All Questions
Tagged with ag.algebraic-geometry sheaf-theory
493 questions
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 $\...
- 28.2k
1
vote
0
answers
200
views
Question regarding affine fibre bundles
Let $f:X\to Y$ be a morphism of affine varieties such that it is a fibre bundle with fibre $F$. Let $\pi_1(Y)=\Gamma$ be a free group (non abelian) of finite rank and $\pi_1(F)$ is a finite group $G$ ...
- 585
1
vote
0
answers
215
views
Artin-Winters proof of semi-stable reduction theorem: details
I've been reading through Artin-Winters proof of the semi-stable reduction theorem (Degenerate fibers and stable reduction of curves) and found myself confused about the following detail—
Let $\...
3
votes
0
answers
530
views
Flasque sheaves on a site
This is a cross-post from MathStackexchange.
We define a flasque sheaf on a site as one whose first Čech cohomology vanishes for every covering of every object of the site. I know this definition is ...
- 15.9k
2
votes
1
answer
130
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Sheaves on families of genus 2 curves in Hassett's paper
Sorry for a maybe stupid long question but I'm reading the paper "Classical and minimal models of the moduli space of curves of genus two" by Brendan Hassett and I'm not able to unravel a ...
- 148k
1
vote
0
answers
355
views
Global section of pullback of an ideal sheaf
For a local ring $R$ with maximal ideal $\mathfrak{m}\subset R$ and residue field $\kappa$, and a flat morphism $f\colon X\rightarrow \mathrm{Spec} R$ of schemes, we consider the short exact sequence ...
- 119
18
votes
4
answers
6k
views
Derived categories of coherent sheaves: suggested references?
I am interested in learning about the derived categories of coherent sheaves, the work of Bondal/Orlov and T. Bridgeland. Can someone suggest a reference for this, very introductory one with least ...
- 4,713
3
votes
1
answer
249
views
Sheafifcation for the étale site
Let $X$ be a scheme and $\mathcal{F}$ a presheaf on $X_{ét}$.
For each $x_{i}\in X$, pick a geometric point $\bar{x}_{i}$ over $x$ and denote by $i_{\bar{x}_{i}}:\text{Spec}(k_{i})_{\text{ét}}\...
- 148k
15
votes
1
answer
1k
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Grothendieck - sheaves as meter sticks
I'm trying to read parts of McLarty's Grothendieck on Simplicity and Generality. In the article, I read Grothendieck thought of sheaves over some topological space as meter sticks measuring it.
...
CommunityBot
- 1
4
votes
1
answer
515
views
Sheaf cohomology in non-commutative setup
Let $X$ be a topological space and $A$ a sheaf of noncommutative associative algebras over a fixed field $k$. My questions are:
1) Does the category of modules over A have enough injective?
2) If we ...
CommunityBot
- 1
5
votes
0
answers
290
views
About the left adjoint of $f^*$
In lots of different cases (Verdier duality, Grothendieck duality, étale cohomology, ...) the very existence of a (right) adjoint to the sheaf functor $f_!$ gives useful information. (I'm going to ...
- 721
7
votes
1
answer
291
views
Direct and inverse image terminology
Let $f\colon X\to Y$ be a continuous map. Then $f$ induces a geometric morphism $f^\ast\dashv f_\ast\colon \mathrm{Sh}(X)\leftrightarrows\mathrm{Sh}(Y)$, whose left adjoint is called inverse image and ...
- 15.9k
5
votes
3
answers
680
views
Deequivariantisation of indecomposable sheaves
Let G be a connected group acting on a space X. All spaces should be reasonable, so e.g. G is a complex affine algebraic group acting on an algebraic variety X, with everything done using the usual ...
- 563
2
votes
0
answers
372
views
How to deduce Künneth from its relative version (in cohomology of sheaves)
Let $p:X\to S$ and $q:Y\to S$ be morphisms of "spaces" over $S$. We have an isomorphism
$$f_!(M\boxtimes N)=p_! M\otimes q_!N$$
in the derived category of "sheaves" over $S$, where ...
CommunityBot
- 1
4
votes
0
answers
278
views
Are manifolds "naturally" ringed or locally ringed spaces?
My question is about whether it's more natural to see manifolds as ringed spaces or as locally ringed spaces. I think I have arguments for both points of view.
On the one hand, it's reasonable to ...
- 721
2
votes
0
answers
167
views
Explicit construction of a presentation of a constructible sheaf of $\mathbb{Z}$-modules
This question was prompted by the two following:
Constructible étale sheaves on X are étale algebraic spaces over X
Naive question about constructing constructible sheaves
If I have a ...
- 617
1
vote
1
answer
305
views
Interesting examples of direct image bundles
Let $\pi : W \rightarrow Y$ be a holomorphic fibration of complex manifolds. Let $L\rightarrow W$ be a holomorphic line bundle on its total space and denote by
$$E^k_q := R^q \pi_*L^k$$
the direct ...
- 42.9k
5
votes
0
answers
250
views
Formality of a category of constructible sheaves
Let $X= S^1 \wedge S^1$ be a wedge of circles. Then $X$ admits a natural stratification $\mathcal{S}$ as a union of two disjoint open intervals $I_1, I_2$ and a point $\{*\}$.
