All Questions
Tagged with or sheaf-theory sheaf-theory
979 questions
4
votes
0
answers
48
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Resolution of constant sheaf by $L^2$ function sheaves
Let $X$ be a compact Hausdorff space equipped with a Radon measure of full support.
Then $U\mapsto L^2(U)$ is a fine sheaf, hence can be taken for a first step in an acyclic resolution of the constant ...
- 460
13
votes
2
answers
660
views
Categories in which isomorphism of stalks does not imply isomorphism of sheaves
Let $\mathcal{A}$ be a locally small category with colimits of small filtered diagrams.
For the purposes of this question, an $\mathcal{A}$-presheaf on a topological space $X$ is a functor $\Omega (X)^...
- 15.9k
0
votes
0
answers
98
views
A question about the sheaf supported on the zero section
Let $X$ be the total space of the cotangent sheaf on $\mathbb{P}^{2}$ and $i \colon \mathbb{P}^{2} \hookrightarrow X$ be thezero section. Suppose that $E$ is a coherent sheaf on $X$ which is set-...
0
votes
0
answers
123
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Counit map surjective
Let $X \to Y$ a (set theoretically) surjective morphism of schemes, $L$ a line bundle/invertible sheaf on $X$ (maybe more generally a locally free coh sheaf, but let's stick firstly on invertible ...
- 6,028
4
votes
1
answer
323
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Is the pushforward of a closed immersion ever fully-faithful at the level of Derived Categories?
Let $i: Z \rightarrow X$ be a closed immersion of schemes. Then, for any $\mathcal{O}_{Z}$-module $\mathcal{G}$, the counit of adjunction $i^{*}i_{*}\mathcal{G} \rightarrow \mathcal{G}$ is an ...
- 127
5
votes
1
answer
367
views
Check that a Sheaf is Invertible Etale Locally
A question about following statement from Martin Olsson's book on Stacks. In the proof of Proposition 13.2.9. (p 269) is claimed that certain sheaf $K$ on a nodal curve $C$ is invertible it suffice to ...
- 6,028
1
vote
0
answers
92
views
Proposition 6.2.7 from Goss
I'm following David Goss's book Basic structures of function field arithmetic. Let $L$ be an extension field of $L_0$ and $\sigma$ an automorphism of infinite order which fixes $L_0$. Let $L\{\sigma\}$...
- 69
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 $\...
- 261
4
votes
1
answer
435
views
Delta distribution on manifolds
Let $M$ be a manifold. There is the sheaf of distributions $\mathcal{D}'$ and the sheaf of distribution densities $\mathcal{D}(\cdot)'$ on $M$. A delta distribution density $\delta_p \in \mathcal{D}(M)...
- 261
2
votes
1
answer
232
views
existence of a coherent sheaf
I am doing algebraic geometry. My question is the following: Here $X=\mathbb{A}^1-0$, $\mathbb{A}^1 = Spec A[t]$, $A$ is a commutative, noetherian ring with unity, consider $\mathcal{O}_{Spec A}$ as $\...
- 613
5
votes
2
answers
358
views
Canonical conics pulling back to polynomials on rational normal curve
(In following all schemes are formed over $\Bbb C$)
Let $C:=\nu_d(\Bbb P^1)$ the rational normal curve obtained via $d$-folded Veronese map $\nu_d: \Bbb P^1 \to \Bbb P^d$. The quadrics on $\Bbb P^d$ ...
- 6,028
11
votes
1
answer
390
views
Giraud's axioms imply balanced
I'm stuck on the following. Let $\mathcal{E}$ be a category satisfying the Giraud axioms: namely $\mathcal{E}$
is locally presentable,
has universal colimits,
has disjoint coproducts, and
has ...
6
votes
1
answer
290
views
Day Convolution of Sheaves
In Bodil Biering's master's thesis, Conjecture 4.3.3 conjectures that Day convolution of presheaves does not generally preserve sheaf conditions, with an incomplete attempt at a counter-example given. ...
- 989
1
vote
0
answers
73
views
Why can we not convert GATs / EATs / limit sketches to sites?
I think I'm in the process of understanding something very subtle here, and I could use an expert's double check. So basically, my question is whether what I write is correct.
(Non-finitary) GATs, ...
- 511
4
votes
1
answer
371
views
Grothendieck topoi as a constructive property
This question is inspired by Homotopy type theory, but I believe it can be thought about also in other constructive foundations.
In HoTT the question could be stated as follows:
Given a definition of ...
