All Questions
Tagged with sheaf-theory ag.algebraic-geometry
493 questions
13
votes
1
answer
1k
views
Sheaves on Contractible Analytic Spaces
Let $(X,\mathcal{O}_X)$ be a contractible complex analytic space. Suppose that $\mathcal{F}$ is a coherent sheaf of $\mathcal{O}_X$-modules. Can we invoke the fact that $X$ is contractible to conclude,...
- 4,920
13
votes
2
answers
3k
views
Wikipedia's definition of 'locally free sheaf'
Let $R$ be a, say, noetherian ring and $M$ an $R$-module. The Wikipedia article on 'locally free sheaf' tells me that the following two statements are equivalent:
The module $M$ is locally free (Edit:...
- 2,782
13
votes
1
answer
1k
views
How is a Stack the generalisation of a sheaf from a 2-category point of view?
A stack is usually given in terms of:
-A category $F$ fibered over another $C$ such that the functor $Hom(x,y), x,y \in F(\alpha), \alpha \in C$ is a sheaf
-The descent data are effective.
There ...
- 433
13
votes
2
answers
3k
views
Interpreting $f^*f_*$
For a morphism of schemes $f: X\rightarrow Y$, one often considers the function $f^*f_*$ on sheaves. For example, this appears in adjunction for sheaves of $\mathcal{O}_X$-modules, the projection ...
- 3,555
13
votes
1
answer
583
views
Which nice/deep elaborations on the (operators <-> sheaves) / (endomorphisms <-> objects) theme are there?
A linear operator $T:V\to V$ on a (say) vector space over a field $k$ is just a $k[T]$-module, and may be viewed as the sheaf $\mathscr F_T$ over $\mathbb A^1_k$, with fibre over $\lambda\in k$ equal ...
- 18.4k
13
votes
0
answers
524
views
Is the morphism of sheaves $(R \mapsto GL(R((h)))) \rightarrow (R \mapsto PGL(R((h))))$ surjective in Zariski topology?
Consider two functors given by $R \mapsto GL(R((h)))$ and $R \mapsto PGL(R((h)))$ for a ring $R$. It is easy to see that these functors are sheaves in Zariski topology (in fact for any affine variety $...
12
votes
1
answer
2k
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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 ...
- 545
11
votes
3
answers
6k
views
Serre's FAC versus Hartshorne as an introduction to sheaves in algebraic geometry
I just found an English translation of Serre's FAC at Richard Borcherds' Algebraic Geometry course web page. I really want to read it sometime. I am beginner in Algebraic Geometry, just started ...
11
votes
5
answers
8k
views
When is the push-forward of the structure sheaf locally free
Let $f:X\longrightarrow Y$ be a morphism of noetherian schemes. Under what conditions is $f_\ast \mathcal{O}_X$ a locally free $\mathcal{O}_Y$-module?
Example 1. Suppose that $f$ is affine. Then $f_\...
- 9,497
11
votes
4
answers
5k
views
When does the sheaf direct image functor f_* have a right adjoint?
Say f: X → Y is a morphism of schemes. The sheaf direct image functor f★ always has a left adjoint, namely the sheaf inverse image functor f★ (with tensoring).
Under what (...
- 11.3k
11
votes
3
answers
935
views
Examples of calculating perverse sheaves on algebraic varieties with easy stratification
I have been learning intersection homology and perverse sheaves in the following way. I started by reading the first $7$ chapters of Kirwan and Woolf's book
http://www.amazon.com/Introduction-...
- 123
11
votes
2
answers
2k
views
Finiteness conditions on simplicial sheaves/presheaves
Could someone give an overview, or just some examples, of "finiteness conditions" for simplicial sheaves/presheaves and/or simplicial schemes? Any answer or comment about this would be interesting, ...
- 5,637
11
votes
2
answers
1k
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Is there a relation between Gelfand duality and the spectrum of a ring (with its Zariski topology)?
Compare the following two results:
Thm A) Let $A$ be a commutative $C^*$-algebra and let $X$ be its Gelfand spectrum. Gelfand duality says that there's a natural isometric $*$-isomorphism from $A$ to ...
- 721
11
votes
1
answer
2k
views
Locally free extension of locally free sheaf
Given a coherent sheaf $F$ on a smooth variety $X$, we know that $F$ is locally free on an open subset $U$ in $X$ outside a codimension two subset. Say the rank is $k$.
Is there a locally free sheaf $...
- 457
11
votes
1
answer
1k
views
Etalé 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 (cf....
