All Questions
Tagged with sheaves or sheaf-theory
979 questions
2
votes
1
answer
374
views
Local question and descent category for a quasi-coherent sheaf on $\mathbb{G}_m$-gerbe
Update: I removed what I thought was unecessary and tried to be more straightforward in the hope to get an answer.
Context:
Suppose I have a $\mathbb{G}_m$-gerbe $\mathcal{G}$ over a scheme $X$ with ...
- 65
2
votes
1
answer
212
views
Leray spectral sequence for continuous functions on pairs of topological spaces
Let $A \subset X$ and $B \subset Y$ be topological spaces, with $A$ and $B$ closed, and $f: X \to Y$ be a continuous functions such that $f(A) \subset B$.
The Leray spectral sequence (with complex ...
- 1,227
2
votes
1
answer
593
views
Example for pullback of stable sheaf not stable
Suppose $C$ is a complete algebraic curve.
Define a coherent locally free sheaf $\mathcal{F}$ over $C$ to be stable if $\mu(\mathcal{E})<\mu(\mathcal{F})$ for any subsheaf $\mathcal{E}$, where $\...
user39380
2
votes
1
answer
429
views
Nisnevich points
Here is a probably stupid question : If $F$ is a sheaf on the big Nisnevich site, then is the morphism $F(X) \to \amalg F(x)$ injective where the sum is over ALL the points of $X$ (not just the closed ...
- 1,347
2
votes
1
answer
250
views
Behaviour of Morita equivalence in families of sheaves
Given a projective scheme $X$, say over $\mathbb{C}$, another $\mathbb{C}$-scheme $S$
and a coherent and torsion free (as an $O_X$-module) $M_n(O_X)$-module $F$.
Now we can use Morita equivalence to ...
- 1,391
2
votes
0
answers
75
views
Pullback by surjective submersion is injective?
Denote by $\mathcal{D}'_X$ the sheaf of distributions on a smooth manifold $X$.
Let $M$ and $N$ be smooth manifolds and $\Phi: M \to N$ a submersion. Then $\Phi$ defines a unique morphism of sheaves $\...
- 311
2
votes
0
answers
121
views
Singular cohomology as a sheaf of $\infty$-categories
In several expositions of $\infty$-categories, I read that singular cohomology of a topological space with integral coefficients is a sheaf valued in $D(\mathbb{Z})$, if we consider Top and $D(\mathbb{...
2
votes
0
answers
137
views
details of a dévissage argument for constructible sheaves
I am working on the following Künneth-type isomorphism from [SGA5, exposé III, 2,3]:
$\mathrm{Settings}.$ Let $X_1, X_2$ be separated finite type schemes over the spectrum of a field $S=\mathrm{Spec}...
- 375
2
votes
0
answers
75
views
Differential equations where we are tryng to find a smooth space(maybe with additional structure)
Is there literature on the differential equation $TM=N$ where $M$ and $N$ are two smooth spaces (maybe with additional structure) and $T$ stands for the tangent functor?
- 35
2
votes
0
answers
70
views
Does every SES of injective bounded cochain complexes split?
Question: Does every short exact sequence of injective bounded cochain complexes, $0\rightarrow I^\bullet\rightarrow J^\bullet\rightarrow K^\bullet\rightarrow 0$, split?
I am interested in a discrete ...
- 213
2
votes
0
answers
148
views
Equivalence of cohomology with compact support
Let $𝑋$ be a connected CW complex, $𝜌:\pi_1(𝑋)→\mathrm{Aut}(G)$ a representation, and $G_\rho$ the associated sheaf. It follows from here, that the following two cohomologies are isomorphic.
(1)The ...
- 241
2
votes
0
answers
170
views
Hodge bundles associated to a family of complex manifolds
I'm reading Voisin's books on Hodge theory. In the first volume she claimed but didn't prove this theorem:
Theorem 10.10 (Voisin) Let $\varphi:\chi\rightarrow B$ be a family of compact complex ...
- 21
2
votes
0
answers
65
views
Coequalizers and pullbacks in $\infty$-topoi
In an $\infty$-topos, suppose we have two cartesian diagrams of the form
$$
\require{AMScd}
\begin{CD}
\overline{A} @>>> \overline{B} \\
@VVV @VVV \\
A @>>> B .
\end{CD}
$$
Let
$$
\...
- 66
2
votes
0
answers
253
views
The conormal sheaf is the sheaf of sections of the conormal bundle for smooth manifolds
$\def\sO{\mathcal{O}}
\def\d{\mathrm{d}}$In ringed spaces theory, there is a notion of “conormal sheaf of an immersion” (mainly used in scheme theory), whereas in smooth manifold theory, there is the ...
2
votes
0
answers
126
views
Compact generators of $\mathcal{D}\text{-Mod}$ via Mayer Vietoris
Let $X$ be a complex variety (or any space for whom a category of sheaves with the six functors is defined), and
$$i\ :\ Z\ \to\ X\ \leftarrow\ U\ :\ j$$
be complementary open and closed embeddings.
