All Questions
Tagged with ag.algebraic-geometry sheaf-theory
493 questions
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193
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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
11
votes
1
answer
856
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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 $\mathcal P$ ...
- 42.4k
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0
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303
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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
16
votes
3
answers
5k
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Do we have non-abelian sheaf cohomology?
Lets $X$ be a complex manifold (algebraic variety), $N$ an integer, and consider the sheaf $F$ defined by:
$F(U)$ ={ holomorphic maps $f: U\rightarrow GL(N,\mathbb{C})$ } with multiplicative ...
- 35.5k
4
votes
1
answer
895
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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 $f^*$...
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6
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1
answer
798
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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 ...
- 1,488
3
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1
answer
479
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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
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1
answer
297
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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 ...
- 16.1k
1
vote
1
answer
310
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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 ...
- 42.8k
2
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0
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1k
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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), ...
CommunityBot
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1
vote
1
answer
507
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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
18
votes
3
answers
2k
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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 ...
CommunityBot
- 1
16
votes
3
answers
3k
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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 ...
- 1,996
26
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1
answer
1k
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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 ...
- 6,262
10
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1
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495
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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, ...
CommunityBot
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3
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0
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978
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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
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1
answer
1k
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"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 ...
- 12.9k
11
votes
3
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935
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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-...
- 27.9k
0
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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
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1
answer
812
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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 ...
CommunityBot
- 1
2
votes
0
answers
272
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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
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1
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529
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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{...
- 17.4k
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^...
- 8,529
2
votes
1
answer
163
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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?
- 23k
6
votes
2
answers
957
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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?
...
CommunityBot
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9
votes
0
answers
378
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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{...
- 26.3k
3
votes
1
answer
1k
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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 $p\...
- 19.3k
45
votes
8
answers
14k
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How should one think about sheafification and the difference between a sheaf and a presheaf
The first time I got in touch with the abstract notion of a sheaf on a topological space $X$, I thought of it as something which assigns to an open set $U$ of $X$ something like the ring of continuous ...
- 251
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0
answers
191
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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{...
CommunityBot
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9
votes
1
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804
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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 ...
CommunityBot
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8
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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 ...
- 4,949
5
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0
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310
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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
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1
answer
159
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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 ...
- 16.1k
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
2
answers
276
views
Sections of a sheaf of differentials on a weighted complete intersection
Let $X\subset\mathbb{P}(a_0,...,a_N)$ be a smooth $n$-dimensional weighted complete intersection in a weighted projective space $\mathbb{P}(a_0,...,a_N)$.
Is it true that if $q\geq 1$ then $H^0(X,\...
- 776
3
votes
2
answers
489
views
Application of sheaves theory in ring theory
Is there any text that gives some applications of sheaves theory in commutative ring theory? In the other word, is any results in commutative ring theory that be verified by sheaves method?
- 11
6
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1
answer
2k
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Information and intuition packed in the Chern character for coherent sheaves
even after quite some time learning it, I still get somehow puzzled by the Chern character. Let me recall some stuff to get notation and setting.
Let us consider a smooth projective algebraic variety ...
- 1,030
13
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1
answer
1k
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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,...
- 66.3k
3
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0
answers
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
4
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1
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409
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Does the nearby cycle functor commute with the Verdier duality?
I would be interested to know the answer to the above question for the constructible bounded derived category on complex analytic or complex algebraic manifolds (or some other context). A reference ...
- 40.2k
6
votes
1
answer
752
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Sections of the conormal bundle
Let $X\subset\mathbb{P}^N$ be a quadratic manifold. That is $I(X)$ is generated by quadratic polynomials $Q_1,...,Q_m$.
Let $\mathcal{I}_X$ be the ideal sheaf of $X$ and $\mathcal{I}_X/\mathcal{I}_X^...
- 38k
10
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2
answers
5k
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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 ...
CommunityBot
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3
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0
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334
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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 ...
CommunityBot
- 1
3
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1
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515
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For what kind of sheaves can we always extend a sheaf map from a closed subset to the whole space?
Let $X$ be a topological space. We know that a sheaf on $X$ is call soft if for any closed subset $Z$ of $X$, a section on $Z$ can be always extend to a section on $X$.
Now we consider a similar ...
- 5,534
3
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0
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102
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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. ...
CommunityBot
- 1
3
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0
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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
2
votes
0
answers
272
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local universal sheaf (moduli of stable sheaves)
I do not know much about moduli of sheaves and I wanted to shows that for a smooth (projective) family over a discrete valuation ring of mixed characteristics (relative dimension 3), the locally free ...
- 101
13
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1
answer
583
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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 ...
CommunityBot
- 1
2
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0
answers
231
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Surjectivity locus of a morphism of families of sheaves
Let $X$ and $T$ be schemes and assume we have two coherent sheaves $\mathcal{F}$ and $\mathcal{G}$ on $X\times T$ which are flat over $T$, that is these are families of sheaves parametrized by $T$.
...
- 1,025
2
votes
2
answers
607
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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 ...
- 1,030