- 623
11
votes
1
answer
812
views
Understanding the purely formal part of the sheaf theoretic (cohomological) framework for representation theory
By now I have the impression that many statements in representation theory can be phrased a lot more elegantly using cohomological language. Therefore I'm trying to understand a bit better the sheaf ...
- 7,789
11
votes
1
answer
856
views
Sheaf associated to presheaf Aut
Let $S$ be a scheme and let $C$ be the category of schemes flat and locally of finite presentation over $S$. Endow $C$ with the fppf topology (or perhaps any subcanonical topology). Let $\mathcal P$ ...
10
votes
4
answers
4k
views
Why is an injective quasi-coherent sheaf's restriction to an open subset still an injective object?
X is a Noetherian scheme, F is an injective object in the category of quasi-coherent sheaves on X. U is an open subset of X. Why F's restriction on U is still an injective object in the category of ...
- 1,091
10
votes
2
answers
5k
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 ...
- 3,830
10
votes
2
answers
2k
views
Geometric interpretation of sheaf cohomology
Please forgive me for the informal and naïve nature of my question, as I am a beginner in algebraic geometry.
In the famous book by Hartshorne, sheaf cohomology is defined as a certain derived functor....
- 328
10
votes
2
answers
1k
views
Is the ideal product presheaf a sheaf? Do we have any reasons to believe it will be / it won't?
My question is about one of those several concepts in algebraic geometry who everybody uses but nobody defines or introduces properly.
Given a ringed space $(X,\mathcal{O}_X)$ and ideal sheaves $\...
10
votes
1
answer
777
views
Why is there no stack of $\ell$-adic sheaves on a curve?
One of the main players in the categorical geometric langlands correspondence is the moduli stack of rank n integrable connections on a complex curve. The reason for considering such objects is that ...
- 101
10
votes
1
answer
495
views
Properties of the petit Zariski topos
What are some (intrinsically formulated) properties of the locally ringed topos $(\mathbf{Sh}(X),\mathcal{O}_X)$ for some scheme $X$, which do not hold for arbitrary locally ringed toposes?
Is there, ...
- 5,482
10
votes
3
answers
2k
views
Representable Presheaf
I have a very quick question. Is there an easy example of a representable presheaf on a site that is not a sheaf? This certainly can't happen on a small FPPF site so I would expect a counterexample to ...
- 565
10
votes
1
answer
529
views
Polylogarithm sheaves
In many different places, I could find the notion on ''(poly)logarithm sheaves''. As is indicated in the name of it, I guess that it should have something to do with (poly)logarithm function: $\mathrm{...
- 1,428
10
votes
0
answers
361
views
How to model (affine) schemes with a large sketch?
Guitart states in "Toute theorie est algebrique et topologique" as Proposition 17 that the category $\mathbf{Sch}$ of schemes is the category of models of a large mixed sketch. Presumably, ...
- 63.1k
10
votes
0
answers
958
views
intuition about perverse sheaves
firstly, I would know if my very basic intuition on perverse sheaves is correct .
secondly, I would have some clarification in what perverse sheaves behaves better than regular sheaves .
my intuition ...
- 546
9
votes
1
answer
1k
views
Kozsul resolution of $\mathcal{O}_X$
Let $i: X \hookrightarrow Y$ be a closed embedding of smooth algebraic varieties. In the book D-modules, perverse sheaves and representation theory the authors say that there exists a locally free ...
- 1,325
9
votes
1
answer
400
views
The (co)tangent sheaf of a topological space
Let $X$ be a topological space (assume additional assumptions if needed) and denote by $\mathcal O _X$ its sheaf of $\Bbbk$-valued continuous functions where $\Bbbk$ is $\mathbb{R}$ or $\mathbb{C}$ ...
- 23.3k
9
votes
1
answer
447
views
Results in “generalised smooth spaces” that did not hold in the case of smooth manifolds
Consider the category of smooth manifolds $\text{Man}$. I quote from n-lab page:
Manifolds are fantastic spaces. It’s a pity that there aren’t more of them.
I understand that this category $\text{...
- 6,023
9
votes
1
answer
804
views
Is the analytification functor part of a geometric morphism of topoi?
Let $Sh(\mathsf{\mathbb{C}-fAlg}^{op})$ be the topos of zariski sheaves on finitely genertaed $\mathbb{C}$-algebras.
A complex analytic space for our purpose is a locally ringed space locally ...
- 7,789
9
votes
1
answer
4k
views
surjective morphism of schemes or epimorphism of sheaves?
I have a technical question coming from reading Toen's master course on stacks.
If we view schemes as locally ringed spaces then there we could define a morphism to be surjective if it the underlying ...
