All Questions
Tagged with ag.algebraic-geometry sheaf-theory
493 questions
5
votes
0
answers
904
views
Injective sheaf of $\mathcal{O}_X$ modules
I am trying to verify that:
A sheaf of $\mathcal{O}_X$ modules $\mathcal{F}$ is an injective object in the category of $\mathcal{O}_X$ modules iff its local rings $\mathcal{F}_x$ are injective $\...
user37663
2
votes
0
answers
126
views
Local cohomology with supports in a constructible set
Let $X$ be a topological space (I'm interested in the case of $X$ being a complex algebraic variety with the Zariski topology) and $Z$ a constructible subset (i.e. a finite union of locally closed ...
- 3,079
2
votes
0
answers
101
views
Sheaves of functions on open semi-algebraic sets
Let $U\subsetneq\mathbb{R}^n$ be an open semi-algebraic set which is not Zariski open (the case I have in mind is that of an open ball $B(0,1)$). A function $f:U\rightarrow \mathbb{R}$ is called
(1) ...
- 953
2
votes
0
answers
35
views
If the set of non-0 stalks of F is relatively open, is the same true of its Verdier dual?
Let $X$ be a complex manifold, $F$ a bounded complex of $\Bbb C_X$-modules with constructible cohomology. If the set $\{x: F_x\neq0\}$ is relatively open (i.e. open in its closure), is the same true ...
- 3,079
2
votes
1
answer
177
views
Is the restriction of a simple sheaf of modules simple?
Let $X$ be a topological space, $A$ a sheaf of (unital and associative but not necessarily commutative) rings on $X$. Suppose $M$ is a simple quasicoherent $A$-module and $U$ an open subset of $X$. Is ...
- 3,079
7
votes
0
answers
574
views
What is the geometric intuition for the sheaf-theoretic terms "soft", "fine", and "flabby"?
The sheaf-theoretic terms "soft", "flabby", and "fine" are of an obviously geometric character, and suggest opposition with "hard", "rigid", and "coarse" sheaves (I'm just inventing these terms here).
...
- 6,025
14
votes
1
answer
702
views
Donaldson and DT invariants
Let $X$ be a smooth projective surface. Then, using the compactified moduli space of anti self-dual connections or torsion free sheaves we can construct Donaldson invariants of $X$. Similarly, one can ...
- 587
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
7
votes
2
answers
839
views
What is the need for torsion in the definition of lisse sheaves?
I am studying the basics of constructible and lisse sheaves, and am trying to understand SGA 4, IX. As Grothendieck himself observes at the beginning of the chapter, one is forced to work with torsion ...
- 5,670
15
votes
2
answers
2k
views
Meaning of the determinant of cohomology
The Arakelov intersection number on arithmetic surfaces is defined as an "extension" of the classical intersection number on algebraic surfaces. It was introduced to get a nice intersection theory ...
- 321
2
votes
0
answers
258
views
Proj construction and pushforward of line bundles
Let $X$ be a variety of dimension $d \geq 2$ (over a field), consisting of two irreducible components meeting transversely in a divisor $D$. (We can assume these components and $D$ are as nice as we ...
- 21
4
votes
0
answers
536
views
When is a coherent subsheaf determined by its global sections
I am reading an article in which a proof is based on defining a subsheaf by only giving its global sections.
The exact setting is that, one has a surjective finite morphism $f:Y\to X$ between ...
- 333
2
votes
0
answers
169
views
Restriction of scalars from an Azumaya algebra and preservation of perfect/compact objects of the derived categories
An Azumaya variety over a field is by definition a pair $(X,\mathcal A_X)$, where $X$ is an algebraic variety of finite type over that field and $\mathcal A_X$ is a sheaf of Azumaya algebras, namely ...
- 2,079
2
votes
0
answers
397
views
Terminology for "global sections" when sheaf is valued in general category
Let $\mathcal F$ be a sheaf (say on a topological space $X$) valued in some category $\mathcal C$.
What do we call $\mathcal F(X)$?
