All Questions
Tagged with sheaves or sheaf-theory
979 questions
3
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285
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What is the logical progression in algebraic tools for studying spaces (varieties -> schemes, sheaves, topos etc.)?
Some algebraists (Cartier, Weil, Atiyah, etc.) sometimes speak of geometry as a long history of essentially asking the same question—"what is space, and how would one describe a space uniquely". ...
- 39
3
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0
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430
views
Computing injective resolution of some constant sheaves
I follow the notations on "Sheaves on manifolds" written by Kashiwara-Schapira. Let $\mathbb{K}$ be a ground field and $X$ be a smooth manifold. Let $D(X)$ be the derived category of sheaves of $\...
- 213
3
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0
answers
375
views
Equivariant sheafs and $G$ actions on modules
I am reading Simpson's paper on The Hodge filtration on nonabelian cohomology. In particular Chapter 5 (p.24) and I am confused about the notion of a group acting on an equivariant sheaf.
The set up ...
- 296
3
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0
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212
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Homotopy Colimit of Čech Complex
I am studying homotopical cosheaves, and I came up with the following "conjecture".
We can see an "additive" precosheaf in chain complexes (such that corestrictions do not commute on the nose) as a ...
- 2,152
3
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0
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95
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On the degrees of the normal and tangent sheaves of Gorenstein curves in Pn
It is known that if an irreducible curve $C$ is a local complete intersection in $\mathbb{P}^n$, then
$\wedge^{n-1}\mathcal{N}\otimes\omega_{\mathbb{P}^n}$ is isomomorphic to the dualizing sheaf $\...
3
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0
answers
84
views
Could we have the simplicial definition of equivariant derived category of sheaves with arrow direction inversed?
Let $X$ be a topological space and $G$ be a topological group acting on $X$ from the left. We consider the simplicial space $[G\backslash X]_{\cdot}$ where
$$
[G\backslash X]_n=\underbrace{G\times \...
- 9,019
3
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0
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100
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Relation between the orientation sheaves of the interior and the boundary of a topological manifold
Let $(M, \partial M)$ be an $n$-dimensional topological manifold with boundary. Let $\mathcal{O}_{M \setminus \partial M}$ and $\mathcal{O}_{\partial M}$ denote the orientation sheaves of $M \setminus ...
- 165
3
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308
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Quotient of a sheaf by group action and representabillity
Let $X$ be a scheme and $S$ be a sheaf of sets over the fppf topology of $X$. Let $G$ be a group scheme over $X$ and there is an action of $G$ on $S$. Now, I want to look at the quotient $G \setminus ...
- 1,277
3
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64
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Canonical map in the direct image of $\mathscr{D}_X$
Let $f : X \to Y$ be a proper holomorphic map between holomorphic manifolds. We work with $\mathscr{D}$-modules. Consider the transfer bi-modules $\mathscr{D}_{Y\leftarrow X}.$ Can one find a ...
- 193
3
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1
answer
360
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Why are sheaves of a coverage the same as those on its generated Grothendieck coverage?
Preliminaries: There are lots of variations on the settings in which we define sheaves. I am concerned with the details linking the general definition below to the Grothendieck topology it generates; ...
- 352
3
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307
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Locality in Grothendieck Topologies
Let $\mathcal{C}$ be a category and $\mathcal{J}$ be a Grothendieck topology on it (i.e., $(\mathcal{C},\mathcal{J})$ is a site). Then what is a good notion of locality in it?
I came up with the ...
- 438
3
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0
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163
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Question about the precise statement of Leray spectral sequences and a simple example
On Bott's paper "Homogeneous vector bundles" there is the following statement of Leray spectral sequence:
Let $X$ and $Y$ be paracompact and locally compact spaces and $f : Y \to X$ be a proper map....
3
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0
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978
views
How does one compute the Chern classes of the dual sheaf from the Chern class of the original sheaf?
Let $X$ be a smooth projective 4-fold (over $\mathbb{C}$). Let $Z \subset X$ be a codimension two subscheme. Let $I_{Z}$ denote the ideal sheaf of
$Z$. How does one compute the Chern classes of $I_Z^{...
- 3,245
3
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240
views
"2-Sheafification" with Values in non $Cat$ categories?
Let $X$ be a 2-site and consider the category of 2-presheaves over $X$, which will be denoted as $Pshv(X;Cat)$. These are $Cat$-valued 2-functors, where $Cat$ is the 2-category of categories. There is ...
