All Questions
Tagged with sheaves or sheaf-theory
356 questions with no upvoted or accepted answers
6
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0
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226
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Is the right adjoint to presheaf direct image interesting?
Let $X\overset{f}{\to}Y$
be a continuous map. It induces on presheaves a classical adjunction inverse image ⊣ direct image. However, the direct image functor has a further right adjoint, defined by ...
- 10.5k
6
votes
0
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322
views
What's the point of fine sheaves? (As opposed to soft ones)
Why do people care to define fine sheaves? What useful property do they have for which softness is not sufficient?
some observations (because I feel guilty about a the one-line question):
The point ...
- 1,141
6
votes
0
answers
212
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G-sheaves on spaces with a free G-action
Let $X$ be a topological space
equipped with an action of a group $G$. In the Tohoku paper, Grothendieck defined
"$G$-sheaves" on $X$ as sheaves equipped with $G$-action on the etale space,
...
- 9,237
6
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0
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179
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Is the category of diffeological spaces a full subcategory of locally ringed spaces?
It is known that the natural functor of smooth functions from the category of smooth manifolds into the category of locally ringed spaces is a full embedding (see, for example, here).
Is a similar ...
- 6,962
6
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452
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Yoneda product on Ext
Let $M$ be a closed manifold and consider the constant sheaf $\mathbb{R}_M$. I have heard that the Yoneda product on $\operatorname{Ext}(\mathbb{R}_M, \mathbb{R}_M)= H^*(M; \mathbb{R})$ coincides with ...
- 177
6
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0
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889
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On a weak notion of sheaves on topological spaces
First of all, I give my definition of weak sheaves:
By a weak sheaf on a topological space $ X $, we mean a presheaf
$F$ such that for all open covering $\{ U_i\}_{i\in I} $ of $X$ sheaf
...
- 751
6
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0
answers
239
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Sheaves on Rectifiable Sets
Basic question: are there (co)homological or sheaf-based tools which might be useful in geometric measure theory?
Background: The jumping off point here is a simple analogy - geometric measure ...
6
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152
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On computability of sheafification
The question will feature some imprecise words but I believe that an expert could parse it to a precise question more or less uniquely.
Assume I have a reasonable topological space (say the ...
- 257
6
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0
answers
171
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Could we characterize injective objects in the category of $G$-equivariant sheaves?
Let $k$ be a field and $X$ be a topological space. We consider Sh$(X)$, the category of sheaves of $k$-vector spaces on $X$.
Let $G$ be a topological group which act on $X$ continuously from the left....
- 9,019
6
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0
answers
187
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Algebraic model for the abelian category of descent data for modules in the non-affine case
Let $f: X \to Y$ be a morphism of schemes. I'd like to have a completely algebraic description of the belian category of descent data for modules along $f$. Here's my attempt:
The category of quasi-...
- 7,789
6
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0
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471
views
What are the most general conditions under which the inverse image of sheaves of abelian groups has a left adjoint?
If $f: E \to X$ is an étale map (a local homeomorphism), then the inverse image of sheaves of abelian groups $f^{-1}$ has a left adjoint, as shown by Roland in his answer here. This subsumes as a ...
- 6,025
6
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0
answers
183
views
Dense (∞,1)-subsites
So if $C$ is a 1-site and $D$ is a subsite (with the induced coverage), there are some conditions that ensure that the pre-composition and right Kan extension functors yield an equivalence of ...
- 439
6
votes
0
answers
141
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Expressing the stack of sheaves with 1-limits
Zhen Lin's answer to the MSE question mentions that $\mathsf{Sh}(-)$ is a stack for the canonical topology on the site of open subsets of a space. One of the comments asks whether the stack descent ...
- 10.5k
6
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0
answers
324
views
Many-sorted nominal sets as sheaves
The category of nominal sets can also be presented as the Schanuel topos, the category of sheaves on $\mathbb{I}^\mathsf{op}$ under the atomic topology, where $\mathbb{I}$ is the category of finite ...
5
votes
0
answers
163
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Why equaliser of product and terminal object is coproduct?