- 1,889
9
votes
1
answer
426
views
How to view $\textbf{Sh}(\textbf{CartSp})/X$ as "space" in its own right, étale machinery from abstract nonsense perspective for smooth manifolds
Let $\textbf{CartSp}$ be the category of spaces of the from $\mathbb{R}^n$ with smooth maps between them. This is a site with respect to (differentially) good open covers, so consider the Grothendieck ...
- 221
9
votes
0
answers
378
views
Is there any notion of "smoothification" from $\mathbb{R}$-schemes to generalized smooth spaces?
I will write $\operatorname{Diff}$ to denote a category of generalized smooth spaces e.g. $Sh(\mathsf{CartSp})$. Is there a version of $\operatorname{Diff}$ for which there exists a functor $\mathcal{...
- 7,789
8
votes
2
answers
4k
views
Closed subschemes and pulling back the structure sheaf via the inclusion map
I would just like a clarification related to closed subschemes.
If $(X,{\cal O}_X)$ is a locally ringed space and $A\subset X$ is any subset with the subspace topology then $i^{-1}{\cal O}_X$ will be ...
- 2,814
8
votes
2
answers
684
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 $\...
- 3,758
8
votes
2
answers
4k
views
Sheaf cohomology question
For a topological space $X$ and a sheaf of abelian groups $F$ on it, sheaf cohomology $H^n(X,F)$ is
defined.
Singular cohomology of $X$ can be expressed as sheaf cohomology if $X$ is locally ...
- 221
8
votes
1
answer
2k
views
Sheaves of $\mathbb Z$-modules = sheaves of abelian groups
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 take $\...
- 1,368
8
votes
2
answers
2k
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 $X$, ...
- 6,792
8
votes
1
answer
1k
views
Fpqc sheafification and localisation
I am slightly confused about sheafification at the moment.
I first learned sheaves defined as a subcategory of presheaves, then I was told that sheaves are also a localisation of presheaves, then I ...
- 1,273
8
votes
1
answer
529
views
Topology on the space of constructible sheaves
Let $X$ be a nice compact topological space with a fixed finite stratification by locally closed topological manifolds. At the beginning one may assume that $X$ is a complex algebraic manifold with ...
- 21.8k
8
votes
1
answer
319
views
How are the left and the right group of a bitorsor related?
This question arose from my answer to To what extent does a torsor determine a group: it turns out that I do not know one thing about it.
Let $G$, $G'$ be groups in some nice enough category (you may ...
- 18.4k
8
votes
1
answer
1k
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 $...
- 16.9k
8
votes
1
answer
562
views
Category of copresheaves over commutative monoids
Let C be a symmetric monoidal category. Let Comm(C) be the category of commutative monoids in C. Consider the topos X = CoPSh(Comm(C)) of covariant functors from Comm(C) to the category Set of sets.
...
- 2,399
8
votes
0
answers
644
views
Trying to understand "Shtukas"
I'm studying Goss' Basic structures of function field Arithmetic, chapter 6 about Shtukas. I'm trying to understand some details about some concepts. This chapter is based on a Mumford's paper An ...
- 69
8
votes
0
answers
333
views
Who introduced the notion of ringed spaces?
My question is very concise, please forgive it.
Who introduced the concept of ringed space?
My first try would be that they were introduced by Cartan in his study of analytic functions with sheaves. ...
- 894
8
votes
0
answers
681
views
Stalks of limit sheaves
Let $\{\mathcal{F}_i\}_{i\in \mathbb{N}}$ be an inverse system of sheaves of abelian groups on a space $X$. Then for any $x\in X$ we have a natural map
$$\left(\lim_i \mathcal{F}_i\right)_x\rightarrow ...
- 4,418
8
votes
0
answers
470
views
Sheaf whose singular support is not Lagrangian
For constructible sheaves $\mathcal F$ on real analytic manifolds $X$, there is a notion of the singular support $SS(\mathcal F)$ which is a radially invariant singular Lagrangian subset of the ...
- 18.7k
7
votes
1
answer
2k
views
The Serre duality theorem intuition
It is a well known fact that proper scheme $X$ over $k$ has a up to isomorphism unique dualizing sheaf (EGA I, Hartshorne).
This dualizing sheaf $\omega_X$ comes with two striking properties:
(i) ...
- 6,038
7
votes
2
answers
458
views
A technical question about derivations of sheaves on group schemes
Let $G$ be a group scheme (for instance, over $k$ a field of characteristic 0).
Let $e$ be its unit.
I denote by $O_G$ the structural sheaf of $G$.
Let $D_e : O_{G,e} \to k$ a derivation.
I would ...
- 1,320