Let $D_{\mathcal{S}}(X)$ ...
- 373
2
votes
1
answer
423
views
Purity of perverse cohomology sheaves
Let $f\colon X\to Y$ be a morphism of projective varieties over a finite field. Let $K$ be a perverse pure sheaf on $X$.
Are the perverse cohomology sheaves of $f_*(K)$ pure?
I am just learning the ...
- 148k
2
votes
1
answer
177
views
Are vector bundles acyclic for $\Gamma_c$?
Let $X$ be a paracompact topological space or a manifold (which is not a particular case since the structure sheaves are different). It is well-known that vector bundles (more generally, $\mathcal{O}...
- 40.2k
3
votes
0
answers
83
views
Do rationally contractible presheaves have rationally contractible injective resolution
Given a presheaf $\mathcal{F}: Sm/k\rightarrow Ab$ we define a new presheaf $C\mathcal{F}= \varinjlim\limits_{X\times \{0,1\}\subset U \subset X\times \mathbb{A}^1}\mathcal{F}(U)$. The presheaf $\...
- 63.9k
4
votes
2
answers
419
views
Why abelian sheaves instead of $\mathcal{O}_X$-modules in topology and étale stuff?
Most often than not, the sheaves appearing in algebraic geometry (with the Zariski topology) are $\mathcal{O}_X$-modules, instead of simple abelian sheaves.
Now, when dealing with topological spaces (...
- 156k
25
votes
5
answers
3k
views
Applications of the "other" definition of sheaves
In most literature, when you try to look for the definition of sheaves you will see the usual definition for presheaves as a functor from a topological space (or from a Grothendieck topology) to some ...
- 63.9k
2
votes
1
answer
474
views
How to find the smallest flabby sheaf containing a given sheaf?
None of the spaces $C^k(\mathbb{R}^n)$, with $0 \leq k \leq \infty$, is a flabby sheaf. However, they are respectively contained in the smallest flabby sheaves $C^k_{nd} (\mathbb{R}^n)$ of functions $...
- 19.6k
16
votes
6
answers
14k
views
What books should I read before beginning Masaki Kashiwara and Pierre Schapira's "Sheaves on Manifolds"
I am a beginner trying to learn about sheaves. I am trying to read Masaki Kashiwara and Pierre Schapira's book "Sheaves on Manifolds", but I find it is not easy for me to understand.
What ...
- 5,534
2
votes
0
answers
265
views
Why does nearby cycles of a local system on $\mathbb{G}_m$ have same monodromy as local system?
Apologies if this belongs on MSE, but none of the tags I wanted existed, so I took it as a hint to post on MO.
Edit: here is my definition of nearby cycles. Suppose $X$ is a complex analytic space ...
- 272
6
votes
2
answers
631
views
Geometric meaning of coherent sheaves $\mathcal{F} \otimes \mathcal{O}_{\mathbb{P}^n}(d)$ over $\mathbb{P}^n$
Maybe it sounds like a silly question but I'm not able to figure out in my head the geometric meaning of "twisting" vector bundles (or more generaly coherent sheaf) over $\mathbb{P}^n$.
I ...
- 9,650
2
votes
1
answer
383
views
Some facts about sheafification functor on étale site
I'm studying the book Etale cohomology and the Weil conjecture by Freitag, Kiehl and I have some questions on the subchapter introducing the machinery associating to an étale presheaf
a sheaf (that is ...
- 148k
6
votes
1
answer
727
views
What do nearby/vanishing cycles have to do with Fourier transforms?
Let $E$ be a vector bundle on some smooth algebraic variety and $E^*$ its dual. Suppose $A$ is a sheaf (constructible or a $D$-module) on $E$. Given a linear function $f$ on $E$, we may compute the ...
- 148k
2
votes
0
answers
115
views
About condition for structure sheaf of a scheme being compact object in a category of sheaf of module over X [duplicate]
I found the condition for one direction :
Categorical interpretation of quasi-compact quasi-separated schemes
This article said that if $X$ is quasi compact and quasi separated, $\mathscr{O}_X$ is a ...
- 121
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 $\...
- 4,492
1
vote
0
answers
172
views
When does a $D$-module think it’s a pullback along a smooth morphism?
Let $X$ and $Y$ be two algebraic varieties, and let $f: X \to Y$ be a morphism. Suppose $A$ is a holonomic $D$-module on $Y$. In this situation we can pull $A$ back to $X$ using either the $!$ or $*$ ...
- 3,019
2
votes
1
answer
271
views
Local extension of holomorphic vector fields
Let $X$ be an open complex manifold, e.g., the complement of a simple normal crossing divisor $D$ in a (smooth) projective manifold $M$. Let $T^{1,0}X$ be the holomorphic tangent bundle of $X$. Let $K ...
- 37.8k
2
votes
0
answers
158
views
Torsors for nonabelian groups and maps to contracted products
$\newcommand\op{\mathrm{op}}$My question concerns torsors for a sheaf of groups $G$ that is not commutative, and left/right are messing me up. A left $G$-torsor is equivalent to a right $G^{\op}$-...
