All Questions
Tagged with sheaves or sheaf-theory
979 questions
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
8
votes
0
answers
750
views
What's the definition of a microlocal sheaf?
I'm slowly becoming familiar with what microsupport of a sheaf is, but none of the references I've seen give a definition of what a microlocal sheaf should be in general.
In this paper of ...
- 191
10
votes
1
answer
460
views
Taking the category of sheaves is symmetric monoidal
Let $M$ and $N$ be topological spaces.
Let $\operatorname{Sh}(M)$ denote the presentable $\infty$-category of space-valued sheaves on $M$.
It seems to me that the equivalence
$$\operatorname{Sh}(M) \...
- 525
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 $\...
- 5,901
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 ...
- 6,028
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 ...
- 3,019
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
2
votes
1
answer
501
views
Higher direct image with compact support of a constant sheaf
Let $f: X \to Y$ be a locally trivial fibration between locally compact spaces with fiber $F$. It is well known that for a constant sheaf $A_X$ on $X$, the higher direct images $R^n f_* A_X$ are ...
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 ...
user105074
18
votes
2
answers
2k
views
What are the points (and generalized points) of the topos of condensed sets?
What are the topos points of $\mathrm{CondSet}$, i.e. the geometric morphisms $\mathrm{Set} \to \mathrm{CondSet}$?
More generally, is there a concise description of the geometric morphisms $\mathcal{E}...
- 1,110
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}$-...
- 1,084
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 ...
- 1,343
1
vote
1
answer
202
views
Decomposing an analytic function into two functions which vanish at $0$ and $\infty?$
This question comes from exercise I-10 of The Geometry of Schemes by Joe Harris (although this question is not about schemes). It is translated to less abstract language below:
Consider the Riemann ...
- 1,755
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
225
views
Sheaves on sites given by a (regular) cd-structure
Let $C$ be a category equipped with a Grothendieck topology generated by a cd-structure (see https://ncatlab.org/nlab/show/cd-structure or Voevodsky's paper Homotopy theory of simplicial presheaves in ...
- 33
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 ...
- 711
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 ...
- 6,028
9
votes
0
answers
308
views
Refinement of hypercovers by ordinary covers
I am asking for references and discussions of statements of the form
Every bounded hypercover can be refined by an ordinary cover
By "bounded" I mean "finite height". E.g., are ...
- 4,519
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}^{+} $ ...
- 41
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 ...
- 1,479
9
votes
2
answers
736
views
Does inclusion from n-stacks into (n+1)-stacks preserve the sheaf condition?
I'm going to describe two situations that seem to contradict each other, and I'm interested to know precisely what's wrong with this reasoning.
Let $M$ be a manifold, and consider the presheaf $C^*(-,...
- 15.4k
29
votes
3
answers
4k
views
Is there a good general definition of "sheaves with values in a category"?
Let $\mathcal{A}$ be a category.
There is a common definition of "sheaves with values in $\mathcal{A}$", which is what one obtains by taking the Grothendieck-style definition of "sheaf ...
- 15.9k
3
votes
0
answers
1k
views
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
5
votes
0
answers
154
views
Sheaf-like reconstruction of a continuous function
Let $X$ and $Y$ be topological manifolds and let $\{(\phi_x,U_x)\}_{x \in X}$ and $\{(\psi_y,Y_y)\}_{y \in Y}$ be respective atlases of $X$ and $Y$; with each $\phi_x:U_x\rightarrow \mathbb{R}^n,\...
- 5,405
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_{\...
- 6,028
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 ...
- 565
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....
- 395
6
votes
1
answer
414
views
Is there a Grothendieck correspondence for sheaves/stacks?
Given a category $\mathcal{C}$, the category of elements functor sets up an equivalence of categories
$$
\mathsf{DFib}(\mathcal{C})
\cong
\mathsf{PSh}(\mathcal{C}),
$$
whereas the Grothendieck ...
- 11.8k
4
votes
1
answer
191
views
Checking that (hyper) sheafification is fibrant in local projective model structure on simplicial presheaves
Context and Notation
Let $X$ be a manifold and $\mathcal{C} = Op(X)$ be the category of open subsets of $X$ with inclusions. I then consider the projective model structure on the (simplicial model) ...
- 1,611
1
vote
1
answer
116
views
Disjoint union of clopen sets such that the fibers has constant cardinality [closed]
Let $Z$ a compact set and $X$ a locally compact set. Let $p:Z\to X$ a local homeomorphism. Show that there exists $n≥1$ and $U_1,…,U_n$ open and closed sets of $X$ such that :
$X=\sqcup_{i=1}^{n}U_i$
...
- 53
1
vote
1
answer
749
views
Computing Ext sheaves over complex projective plane
Let $X:=\mathbb{P}^2_K$ with $K$ algebraically closed field. Take $p\in X$ a point and $\mathcal{I}_p$ its ideal sheaf. One can prove (using Serre Duality and the exact sequence defining $\mathcal{I}...