I’m reading “Sheaves in geometry and logic”, in page 80:
Please refer to [1]: https://i.sstatic.net/INrU0.jpg
It says “…,therefore $FU=\coprod_{x\in U} fx$. The space…”.
So could anyone please explain ...
- 51
5
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220
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Is there a simple counterexample to étale proper base change on the unbounded derived category?
The best non-derived version of proper base change on the étale site of a scheme I know is that for $f : X \to Y$ proper and $g : Y' \to Y$ arbitrary, the base change morphism $g^{-1} R f_\star \...
- 605
5
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0
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146
views
Do presheaf toposes satisfy the full fan theorem?
Presheaf toposes satisfy LPO and (edit: if over categories with binary products) PAx and countable choice internally, so they automatically satisfy the stable fan theorem (every bar which is the ...
- 1,947
5
votes
0
answers
269
views
Line bundle whose pushforward is a complex of vector bundles
If $E\to X$ is a holomorphic vector bundle, it is well known that the tautological line bundle $\mathcal{O}_E(1)$ over the projectivization $\pi:\mathbb{P}(E^*)\to X$ satisfies
$$\pi_*\mathcal{O}_E(1)=...
- 789
5
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0
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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 ...
- 711
5
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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
5
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0
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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
5
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0
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346
views
Properties of microlocalization
Let $i: M\hookrightarrow X$ be the inclusion of a closed submanifold in a smooth manifold $X$. I denote by $T_MX$ the normal bundle to $M$ in $X$, by $T^{\ast}_MX$ its dual bundle, and by $D^b(X)$ the ...
- 303
5
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0
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268
views
Coherent cohomological dimension and affine morphisms
For simplicity, all varieties in this question are quasiprojective varieties over an algebraically closed field of characteristic $0$.
The coherent cohomological dimension $cd(X)$ of a variety $X$ is ...
- 59
5
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0
answers
375
views
What is an example of a cokernel $B/\phi(A)$ in group schemes which does not have $A=\mu_d$ and requires the fppf topology to be a sheaf?
Let $S$ be affine. A bit of background: Let us think of $S$-group schemes as abelian sheaves over a given site (etale, Zariski, fppf, etc). When we take a cokernel of a morphism $\phi$ this category: $...
- 3,539
5
votes
0
answers
859
views
How to construct the espace étalé (space of sections) for an arbitrary category?
I want to consider the sheaf valued in an arbitrary category (not only of sets, groups, modules and so on) on a topological space, using the language of étalé space.
In all references I am reading (...
5
votes
0
answers
720
views
What is the filtration in Leray's spectral sequence?
Leray's theory of spectral sequences considers a continuous map $f : X \to Y$ between topological spaces. The statement is that there exists a filtration
$$H^n(X,k) = F^0H^n(X,k) \supseteq ... \...
- 1,227
5
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0
answers
113
views
How "commutative" are Cech cochains of a sheaf of commutative (dg) algebras?
Let $X$ be a topological space and $\mathcal{F}$ be a sheaf of commutative dg-algebras over $X$. Let $\mathfrak{U}$ be a fixed open covering of $X$ and $C^\bullet(\mathfrak{U},\mathcal{F})$ be the ...
5
votes
0
answers
271
views
K-flat, K-flabby resolution
Let $X$ be a topological space and $F$ a flat sheaf of abelian groups. It is well known that taking the Godement resolution of $F$ gives rise to a "flabbyflat" or "flasqueflat" resolution, in the ...
- 501
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 $\...
user37663
5
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0
answers
380
views
Does the de Rham complex induce a functorial soft resolution of the category of cochain complexes of sheaves of vector spaces on a smooth manifold?
I apologize in advance if this is pretty straightforward; I'm a differential geometer and physicist by training so my homological algebra and homotopy theory are a bit weak.
Question: Let $M$ be a ...
- 6,025
5
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0
answers
448
views
Examples of nonstable ∞-categories in which sifted colimits commute with finite limits
What are some natural examples (if any) of nonstable ∞-categories in which finite limits commute with sifted colimits (or rather just colimits over Δ^op)?
Stable ∞-categories do satisfy this property,...