...
- 5,701
2
votes
0
answers
239
views
Do we have $\underline{\operatorname{Ext}}^i_\text{fppf}(\widehat{\mathbb{G}}_a,\mathbb{G}_m)=0$ for $i>0$?
Let $k$ be a characteristic zero field and consider the category $(\mathsf{Sch}/k)_\text{fppf}$ of schemes over $k$ with the fppf topology. I know that $\underline{\operatorname{Hom}}(\mathbb{G}_a,\...
- 721
2
votes
0
answers
100
views
Global sections of relative characteristic of log-smooth curves
$\DeclareMathOperator\Spec{Spec}$I am currently learning about log-geometry and try to understand the theory in the example of curves with basic log-structure over a general base $S$. Especially, I am ...
- 223
2
votes
0
answers
175
views
Is this double quotient of $\operatorname{SL}_2$ representable by an algebraic space or a scheme?
$\DeclareMathOperator\SL{SL}$Let $B$ be a Borel subgroup (upper triangular matrices), and let $\Gamma := \langle \sigma\rangle$ be the group generated by a (hyperbolic) element of $\SL_2/\mathbb{Q}_p$ ...
- 2,907
2
votes
0
answers
148
views
Push-forward of a locally constant sheaf using two homotopic maps
Let $X,Y$ be compact smooth manifolds. Let $f,g\colon X\to Y$ be smooth submersions
(in particular, locally trivial bundles) which are homotopic to each other (in the class of smooth maps, not ...
- 21.8k
2
votes
0
answers
111
views
Canonicity in split sequence in cotangent spaces
Let $X$ be a locally ringed space. We have for a point $p$ the exact sequence
$$0\to \mathfrak{m}_p^2\to \mathfrak{m}_p\to \mathfrak{m}_p/\mathfrak{m}_p^2 \to 0$$
where $\mathfrak{m}_p$ is the maximal ...
- 167
2
votes
0
answers
128
views
On the generalization of a Cech-to-sheaf type spectral sequence
Let $(X,O_{X})$ be a ringed space and $\mathcal{F^{\bullet}}$ be a bounded below complex of $O_{X}$-modules on $(X,O_{X})$. On [Stacks Project 20.25.1][1] it is shown that there is a weakly convergent ...
- 541
2
votes
0
answers
121
views
Norm of sections on $T$-invariant subspace of a homogenous space over $\mathbb{Q}_p$
Let $K=\mathbb{Q}_p$ and $G$ a split reductive group over $K$ with split maximal torus $T$. Furthermore, $X$ is a smooth projective variety over $K$ with a free $G$-action. Let $U \subset X$ be a $T$-...
- 473
2
votes
0
answers
114
views
Two natural morphisms of sheaves with the same source and target; do they agree?
Suppose we have a diagram
$\require{AMScd}$
\begin{CD}
A @>a>> B\\
@V b V V @VV c V\\
C @>>d> D @>e>> E \\
@VfVV @VVgV @VVhV \\
F @>>i> G @>>j> H
\end{CD}...
- 125
2
votes
0
answers
168
views
Criteria for a sheaf to be locally free over subvariety
Let $X$ be a compact complex manifold and $\mathcal{F}\to X$ a sheaf.
Is there a regularity criteria (or a condition) for $\mathcal{F}$ that determines whether we there exists a closed subvariety $i:...
- 789
2
votes
0
answers
136
views
Homotopy fixed points vs coalgebras
Referring to the last part of this answer https://mathoverflow.net/a/225403/170683, I would like to understand how in the case of a Galois cover $f\colon X\to Y=X/G$ with Galois group $G$ (I guess ...
- 351
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
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 ...
- 721
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
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
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}$-...
- 1,094
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
2
votes
0
answers
337
views
High direct image of dualizing sheaf
I'm reading the paper "High direct image of dualizing sheaf" of professor Kollar. I summarizing my questions as follows:
Let $f:X\rightarrow Y$ be surjective projective morphism between ...
- 623
2
votes
0
answers
92
views
Cone of morphism in families
I am working in derived category $D^b(X)$ of coherent sheaves on a smooth projective varitey.
Let $E,F$ be two sheaves on $X$, with $\mathrm{R}Hom(E,F)=k\oplus k[-1]$, I consider the following ...
- 1,982
2
votes
0
answers
144
views
Local freeness of $\pi_*F(r)$ from flatness of $F$
In 'Fundamental Algebraic Geometry' by Fantechi there is a lemma in section 5.3.2, page 119:
LEMMA 5.5 Let $S$ be a noetherian scheme and let $F$ be a coherent sheaf on $\mathbb{P}^n_S$. Suppose there ...
- 6,038
2
votes
0
answers
296
views
Small sheaves on big sites
Background: If one works with sheaves on small etale site over a fixed scheme (which is really an essentially large category), one can instead work with sheaves on the affine etale site (which turns ...
