All Questions
Tagged with sheaves or sheaf-theory
979 questions
2
votes
1
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290
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Calculate stalk of etale derived pushforward sheaf (Milne's LEC)
Assume $X=\operatorname{Spec}(A)$ is connected and normal (especially integral), and let $g:\eta \hookrightarrow X$ be the inclusion of the generic point of $X$. In Milne's LEC script on Etale ...
- 6,028
2
votes
2
answers
429
views
If $\mathcal{F}$ globally generated, then counit map $f^*f_* \mathcal{F} \to \mathcal{F} $ surjective
Let $f: X \to S$ be a morphism, and $\mathcal{F}$
be quasi-coherent $\mathcal{O}_X$-module generated by global sections (eg if
$X$ projective, then this holds for the twisted sheaf $\mathcal{F}(n)$ ...
- 39.3k
1
vote
1
answer
271
views
Second fundamental exact sequence of sheaves of differentials. Sufficient condition to get a splitting s.e.s
$\def\spec{\operatorname{Spec}}$I am trying to understand the proof of Lemma 0474 of the Stacks Project. I'll give some context to its statement before discussing its proof: In commutative algebra, if ...
5
votes
2
answers
949
views
Can a nontrivial presheaf have a trivial sheafification?
I'm studying Vakil's Foundations of Algebraic Geometry, and working through the exercises in 2.4 about compatible germs as a method for constructing a sheafification of a given presheaf. Can this ...
- 63.9k
7
votes
1
answer
490
views
Equivariant perverse sheaves and orbit stratification
Let $X$ be a complex algebraic variety with an action of a connected algebraic group $G$.
The forgetful functor from the category of $G$-equivariant perverse sheaves on $X$ to the category of perverse ...
- 12.9k
4
votes
1
answer
168
views
Sheaves and gratings
A grating is a notion in algebraic topology from the 1940 introduced by Alexander. Cartan extended it as follows.
A grating (carapace in french) is defined by a topological space $X$, a module (or a ...
- 12.9k
5
votes
0
answers
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 \...
- 12.9k
5
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0
answers
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
3
votes
0
answers
215
views
How to read the definition of Grothendieck Pretopology in SGA4?
In SGA4, the first axiom of a Grothendieck pretopology is given as:
PT0: Pour tout objet $X$ de $C$, les morphismes des familles de morphismes de $Cov(𝑋)$
sont quarrables. (Rappelons qu’un morphisme ...
- 237
7
votes
2
answers
1k
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Non-zero sheaf cohomology
Let $\mathbb{R}$ denote the real line with its usual topology. Does there exist a sheaf $F$ of abelian groups on $\mathbb{R}$ whose second cohomology group $H^{2}\left(\mathbb{R},F\right)$ is non-zero?...
- 6,367
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
4
votes
1
answer
541
views
Clarification on smooth de Rham theorem
I am misunderstanding something in Theorem 2.1.9 in Dimca’s Sheaves in Topology:
Let $X$ be a real smooth manifold. Then the natural morphism from the constant sheaf to the de Rham complex
$$\mathbb{R}...
- 37k
1
vote
0
answers
91
views
Quotient sheaf obtained from a quotient of $\mathrm{SL}_2$ having a section
$\DeclareMathOperator\SL{SL}$Let $K$ be a field and let $G$ be a $K$-defined, closed, algebraic subgroup of $\SL_2$. Denote by $\mathcal{C}$ the site whose objects are $K$-schemes, with your favorite $...
- 63.9k
3
votes
0
answers
177
views
Sheaf theory in TDA
I was wondering wether anyone had any examples as to why it more useful to consider a sheaf theory approach to TDA problems.
I am familiar with some of the benefits of using cellular cosheaves to ...
- 343
8
votes
0
answers
333
views
Who introduced the notion of ringed spaces?
My question is very concise, please forgive it.
Who introduced the concept of ringed space?
My first try would be that they were introduced by Cartan in his study of analytic functions with sheaves. ...
- 4,713
3
votes
0
answers
280
views
Cellular (co)Sheaves and applications
In the last few years there have been efforts made to generalise the theory of peristence homology and cohomology to deal with sheaves for example Russold - Persistent sheaf cohomology. This as i ...
- 12.9k
11
votes
2
answers
664
views
Equivalence of the definitions of a sheaf in SGA4 and in "Categories and Sheaves"
I asked this question on Mathematics Stack Exchange, but got no answer.
I don't understand why the definition of a sheaf (Definition 17.3.1 (ii)) given in the book
[KS] Categories and Sheaves by ...
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,\...
- 711
1
vote
0
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67
views
Question of deforming a sheaf
Consider over $\mathbb C$. Let $Artin/\mathbb C$ denote the category whose objects are Artinian local $\mathbb C$-algebras with residue field $\mathbb C$, and morphisms are local ring maps preserving ...
- 930
7
votes
3
answers
924
views
not quite the sheaf condition
Let $Sets$ be the category of finite sets and all maps. I have come accross several example of functors $F : Sets^{op} \to Sets$ which satisfy the condition below:
-- There exists an integer $k$ such ...
- 15.9k
5
votes
1
answer
512
views
Do we have $\underline{\operatorname{Ext}}^i_\text{fppf}(\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,\...
- 711
1
vote
1
answer
251
views
Some question about (semi-)stable sheaves
Let $X$ be a projective normal variety over $\mathbb C$, I have several questions about semi-stable sheaves:
Question 1. Suppose that $E$ is a pure sheaf such that $HN_*(E)$ is the Harder-Narasimhan ...
- 26
2
votes
1
answer
193
views
Unifying categorical equivalences and dualities for the functors : Gelfand spectrum and Zariski spectrum, Structural sheaf and Continuous sections
I would be very grateful for any references I might be led to, from a categorical point of view for the functors:
$\textsf{Spec}_{\mathscr{Z}\textrm{arisky}}(-)$, related to $\mathcal{O}(-)$, which ...