- 395
1
vote
0
answers
131
views
Schur's lemma for sheaves with different reduced Hilbert polynomials
Recall Schur's Lemma for Gieseker-semistable sheaves, in particular the injectivity statement:
Let $\psi : F \to G$ be a morphism of Gieseker-semistable sheaves. If $p(F)=p(G)$ and $F$ is stable, ...
- 123
3
votes
1
answer
258
views
Automorphisms of Frobenius liftings and degeneration of the Hodge-de Rham spectral sequence
I am still studying Deligne and Illusie's paper (https://eudml.org/doc/143480), and I am again stuck, this time on pages 262/263.
Assume $X\longrightarrow S$ is a smooth morphism of $\mathbb{F}_{p}$-...
- 1,805
6
votes
2
answers
566
views
commutativity of restriction and Gysin morphisms in a cartesian square
Let $X, Y, Z$ be compact topological manifolds $f: Y \to X, g: Z \to X$ be embeddings of submanifolds meeting transversely and let $W = Y \times_X Z$:
$$
\begin{array}{ccc}
Y & \to^f & X \\...
- 4,070
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 ...
- 1,221
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:\...
- 1,805
2
votes
0
answers
159
views
Sheafification in an arbitrary category
Let $\mathcal{F}$ be a presheaf valued in an arbitrary category $\mathcal{C}$ on a topological space $X$, with $\mathcal{C} $ has limits(or, $\mathcal{C}$ has equalizers so that a sheaf valued in $\...
- 111
49
votes
4
answers
7k
views
Sheaf-theoretic approach to forcing
Inspired by the question here, I have been trying to understand the sheaf-theoretic approach to forcing, as in MacLane–Moerdijk's book "Sheaves in geometry and logic", Chapter VI.
A general ...
- 21.3k
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
2
votes
1
answer
476
views
On the definition of a principal ideal sheaf
In his book Algebraic Geometry and Arithmetic Curves Qing Liu claims in Exercise 3.4, page 56, the following for a scheme $X$ and a global function $f\in \mathcal O_X(X)$:
"The map $U\mapsto f\...
- 417
0
votes
0
answers
100
views
Quotient of $\text{Proj}(A)$ by the action of a finite group
Let $X$ be $ \operatorname{Proj}(A)$ for some graded ring A, and let $G$ be a finite group acting on $A$ with morphisms of graded rings; consequently $G$ acts on $X$.
I know I can write $X = \bigcup_{...
- 1
1
vote
1
answer
427
views
Flat familiy of coherent sheaves over a scheme
I'm studying the moduli problem of locally free sheaves over a connected smooth projective curve on an algebraically closed field, from the Lecture Notes of Victoria Hoskins, and I cannot fully ...
- 395
2
votes
1
answer
399
views
Locally free sheaves and vector bundles over smooth connected projective curve
Let $X$ be a connected smooth projective curve over an algebraically closed field $K$. Let $\mathcal{F}$ be a locally free sheaf on $X$ and $\mathcal{E}$ a subsheaf of $\mathcal{F}$, which is again ...
- 395
4
votes
1
answer
435
views
Push-out in the category of coherent sheaves over the complex projective plane
I'm trying to deal with an example of a rank two vector bundle over the complex projective plane which is non slope-stable (because the associated sheaf of sections has a coherent subsheaf of equal ...
- 395
18
votes
2
answers
618
views
In the internal language of the topos of sheaves on a topological space, can we define locally constant real-valued functions?
For the purposes of this question, in a Grothendieck topos, we will call “definable” the objects and relations obtained from the terminal object, the natural numbers object and the subobject ...
- 32.5k
2
votes
1
answer
202
views
How to compute cup product of derived limits / presheaf cohomology
I have a finite category $\mathcal{C}$, along with a functor $F \colon \mathcal{C}^{\mathrm{op}} \to \mathsf{GradedCommRings}$. If $F_j$ is $j$-th graded piece of $F$, then I write $H^i(\mathcal{C},...
- 317
5
votes
1
answer
209
views
Cohomology of doubly pinched torus via spectral sequences
Let $f:T^2\to Y$ be a resolution of singularities where $Y$ is a torus with two "pinched" points (or, if you prefer, two copies of $\mathbb{P}^1$ meeting at two points). I'm interested in ...
- 191
3
votes
1
answer
549
views
Question about adjoint of forgetful functor from condensed abelian groups to condensed sets
There is a forgetful functor from condensed abelian groups to condensed sets. According to Scholze's notes, this has an adjoint $T \mapsto \mathbb{Z}[T]$ (which is the sheafification of the functor ...
7
votes
1
answer
1k
views
Different definitions of condensed sets
The $\kappa$-condensed sets are defined as the sheaves on the site of profinite spaces of cardinality less than $\kappa$ (with $\kappa$ an uncountable strong limit cardinal) with morphisms the ...
- 12.1k