- 913
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108
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When is a functorial coverage a sheaf, and what universal property does it have?
In The Elephant (A.2.1.9), Johnstone defines the notion of a coverage on a category $\mathcal{C}$. Quoting verbatim, a coverage on $\mathcal{C}$ is
a function assigning to each object $A$ of $\...
- 8,659
3
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579
views
A question about the adjunction between pushforward and pullback of sheaves
I am reading this article: http://arxiv.org/pdf/1310.5978.pdf. In definition 2.6 on page 4 there is claim that is made and I don't see why it is true. I will recall it here:
Let $X$ be an integral ...
3
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0
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551
views
Defining Inertia Stack
Let $X$ be a topos and $F: \zeta \rightarrow X$ a stack on $X$. Now in the paper http://arxiv.org/pdf/math/0411337v2.pdf ( Definitions 2.1.1.5 and 2.1.1.1) of Lieblich, he describes the inertia stack ...
- 31
3
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334
views
Which sheaves on a projective bundle are flat over the base scheme?
Assume $X$ is a noetherian scheme over $\mathbb{C}$ and $E$ a locally free sheaf of finite rank on $X$. Denote the the associated projective bundle by $f: \mathbb{P}(E)\rightarrow X$.
Are there any ...
- 1,025
3
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0
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102
views
Can we extend a homotopy invertible chain morphisms between complexes of sheaves from a closed subspace to the whole space?
Let $X$ be a (say, topological) space and $i: Z\hookrightarrow X$ be a closed subspace. Let $Sh(X)$ and $Sh(Z)$ denote the categories of sheaves of abelian groups on $X$ and $Z$ respectively. ...
- 9,019
3
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155
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Elementary examples on sheaf extension
Let $V\subset\mathbb{P}^n$ be a projective variety and $C_V$ its conormal subvariety in $T^\ast\mathbb{P}^n$. Denote by $\mathscr{O}_{C_V}$ its structure sheaf, then when will the condition
$\mathit{...
- 3,187
3
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293
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Is this diagram of sheaves actually Cartesian as claimed?
The question is about Corollary 1.6.2 (b) in the book by Laumon and Moret-Bailly on algebraic stacks.
There we have a scheme $S$ and morphisms $X \xrightarrow{f} Y \xrightarrow{g} Z$ of sheaves on a (...
- 825
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0
answers
422
views
What kind of ringed space $X$ has the property that a locally free sheaf is projective in Qcoh$(X)$?
It is well known that for an affine scheme $X$, every finitely generated locally free sheaf $\mathcal{E}$ is projective in the category Qcoh$(X)$. i.e. the functor $\text{Hom}_{\text{Qcoh}(X)}(\...
- 9,019
3
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0
answers
217
views
Coherence of $\mathcal O_X[T]$
Let $X$ a complex manifold, and $\mathcal O_X$ the sheaf of holomorphic functions. Oka Coherence Theorem states that $\mathcal O_X$ is coherent (as $\mathcal O_X$-module).
How to prove that also the ...
- 31
3
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0
answers
115
views
Characterization of global sections (which are not products) of a sheaf which is locally a product
In order to compute certain group cohomology sets I have come upon a construction which seems rather general concerning sheaves which are locally products. So I will state the problem here in a ...
- 397
3
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0
answers
716
views
Two functorial definitions of schemes
I have been reading a bit about the "functor of points" theory for schemes. There seem to be two ways of going about defining schemes this way:
Equip the category $\textbf {Psh}=\operatorname{Fun}(\...
- 2,011
3
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0
answers
160
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 ...
- 1,595
3
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0
answers
260
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} \otimes^L_{...
- 7,527
3
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0
answers
152
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 ...
- 8,723
3
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0
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384
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$.
...
- 13.2k
3
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0
answers
334
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 "...
- 945
3
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0
answers
306
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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 ...
- 35.5k
3
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0
answers
134
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 set-...
- 393
3
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0
answers
877
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$:
$$
V\...
- 7,609
3
votes
0
answers
411
views
Two definitions of Čech cohomology
Hello,
I have found different definitions of Čech complex for sheaf $F$ od abelian groups on topological space $X$ with respect to the cover $\mathcal U$. One in Gelfand-Manin says to take product of ...
- 1,368
3
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0
answers
145
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 ...