- 63.9k
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
3
votes
1
answer
129
views
Infinite uniform dimension $\Rightarrow$ infinitely many idempotents in a localization of a quotient
Let $R$ be a commutative ring with $1$ such that its uniform dimension is infinity, equivalently, $$\sup\{k \mid R \text{ contains a direct sum of $k$ nonzero ideals }\}=\infty.$$ How can we ...
CommunityBot
- 1
1
vote
1
answer
749
views
Computing Ext sheaves over complex projective plane
Let $X:=\mathbb{P}^2_K$ with $K$ algebraically closed field. Take $p\in X$ a point and $\mathcal{I}_p$ its ideal sheaf. One can prove (using Serre Duality and the exact sequence defining $\mathcal{I}...
- 2,607
15
votes
1
answer
2k
views
How to motivate constructible sheaves
I'm writing some notes for some students which just finished a first course in scheme theory. There I would like to talk about constructible sheaves, but I found it hard to give a compelling ...
- 24.2k
2
votes
1
answer
332
views
Example of an Algebraic Space ("false" affine line with different tangents at origin)
I have a question about the following example from the Algebraic spaces and quotients by equivalence relation of schemes by Roy Mikael Skjelnes (page 12)
of a presheaf quotient, which
has associated ...
- 6,038
4
votes
1
answer
293
views
Functorial isomorphisms
$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Sh{Sh}\DeclareMathOperator\PSh{PSh}$We know that a presheaf $\mathcal{F}$ on category $ \mathcal{C} $ gives a colimit preasheaf $ \mathcal{F}^{+} $ ...
- 12.9k
4
votes
0
answers
318
views
Is the restriction of an injective sheaf on a closed subscheme still injective?
Let $X$ be a Noetherian scheme, and let $i:Z\to X$ be the inclusion of a closed subscheme $Z$. Let $\mathcal{I}$ be an injective sheaf of modules on $X$.
Question. Is $i^*\mathcal{I}$ still an ...
- 66.3k
1
vote
1
answer
186
views
Dual of stable vector bundle on a Fano threefold
Let $E$ be a rank $2$ stable vector bundle on a prime Fano threefold of genus $8$, with Chern numbers $c_1=1, c_2=6, c_3=0$.
Question. Is it true that $E(-1)=E^*$?
What I am able to show is that ...
- 66.3k
3
votes
0
answers
1k
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Saturation of sheaves
Let $(X, \mathcal{O}_X)$ be a complex manifold, which we can take to be projective. A coherent subsheaf $\mathscr{F}$ of some sheaf $\mathscr{G}$ is said to be saturated in $\mathscr{G}$ if the ...
- 651
1
vote
0
answers
105
views
Interesting property of a divisor contained in special fiber
Let $(A, \mathfrak{m}, \kappa=A/\mathfrak{m})$ be a local ring and $f:X \to \operatorname{Spec} (A)$
a scheme. Let $D \subset X$ a divisor on $X$ contained in special fiber $D \subset f^{-1}(\sigma_{\...
- 6,038
0
votes
1
answer
188
views
Support of a coherent sheaf over a fiber product scheme
I'm trying to prove the following fact which I don't know if it is true since I am not able to find a counterexample:
Let $X,S$ be two $K$-scheme of finite type with $K$ an algebraically closed field....
- 39.3k
3
votes
1
answer
258
views
Automorphisms of Frobenius liftings and degeneration of the Hodge-de Rham spectral sequence
I am still studying Deligne and Illusie's paper (https://eudml.org/doc/143480), and I am again stuck, this time on pages 262/263.
Assume $X\longrightarrow S$ is a smooth morphism of $\mathbb{F}_{p}$-...
- 1,805
1
vote
0
answers
131
views
Schur's lemma for sheaves with different reduced Hilbert polynomials
Recall Schur's Lemma for Gieseker-semistable sheaves, in particular the injectivity statement:
Let $\psi : F \to G$ be a morphism of Gieseker-semistable sheaves. If $p(F)=p(G)$ and $F$ is stable, ...
- 123
6
votes
1
answer
911
views
Putting sheaves to work for algebraic topology?
This is cross-posted from math.se after receiving points and no answers. I apologise if this question is too basic for MathOverflow.
I'm refreshing my memory of ...
- 37.8k
3
votes
1
answer
331
views
Degeneration of Hodge-de Rham spectral sequence, exactness of a pairing and the trace morphism
If $X\longrightarrow S$, where $S=\operatorname{Spec}(k)$, is a proper smooth morphism of dimension $p$ between $\mathbb{F}_{p}$ schemes with Frobenius $F$, then there is an exact pairing $\wedge:\...
- 1,829
3
votes
0
answers
460
views
Does isomorphism on local rings imply the global isomorphism for the sheaf of spectra?
Let's assume we have a sheaf of spectra on some scheme. As an example I will assume that we are working with the $K$-theory sheaf. There are certain local to global spectral sequences, like descent ...
- 5,901