When $\mathcal C$ is some vaguely linear category (e.g. the ...
- 18.7k
0
votes
0
answers
182
views
Analytic-Local Germs of "General" Sections
Let $C$ be an algebraic curve over an algebraically closed field $k$ of characteristic $0$, and let $\mathcal{L}$ be a base-point-free line bundle on $C$. Furthermore, let $p \in C$ be a smooth point, ...
24
votes
1
answer
837
views
Is there a useful theory of D-modules on smooth (non-analytic) manifolds?
D-modules are related to flat connections on vector bundles, end hence local systems. The theory of D-modules (and related notions such as crystals etc.) seems to be very popular in complex analytic ...
- 6,025
0
votes
0
answers
193
views
About the definition of flat twisted sheaves
Flat twisted sheaves are mentioned in Căldăraru's thesis (Lemma 2.1.2 for example), but I'm confused about how they should be defined. I have in mind some possibilities, given an $\alpha$-twisted ...
- 2,079
0
votes
0
answers
303
views
Sheaves invariant for group actions two equivalent definitions?
Given a (topological) group acting on $X$ a topological space continuously.
Then we have the category $Sh_G (X)$, it's the full subcategory of $Sh(X)$ consisting on the sheaves $\mathcal{L}$ such ...
- 335
11
votes
1
answer
855
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 $\mathcal P$ ...
4
votes
1
answer
895
views
When does derived pullback commute with infinite products?
Let $f:X \to Y$ be a morphism of reasonable schemes (qcqs). Let $f^*: D(Y) \to D(X)$ be the pullback defined on the derived unbounded categories of quasi-coherent sheaves.
Question: When does $f^*$...
- 7,789
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
3
votes
1
answer
203
views
Sheaves with $\mathbb{R}$-constructible proper direct image closed under dualizing?
I learned about $\mathbb{R}$-constructible sheaves very recently, so I hope this isn't a silly question:
Let $M$ be a real analytic manifold, and $U\subset M$ an open subanalytic subspace. Denote the ...
- 163
6
votes
1
answer
798
views
Example of non-holonomic D-module and explicit computation of characteristic variety
I'm currently trying to have a better understanding of the concepts of characteristic variety and holonomic $D$-modules (let us assume that they are coherent) on a holomorphic manifold $X$. I know ...
- 1,017
3
votes
1
answer
479
views
K-injective (also known as hoinjective) complexes of sheaves of modules
Let $(X,\mathcal O_X)$ be a ringed space (if necessary, assume that it is a scheme with suitable hypotheses). Given two complexes of sheaves $\mathcal F$ and $\mathcal G$ of $\mathcal O_X$-modules, ...
- 2,079
2
votes
1
answer
297
views
Nearby cycle functor for a family of stable curves
Let $B$ be a smooth algebraic curve over $\mathbb{C}$ (or rather a germ of it at a point $b\in B$). Let $f\colon E\to B$ be a proper flat family of stable curves with smooth generic fiber. Assume that ...
- 21.8k
1
vote
1
answer
310
views
When is a ring or algebra a ring/algebra of functions?
Note: For the record, exterior algebras and derivations are irrelevant to my question. However, I have a hard time assessing what I want to ask and I find it is the easiest to do so using a direct ...
- 3,292
2
votes
0
answers
1k
views
Chern Classes: two approaches
The following question is closely related to this one.
Let $X$ a non singular projective variety over $\mathbb C$, and let $\mathscr E$ a locally free sheaf of rank $r$ (an algebraic vector bundle), ...
- 1,237
18
votes
3
answers
2k
views
Can $\mathcal O_X$ be recognized abstract-nonsensically?
This question has been asked by Teimuraz Pirashvili many years ago. I forgot about it after a while and remembered only now by accident. He probably knows the answer by now, but I still don't.
In the ...
- 18.4k
16
votes
3
answers
3k
views
Physical interpretations/meanings of the notion of a sheaf?
I fairly understand the fiber bundles, both the mathematical concept of fiber bundles and the physics use of fiber bundles. Because the fiber bundles are tightly connected to the gauge field theory in ...