- 1,347
1
vote
1
answer
127
views
Vanishing of higher morphisms for pair moduli
Consider a moduli scheme $P$ of sheaves $F$ on a Calabi-Yau n-fold $X$ with a section $s\in H^0(F)$. Alternatively, such objects can be described as maps $\mathcal{O}_X \xrightarrow{s} F$ called pairs....
- 988
2
votes
0
answers
121
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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
1
answer
326
views
Extension by zero operation
Suppose you have a closed subset $Z$ of a topological space $X$, and $F$ is a sheaf on $Z$. Then one can consider the extension by zero sheaf $F^X$ on $X$.
What are some examples and situations which ...
- 129
4
votes
0
answers
178
views
Splitting of counit-trace map for $\ell$-adic sheaf $\Bbb Q_{\ell}$
I have a question about following argument on page 3 in paper arxiv.org/abs/1702.04404 by Will Sawin (Proposition 3):
The claim is that for a finite, dominant map $f:X \to Y$ between varieties $X,Y$ ...
- 6,028
3
votes
0
answers
199
views
When a fully faithful functor from an abelian category to itself will be an equivalence
Let $A$ be an abelian category. Suppose $i:A\to A$ is a fully faithful functor from $A$ to itself. I wonder when the functor will be an equivalence.
If $A$ is a "nice" category, I think $i$ ...
- 253
6
votes
0
answers
223
views
Under what generality are the compactly supported singular and sheaf cohomologies equal?
Edit: I have since resolved my question.
If X is locally compact Hausdorff in addition to being cohomologically locally contractible with coefficients in $A$ - eg it is a manifold or an open subset of ...
- 1,021
4
votes
1
answer
236
views
Ampleness verifiable over faithfully flat cover
Let $X$ be a Noetherian scheme over a field $k$ and $\mathcal{L}$ an invertible sheaf. Recall $\mathcal{L}$ is called ample iff for every coherent $\mathcal{M}$ there exist a $n_0(M)$ such that for ...
- 6,028
3
votes
0
answers
45
views
Sheaf with no singular support in $\mathbb{R}$-direction decomposes as an external tensor product
Consider $X$ to be a smooth manifold. Denote by $p_1, p_2$ respectively the projections $X\times Y$ to $X$ and $Y$ respectively. Recall the definition of microsupport/singular support of a sheaf (...
- 71
1
vote
0
answers
58
views
Which sheaves are good for calculating extraordinary restriction?
Let $X$ be a sufficiently nice locally compact Hausdorff space and let $i:Y\subset X$ be the inclusion map of a sufficiently nice closed subspace. For example, one could take $X$ to be a locally ...
- 23.5k
2
votes
1
answer
217
views
Dual of slope semistable vector bundle on higher dimensional variety
Recall the definition of slope semistability, taken from section 1.2 of Huybrechts and Lehn's "Geometry of Moduli Spaces of Sheaves" book. Let $X$ be a projective $\mathbb{C}$-scheme and $E \...
- 129
9
votes
0
answers
258
views
Sheaf cohomology of non-paracompact manifolds (e.g. the long line)
I have long heard that manifolds are "affine". If we allow non-paracompact manifolds, then this seems to fail, since as explained in Dmitri Pavlov's answer, the Serre–Swan theorem fails. I ...
- 2,806
3
votes
1
answer
240
views
Cohomology of the complement of a subvariety
Let $X$ be a complex manifold, $Y\subset X$ a subvariety, and $U:=X\setminus Y$ of codimension $d$. It is well known that the restriction map
$$
H^i(X,\mathbb Q)\to H^i(U,\mathbb Q)
$$
is an ...
- 178
0
votes
0
answers
116
views
How can I calculate $\chi(\mathscr{O}(P))$
Let be X a reduced and irreducible curve over a field $L_0$. Let $L$ an extension of $L_0$ and set
\begin{gather*}
\overline{X}=L \otimes X.
\end{gather*}
Assume $\overline{X}$ also irreducible. Now, ...
- 69
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
1
vote
0
answers
141
views
Homeomorphic endomorphism of schemes inducing equivalence of sheaves
Let $F: X \to X$ to be an endomorphism of scheme $X$, which is additionally assumed to induce an universal homeomorphism on the underlying topological space $| X|$. Then it is known that this induces ...
- 6,028
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
8
votes
0
answers
644
views
Trying to understand "Shtukas"
I'm studying Goss' Basic structures of function field Arithmetic, chapter 6 about Shtukas. I'm trying to understand some details about some concepts. This chapter is based on a Mumford's paper An ...
- 69
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
7
votes
1
answer
607
views
Converses to Cartan's Theorem B
Here is a phrasing of some Cartan Theorem B statements:
Consider the following conditions:
$X$ is a {Stein manifold, affine scheme, coherent analytic subvariety of $\mathbb{R}^n$, contractible ...