- 319
2
votes
0
answers
142
views
Degree of a divisor along a subscheme
I'm curious about a computation of Prop2.3 in The gonality conjecture on syzygies of algebraic curves of large degree by Ein and Lazarsfeld. Let $C$ be a smooth projective curve carrying a pencil $\...
- 439
2
votes
0
answers
131
views
$m$-regularity of sheaves
This is Lemma 1.4 on Green and Lazarsfeld's Some results on the syzygies of finite sets and algebraic curves. Let $X$ be a closed subscheme of $\mathbb{P}^r$. Suppose the ideal sheaf $\mathcal{I}$ of $...
- 439
2
votes
0
answers
212
views
Only discrete topology gives trivial topos?
Given a Grothendieck site $\mathsf{(C,\tau)}$, if the associated Grothendieck topos $\mathsf{Shv(C,\tau)}$ is trivial, i.e. $\mathsf{Shv(C,\tau)}$ consists of the terminal sheaf $*$ only, can we ...
- 1,906
2
votes
0
answers
98
views
What does a partial map classifier look like as a sheaf?
[Cross-posted from M.SE, where it didn't get an answer]
In constructive logic, it's possible for a set $X$ to satisfy $$\forall x,y \in X.\, x = y$$ while being non-trivial. Such a set is called a ...
- 4,943
2
votes
0
answers
66
views
For which locally ringed spaces is the structure sheaf given by LRS morphisms to the real line?
Let $\mathsf{LRS}_{\mathbb R}$ denote the category of locally $\mathbb R$-ringed spaces.
Given a locally ringed space $(X,\mathcal O_X)$, write $C_{(X,\mathcal O_X)}^p$ for the hom-sheaf on $X$ of ...
- 10.5k
2
votes
0
answers
645
views
Direct image functor commuting with infinite direct sum of sheaves
Normally I would think this kind of question doesn't belong on overflow, but I haven't been able to find an answer anywhere else, so perhaps it is not so trivial.
Let $f: X \rightarrow Y$ be a ...
- 453
2
votes
0
answers
293
views
Direct sums of invertible sheaves commuting with global sections and the functor of points approach
I am looking at the Stacks Project's treatment of the functor of points for projective space.
Let's restrict to the case that $S$ is a graded ring, generated by $S_{1}$ as an $S_{0}$ algebra. The ...
- 453
2
votes
1
answer
247
views
Quasi-coherent sheaves on affinoid space
From Conrad's notes on rigid geometry:
More specifically, Gabber has given an example of a sheaf of modules $F$ on the closed unit disk $B^1$ such that $F$ is locally a direct limit of coherent ...
user138661
2
votes
0
answers
315
views
Any vector bundle can be twisted to have sections
Let $X$ be an integral scheme proper over $\mathbb{C}$. Let $V$ be a locally free $\mathcal{O}_X$-module of positive finite rank. Does there necessarily exist a locally free $\mathcal{O}_X$-module of ...
user139965
2
votes
0
answers
132
views
Compact generation of quasicoherent sheaves on mapping stack
Let $k$ be a field of characteristic $0$, and let $\mathcal{C}= \mathbf{Vect}_k^{\leq 0}$ be the $\infty$-category of vector spaces concentrated in degrees $\leq 0$. Consider the category $\mathbf{Pr}(...
- 3,019
2
votes
0
answers
139
views
Defineing a Sheaf of rings over a topological space
Let $X$ be a topological space and let $R$ be a commutative ring with $1$ such that for each $x\in X$ there exists a multiplicatively closed subset $S_x$ of $R$ such that for each $a\in R$ if $\frac{a}...
- 31
2
votes
0
answers
119
views
On the limit of a directed system of sheaves
I find myself in the following situation:
Let $(\mathcal{F}_i, f_{i,j})$ be a directed system of sheaves, defined on an exhausting family of closed topological subspaces $$X_i \subset X_{i+1} \...
- 1,227
2
votes
0
answers
143
views
Maps between Leray spectral sequences
Let $f_1 : X_1 \to Y_1$ and $f_2 : X_2 \to Y_2$ be two continuous maps between topological spaces. Suppose that there exists a commutative diagram of singular cohomology groups (say with coefficients ...
- 1,227
2
votes
0
answers
193
views
Usage of Leray spectral sequence
Maybe this is an elementary stuff for experts, I could not figure it out by myself. Let $\pi:G\to S$ be an elliptic curve with zero section $e:S\to G$. Take $\mathcal{H}=(R^1\pi_*\mathbb{Q}_\ell)^\vee$...
- 1,428
2
votes
0
answers
363
views
Singularities of reflexive sheaves
I am studying reflexive sheaves (on $\mathbb{P}^3$) by the Hartshorne's paper ''Stable reflexive sheaves''. As far I understood, reflexive sheaves fail to be locally free at a finite number of points (...
- 556