,F)=0$ for ...
18
votes
0answers
555 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 ...
3
votes
1answer
171 views

flat descent for perverse sheaves

Let $E \in D^{b}_{c}(X,\overline{\mathbb{Q}}_{l})$ where $X$ is a $k$ scheme of finite type for a field $k$. Let $Y\rightarrow X$ a finite flat surjective morphism such that $f^{*}E$ is perverse and ...
1
vote
0answers
51 views

Construction of a Hermitian metric on a locally free subsheaf which is not a subbundle

Assume $X$ is a smooth projective curve over $\mathbb{C}$ and let $\mathcal{E}$ be a locally free sheaf of rank 2 on $X$. We pick a closed point $x\in X$ and a surjection $f: \mathcal{E}\rightarrow ...
4
votes
1answer
192 views

Exercises around Diffeological Spaces or a Diffeologic Atlas Theory

Is there a Book or a bunch of exercises to get used to diffeological spaces from a practical point of view? It seems to me that papers on this topic are mostly concerned with their very good ...
0
votes
1answer
137 views

Sheaffication using a $\lambda$-transfinite colimit

I asked this question on mathstack (long time ago), however I received no answers, so I'm trying it here. I don't know whether it's suitable for this site, anyway. I was reading this article ...
2
votes
0answers
206 views

Cech cohomology.

Let us consider the scheme $X$ and the coherent sheaf $\cal F$ on it. We consider finitely many affine open covers $U_{\lambda}$ with $\lambda \in \Lambda$. When I calculate Cech cohomology with ...
0
votes
1answer
218 views

Global to local for Ext groups and Sheaves

Let $X$ be a projective variety. The sheaf $\mathcal{E}xt^{1}(\Omega_{X},\mathcal{O}_{X})$ is supported on $Sing(X)$. Now, there should be a theorem (perhaps by Schlessinger) that says that if $X$ ...
1
vote
1answer
179 views

What is the connection between direct/inverse image of set maps and direct/inverse image functors of sheaves?

Given a function of sets $f:X\to Y$, one defines the direct image and inverse image maps: $$ f_*:\mathcal{P}(X)\to\mathcal{P}(Y) $$ $$ f^{-1}:\mathcal{P}(Y)\to\mathcal{P}(X) $$ In the usual way. ...
4
votes
1answer
193 views

Relating deformations of a scheme to deformations of its singular locus

Let $X$ be a normal scheme with quotient singularities and $Y\subset X$ its singular locus. The first order deformations of $X$ are parametrized by $\mathcal{E}xt^{1}(\Omega_{X},\mathcal{O}_{X})$. ...
0
votes
1answer
188 views

Birkhoff decomposition vanishing of the Chern numbers

Birkhoff decomposition vanishing of the Chern numbers of the holomorphic line bundles of the Birkhoff-Grothendieck decomposition, is some statement I read off in One of Connes papers. Without going ...
3
votes
0answers
46 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 ...
3
votes
1answer
250 views

Explicit examples presheaves associated to higher direct images which fail to be sheaves

So I would like to have a few simple examples where the presheaf associated to higher direct image of sheaf fails to be sheaf. So I'm looking for two (natural and simple) topological spaces $X$ and ...
0
votes
2answers
245 views

Pullback of a constant sheaf

Let $\varphi:X\to Y$ be a surjective morphism of schemes which are connected and of finite type. Let $A$ be an abelian group, $\mathscr{F}$ be the constant sheaf on $X$ with fibers $A$ and ...
3
votes
0answers
72 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 ...
4
votes
1answer
206 views

Representability of a certain group scheme quotient

Let $k$ be a field. Suppose we have an exact sequence of $k$-group schemes (not finite-type) $$ 1\to H\to G\to K\to 1 $$ In other words, the sheaf quotient $G/H$ is representable by a $k$-group ...
1
vote
1answer
270 views

Cohomology of tangent bundles

Let $X$ be a smooth scheme and $Z\subset X$ a smooth subscheme. Consider the blow-up $$\pi:\widetilde{X}:=Bl_{Z}X\rightarrow X$$ of $X$ along $Z$. What is the relation between the cohomology of the ...
1
vote
0answers
82 views

Pull back of D-modules and Koszul resolution

Consider an holonomic D-module on a smooth algebraic variety $X$ over a field $k$ of caracteristic 0. Let $i: Y \hookrightarrow X$ be a regular embedding. $Li^* M = \mathcal{D}_{Y\to X} ...
1
vote
1answer
124 views

global sections of higher direct images of étale sheaves

Is there a useful criterion for when $\Gamma(X, R^qf_*F) = H^q(X',F)$, $f: X' \to X$, $F$ an étale sheaf on $X'$?
1
vote
1answer
431 views

relation between sheaf of hom and hom of sheaf

If $\mathcal{M,N}$ are the associated sheaf of $A$ modules $M$ and $N$ on $X=Spec A,$ then what is $\mathcal{Hom_{O_X}(M,N)}$?Is that the associated sheaf of $Hom(M,N)\ ?$
3
votes
0answers
170 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 ...
1
vote
0answers
86 views

Proving that two given functionally structured spaces are isomorphic

The relevant definitions are listed below. They can be found in Chapter VI, pages 297-298 of Bredon's Introduction to Compact Transformation Groups; and Section 2, Chapter II of Bredon's Topology and ...
3
votes
1answer
118 views

Does the bundle of germs of functions $f:X\to \mathbb R$ have the same sheaf of sections as $X\times \mathbb R$?

I'm just starting to learn about sheaves, and I'm confused about a certain matter: I've just learned, to my delight, that every sheaf $S$ on a space $X$ is the sheaf of sections of a particular ...
2
votes
0answers
101 views

Reconstructing complexes of sheaves from their cohomology sheaves

If $R$ is an algebra over some field $k$, and $C$ is a complex of modules over $R$, then according to B. Keller's ``Introduction to A-infinity algebras and modules'', one can record the isomorphism ...
5
votes
2answers
376 views

C*-algebras and quantum fields

One can represent a quantum system by the Weyl algebra (which is a C*-algebra). For instance, a 1 degree of freedom system can be represented by the algebra generated by $e^{\imath t Q}, e^{\imath s ...
2
votes
3answers
551 views

Why are (pre)sheaves defined as contravariant functors? Why not just reverse the arrows in the first place?

Why not just have arrows in the category of opens represent coverings instead of inclusions? It seems to me like both conventions (whether presheaves are co/contra and which of the two dual orderings ...
0
votes
0answers
74 views

$\mathcal{F}$— twists of Lie algebras

I am trying to figure out with Drindfel's Opers. Let us consider Lie group $G$ and $G$ -- bundle $\mathcal{F}$ on the smooth algebraic curve $Y$. Can anybody help me and clarify definition of the ...
-1
votes
1answer
165 views

Pure Quotient and pure sub-object

Let $\mathcal{C}$ be the category of modules over a ring. Let also $\mathcal{F}$ be a class of objects in $\mathcal C$ closed under pure subobject (pure quotient) and direct limit. Is $\mathcal{F}$ ...
3
votes
1answer
177 views

Euler Characteristic of Coverings via Sheaf Theory

Let $X$ be a nice space (compact $CW$-complex or triangulated space, compact manifold, whatever works), $f:Y\to X$ be a finite covering of degree $n$, and $\chi(X)$ be the euler characteristic. By the ...