- 1,109
11
votes
1
answer
2k
views
6-functor formalism for topological stacks
I am trying to understand the 6-functor formalism of sheaves on topological stacks. As explained in this answer, there is a 6-functor formalism of sheaves for locally compact Hausdorff spaces, which ...
- 197
5
votes
1
answer
350
views
Dévissage of stratified structures in Grothendieck's "Esquisse d’un programme"
I have a question about the intuition behind Grothendieck's proposed notion of so called "Tame topology" in his Esquisse d’un programme. Grothendieck insisted that theory should admit “...
- 6,028
4
votes
1
answer
355
views
Bott & Tu differential forms Example 10.1
In Bott & Tu's "Differential forms", Example 10.1 states:
$\textbf{Example 10.1}$ Let $\pi: E \to M$ be a fiber bundle with fiber $F$. Define a presheaf on $M$ by $\mathcal F(U) = H^q(\...
- 51
1
vote
0
answers
126
views
Local freeness of dualizing sheaf
I am reading the dualizing sheaf and duality theorems from Hartshorne’s algebraic geometry book. I am wondering about the following.
When does the dualizing sheaf of a projective scheme is an locally ...
- 613
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
3
votes
0
answers
205
views
Category of sheaves of vector spaces on BG
Let $G$ be an affine group scheme over $\mathbb{C}$. I am interested in understanding the differences between different notions of sheaves on the stack $pt/G = BG$. For any algebraic stack $X$ one can ...
- 53
6
votes
1
answer
326
views
Spectral sequence generalizing Čech cohomology
Let $X$ be a 'nice' topological space. Let $\left(U_i\right)_{i\in I}$ be a finite open covering of $X$. Let $\mathcal{F}$ be a sheaf of abelian groups.
For a subset $A\subset I$ denote $$U_A:=\cap_{...
- 21.8k
3
votes
0
answers
99
views
Cohomology of differentiable stacks: should the sheaf be fine?
I'm reading these Behrend's notes on cohomology of stacks, and I can't get over a detail in the fifth page.
Let $X_\bullet=(X_1\rightrightarrows X_0)$ be a Lie groupoid and let $\mathcal{N}$ be its ...
- 105
4
votes
0
answers
102
views
Topos as a totally cocomplete object in a 2-category CART
In the preface to Sketches of an elephant, Peter Johnstone gives a list of characterizations of topos, some applicable to elementary (ii- finite limits and power objects) other to Grothendieck (i- ...
- 1,347
0
votes
0
answers
132
views
Higher direct images of locally constant etale sheaf under smooth proper map locally constant
Let $f:X \to Y$ a surjective smooth proper map between Noetherian schemes and $F$ a locally constant sheaf on small etale site of $X$.
Question: Refering to Donu Arapura's answer here, how to see that ...
- 6,028
5
votes
1
answer
286
views
Origin of the name Trace resp Integral symbol for the trace map of Dualizing Sheaf
Let $X \subset \mathbb{P}^n_k$ be a normal projective subscheme over $k$ of dimension $n$. The dualizing sheaf is in context of Serre duality a pair $(\omega_X,t)$ (which exists in that case) ...
- 6,028
0
votes
2
answers
331
views
Vakil exercise on sheaf associated to the divisor of rational section
This is exercise 15.4.G. of Vakil's notes.
Let $\mathscr{L}$ be an invertible sheaf on an irreducible normal scheme $X$ with $s$ a rational section of $\mathscr{L}$. We want that $\mathscr{O}_X(\text{...
- 29
1
vote
1
answer
122
views
System of local isos gives system of local epis
Suppose that $W$ is a system of local isomorphisms on a presheaf topos $\mathbf{Pre}(\mathcal{C})$. We say a map in $W$ is a $W$-local isomorphism, and we say that a map of presheaves $f: X \to Y$ is ...
27
votes
0
answers
469
views
Are these comparison morphisms between Čech and Grothendieck cohomology the same?
For better or for worse, there is more than one approach to comparing Čech cohomology $\smash{\check{H}}^\bullet(\mathfrak{U},X;\mathscr{F})$ of a sheaf $\mathscr{F}$ on a space $X$ w.r.t the cover $\...
- 1,021
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
6
votes
0
answers
103
views
Is the derived category of sheaves localised at pointwise homotopy equivalences locally small?
In order to define the cup and cross products in sheaf cohomology, Iversen makes computations in an intermediate derived category. If $K(X;k)$ is the triangulated category of cochain complexes of ...
- 1,021