- 91
3
votes
0
answers
341
views
Descent of singular cohomology
When proving that singular cohomology of an appropriate space $X$ equals sheaf cohomology of $X$ with "values" (does one say that?) in the sheaf $\mathbb{Z}_X$ of locally constant functions, the ...
- 319
3
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1
answer
467
views
Restriction of Ext sheaves on closed subschemes
Let $f:X\rightarrow C$ be a morphism, where $C$ is a smooth curve. For $t\in C$ let $i_t:X_t = f^{-1}(t)\rightarrow X$ be the inclusion of the fiber of $f$ over $t$, and let $\mathcal{F}$ a coherent ...
user61586
2
votes
2
answers
451
views
Is a submodule of the sheaf of sections of a smooth vector bundle necessarily finitely generated?
Let $X$ be a finite-dimensional smooth manifold, $\mathcal C^\infty(X)$ its algebra of smooth functions, $V\to X$ a finite-dimensional smooth vector bundle, and $\Gamma(V)$ the space of smooth ...
- 54.6k
2
votes
1
answer
242
views
Derived category of local systems of finite type on a $K(\pi,1)$ space: an explicit counterexample
Let $X$ be a nice enough topological space. I am mostly interested in smooth complex algebraic varieties. One may ask whether the bounded derived category of the category $\mathrm{Loc}(X)$ of local ...
- 1,071
2
votes
1
answer
752
views
why is counit called the trace map
Let $f: X \to Y$ be a morphism of schemes, then
$f_*$ and $f^*$ form an adjoint pair inducing natural
correspondence
$\text{Hom}_{\mathcal{O}_X}(f^*\mathcal{G},\mathcal{F})=
\text{Hom}_{\mathcal{O}_Y}(...
- 619
2
votes
2
answers
607
views
Canonical (tautological) section of a family of sheaves
A couple of months ago, i saw a construction, that somehow looks like the construction of the tautological section of the pullback of a vector bundle to its total space, i am trying to piece it ...
- 1,025
2
votes
1
answer
293
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Co-stalk of co-presheaves and cosheaves
Consider the co-presheaf $\mathcal{F}$ of continous real-valued functions with relatively-compact support on a topological space $X$. Consider a point $x\in X$.
1) When $\mathcal{F}$ is considered a ...
- 229
2
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2
answers
429
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If $\mathcal{F}$ globally generated, then counit map $f^*f_* \mathcal{F} \to \mathcal{F} $ surjective
Let $f: X \to S$ be a morphism, and $\mathcal{F}$
be quasi-coherent $\mathcal{O}_X$-module generated by global sections (eg if
$X$ projective, then this holds for the twisted sheaf $\mathcal{F}(n)$ ...
- 619
2
votes
1
answer
1k
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 $Hom(M,N)\ ?$
- 21
2
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2
answers
1k
views
Simplicial Sheaves?
I recently was wondering if there was a name for sheaves which were locally constant on the open simplexes in a simplicial complex. After some googling I stumbled across simplicial sheaves. I am ...
- 4,842
2
votes
2
answers
219
views
Continuous map with homeomorphic fibers whose associated $H^{k}_c$ sheaf is not a local system?
Let $ f: X \to Y$ be a continuous map between connected manifolds s.t. for all $y \in Y$ the fiber $f^{-1}(y)$ is homeomorphic to some fixed connected manifold $Z$.
Let $k$ be a ring and for every $...
- 7,789
2
votes
2
answers
2k
views
Serre's Theorem for Coherent Sheaves
I recently heard a discussion about a certain of Serre which reconstructs the category of coherent sheaves of a variety $V$ as the category of modules over the homogeneous space of $V$ modulo modules ...
2
votes
1
answer
2k
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 ...
- 329
2
votes
1
answer
151
views
Is the slice of a subcanonical site also subcanonical?
A subcanonical site is one for which every representable functor is a sheaf.
For a subcanonical site $C$, the fundamental theorem of topos theory says that there is an equivalence $Sh(C/c)\cong Sh(C)/...
- 237
2
votes
1
answer
218
views
About the support of a holonomic D-module
Let $X$ be a smooth algebraic variety over $\mathbb{C}$ and let $M^\bullet\in\mathsf{D}^b_\text{h}(\mathcal{D}_X)$ be a complex of D-modules with holonomic cohomologies. We define the support of $M^\...
- 711