- 37.8k
5
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0
answers
189
views
Constructible sheaves on general stratified spaces
I am not an expert in the field, so my question might be rather standard. Let $X$ be a compact metric space. Assume that $X=\cup_{i=1}^NS_i$ is a finite disjoint union of locally closed topological ...
- 21.8k
5
votes
0
answers
377
views
Push forward of the constant sheaf for a Serre's fibration
Let $f\colon X\to Y$ be a proper continuous map of topological spaces which is a Serre's fibration. $X$ and $Y$ may be assumed to be locally compact, $Y$ is connected topological manifold of finite ...
- 21.8k
5
votes
0
answers
310
views
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
5
votes
0
answers
511
views
Applications and main properties of hyperfunctions
I am trying to get familiar with hyperfunctions, and I do have some familiarity with the classical theory of distributions.
I am wondering whether hyperfunctions have any advantages over ...
- 21.8k
5
votes
0
answers
281
views
Mixed structures on Hom spaces induced by mixed sheaves
Let $D^b_m(X)$ (resp $D^b(X)$) denote the derived category of mixed Hodge modules (resp. constructible sheaves) on a complex variety $X$. Let
$rat\colon D^b_m(X)\to D^b(X)$
be the `forgetful' ...
- 1,595
5
votes
0
answers
564
views
About an argument in `Koszul duality patterns in representation theory' by Beilinson-Ginzburg-Soergel.
I am trying to understand Proposition 3.4.2 in `Koszul duality patterns in representation theory' by Beilinson-Ginzburg-Soergel [BGS]. A copy of the paper can be found at http://home.mathematik.uni-...
- 1,595
5
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336
views
Do 'change of coefficients' functors for sheaves commute with the four functors (formalism)?
For a morphism $f$ of varieties over a field of characteristic $\neq l$ I can consider the functors $Rf_*$, $f^\ast$, $f_!$, and $f^!$ both for the corresponding derived categories of 'all' (...
- 16.9k
5
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answers
374
views
Sheaf Cohomology on Zariski-Riemann Spaces
Can sheaf cohomology on the Zariski-Riemann spaces give some sort of classification for field extensions (even just for function fields)? If not, are there any significant or useful results (e.g. for ...
- 468
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answers
2k
views
Monomorphisms of sheaves
The following is surely pretty standard, but I have been unable to prove it or find a proof in the literature (many sources assert it without proof).
Let $\phi : \mathcal{F} \rightarrow \mathcal{G}$ ...
- 433
4
votes
0
answers
48
views
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
4
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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 $X,Y$ ...
- 6,038
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
4
votes
0
answers
216
views
When inverse image presheaf is already a sheaf
Following proof from Milne's Étale Cohomology (page 94) contains an equality I not understand.
Setting: assume $X$ is a variety (=absolutely reduced, irreducible scheme of finite type over base field ...
- 6,038
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 ...
- 711
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
4
votes
0
answers
195
views
Example of pseudocoherent complex which is not locally quasi-isomorphic to a strict pseudocoherent one
I ask this question here even if I posted it also on math.stackexchange (recieving no answer so far) because I have read some analogous question but for perfect complexes on this site, even though ...
- 351
4
votes
0
answers
120
views
Understanding a step in proof of sheaf version Verdier duality
Warning: This question is likely low-level for MathOverflow. My apology that there is almost surely something basic I miss.
So all proofs I can find factors through a particular statement, which goes ...
- 1,856
4
votes
0
answers
201
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Infinity-categorical exceptional push-forward
Classically, if $f:X\to Y$ is a map of locally compact Hausdorff topological spaces, one can define the exceptional push-forward functor $f_!:Sh(X;k)\to Sh(Y;k)$ among $k$-valued sheaves for, say, a ...
- 4,189
4
votes
0
answers
138
views
Examples of non-hypercomplete sheaves on affine schemes
Let $A$ be a commutative ring and let $\mathcal{O}$ be a sheaf of $E_{\infty}$-ring spectra on $\mathrm{Spec} A$ such that $\pi_0\mathcal{O} = \mathcal{O}_{\mathrm{Spec} A}$. Lurie provides a ...
- 12.1k