- 10.5k
26
votes
1
answer
1k
views
Why there is a Quot-scheme, not a Sub-scheme?
Let $X$ be a projective variety, and $E$ be a coherent sheaf on $X$. Grothendieck has proven that there is a scheme $\mathrm{Quot}_X(E)$ parametrizing arbitrary quotient sheaves of $E$. It is probably ...
- 1,980
2
votes
0
answers
163
views
Lifting a local section to a global section along a homomorphism of quasi-coherent sheaves
If $X$ is a scheme, is it always possible to find a basis $\mathcal{B}$ for the topology of $X$ (for example, the affine open subsets) with the following property?
For every quasi-coherent sheaf $M$...
- 5,482
1
vote
1
answer
507
views
Relation between local cohomology and open immersions
Let $X$ be a noetherian scheme $U \subset X$ an open subset with complement $Z = X- U$. Assume $Z$ is cut out by the ideal sheaf $\mathcal{I} \subset \mathcal{O}_X$. We have exact sequences:
$$0 \to \...
- 7,789
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
3
votes
0
answers
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
5
votes
1
answer
1k
views
"Role" of cohomology of coherent sheaves in SGA 4.5, étale cohomology
As the question title suggests, what is the role cohomology of coherent sheaves plays for SGA 4.5, étale cohomology? Why are they so important for the construction and establishing properties of étale ...
user97565
0
votes
0
answers
265
views
Explicit adjunction formula and local top form
I am reading the section 4.2 in Kollar-Mori, where they provide the explicit equations for Du Val Singularities. In the course of the proof, they reduce to studying the equation $x^2+f(x,y)=0$ in a ...
- 625
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
2
votes
0
answers
272
views
Double dual of ample sheaf
Let $X$ be a projective manifold. Then we can define ample sheaves on $X$, and many results of ample vector bundles still hold in this more general case (See K. Kubota, Ample sheaves).
Now I was ...
- 147
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
2
votes
1
answer
163
views
ample subsheaf contained in the tangent bundle of projective space
Let $\mathcal F$ be an ample subsheaf of $T_{\mathbb P^n}$. Is it actually locally free? If not, is there a counterexample?
- 147
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
0
votes
0
answers
191
views
First sheaf cohomology $H^1(\mathscr{O}_D, \mathbb{D})=0$
Given a finite divisor$$D=p_1+\dots +p_m -q_1 -\dots -q_n$$on the unit disk $\mathbb{D}$, does it necessarily follow that the first sheaf cohomology group equals zero, i.e.$$H^1(\mathscr{O}_D, \mathbb{...
user76513
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
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
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
5
votes
0
answers
310
views
Extension of ample vector bundles is ample
As I read Huybrechts-Lehn's book on Moduli of Sheaves, it is making a claim that extensions of several (at least 2) ample vector bundles (on curves) is again ample. Somehow, I am unable to see this ...
- 159
3
votes
1
answer
159
views
Homology in the sections of an infinite exact sequence of injective sheaves of $\mathcal O_X$-modules?
Let $(X, \mathcal O_X)$ be a scheme and the following an infinite, exact sequence of injective sheaves of $\mathcal O_X$-modules:
$$
\cdots \overset{f_5}\longrightarrow I_5\overset{f_4}\longrightarrow ...
- 355
3
votes
0
answers
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 ...
1
vote
0
answers
236
views
Canonicity of Čech cohomology
For a topological space $X$, consider the Leray covering $U_\lambda$ (i.e. $\cap U_\lambda$ is sufficiently fine, e.g. affine for Zariski topology) of $X$.
For a sheaf $F$ on $X,$ the cohomology $H^...
- 781
6
votes
2
answers
957
views
An apparent equivalence of the category of affine schemes over $S$ and the category of quasi-coherent $\mathcal{O}_S$-algebras
I had asked something very similar before on math.se (deleted now) but unfortunately it hadn't received a lot of attention. I decided to re-ask here.
Let $S$ be a fixed scheme. Is the following true?
...
- 7,789