- 37.8k
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
11
votes
2
answers
1k
views
Is there a relation between Gelfand duality and the spectrum of a ring (with its Zariski topology)?
Compare the following two results:
Thm A) Let $A$ be a commutative $C^*$-algebra and let $X$ be its Gelfand spectrum. Gelfand duality says that there's a natural isometric $*$-isomorphism from $A$ to ...
-1
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1
answer
177
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When morphism of complexes is homotopic to 0?
Let $f\colon A^\bullet\to I^\bullet$ be a morphism of bounded below complexes in an abelian category. Assume all $I^i$ are injective objects. Assume also that $f$ induces the zero map on cohomology.
...
- 38k
10
votes
2
answers
2k
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Geometric interpretation of sheaf cohomology
Please forgive me for the informal and naïve nature of my question, as I am a beginner in algebraic geometry.
In the famous book by Hartshorne, sheaf cohomology is defined as a certain derived functor....
- 5,857
5
votes
2
answers
233
views
References on principal $\mathbf{C}$-bundles, where $\mathbf{C}$ is a category?
Is there any treatment on principal "categorical" bundles - principal $\mathbf{C}$-bundles, where $\mathbf{C}$ is some (topological) category?
I know that one can define "categorical ...
- 9,263
1
vote
1
answer
226
views
flatness of restriction of structure sheaf over ring of global sections
Let $X$ be an affine scheme. $U \subseteq X$ open. Then I want to show that $\mathcal{O}_X(U)$ is flat over $\mathcal{O}_X(X)$.
But I want to prove it only by knowing the definition of structure sheaf ...
- 4,111
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$ ...
- 12.9k
14
votes
3
answers
2k
views
Recommendations for getting into sheaves with emphasis on differential geometry and algebraic topology
I want to study the theory of sheaves from a categorical point of view with an emphasis on applications in algebraic topology and differential geometry and I'm looking for a good introductory book to ...
- 7,127
7
votes
1
answer
255
views
Subobject classifier for sheaves on large sites with WISC
Let $\mathsf{C}$ be a possibly large category with a Grothendieck topology satisfying the Weakly Initial Set of Covers condition: there is for each $X$ a set (not a proper class) of covering families ...
- 15.9k
1
vote
0
answers
79
views
What is $\text{Hom}(\mathcal{F}\times\mathcal{G}, \mathbb{A}^1)$?
Let $S$ be an affine scheme and let $\text{Aff}(S)$ be the site of affine $S$-schemes. Let $\mathcal{F}$ and $\mathcal{G}$ be a pair of sheaves on $\text{Aff}(S)$, and let $\mathbb{A}^1_S$ be the ...
- 2,907
14
votes
2
answers
924
views
A bit of history of Verdier duality
I was wondering who originated the presentation of Verdier duality as an equivalence between categories of sheaves and cosheaves ?
I learnt it reading Jacob Lurie's Higher Algebra and Justin Curry's ...
- 12.9k
12
votes
3
answers
845
views
Can one characterize those sheaves which have Hausdorff etale spaces?
Given a sheaf of sets $F$ on a space $X,$ under the equivalence of categories between etale spaces over $X$ and sheaves over $X,$ $F$ is associated to a local homeomorphism $$E\left(F\right) \to X$$ ...
9
votes
1
answer
174
views
W-types and inverse image functor
All sheaf topoi have W-types and in fact there's an explicit construction given by Benno van den Berg & Ieke Moerdijk, but the construction is quite involved.
I would like to know whether the ...
- 3,312
3
votes
1
answer
284
views
Is this functor $\mathcal{F}: \text{Sch}/\mathbb{Q}\longrightarrow \text{Sets}$ a sheaf?
Consider the functor $\mathcal{F}: \text{Sch}/\mathbb{Q}\longrightarrow \text{Sets}$, defined by sending a scheme $X$ with coordinate ring $\mathcal{O}(X)$ to the set of orbits $B(\mathcal{O}(X))\...
- 156k
0
votes
0
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91
views
Why does the associated sheaf vanish?
I am learning local cohomology from Hartshorne’s book Local Cohomology.
My question is about understanding a line in the proof of proposition 1.11 in this book.
The set-up for proposition 1.11 is that ...
- 639
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
1
vote
0
answers
137
views
The stack $\operatorname{GL}_2/B$
Let $F$ be the functor from the category of affinoid Tate algebras over $\mathbb{Q}_p$ to the category $\mathrm{Sets}$, which maps an affinoid $\operatorname{Spm} R$ to the set of orbits $\...
- 12.9k
4
votes
1
answer
2k
views
How is the right adjoint $f_*$ to the inverse image functor $f^*$ described for functor categories $Set^C$, $Set^D$ and $f : C \to D$
For $C,D$ small categories, and $f : C \to D$ a functor between them, there is a precomposition, or "inverse image", functor $f^* = (-) \circ f : Set^D \to Set^C$. It has a left and a right adjoint. ...
- 1,021
3
votes
1
answer
345
views
Normal bundle of a linear subspace
Let $X\subset\mathbb{P}^N$ be a smooth scheme theoretical complete intersection, and $H\subset X$ a linear subspace. Denote by $N_{H,X}$ the normal bundle of $H$ in $X$.
If $\dim(H) = 1$, that is $H$ ...
- 39.3k
2
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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
4
votes
3
answers
483
views
"Quasi-coherent" vector spaces in Sch/S
$\DeclareMathOperator\Vec{Vec}\newcommand\Sch{\mathrm{Sch}}\DeclareMathOperator\Hom{Hom}$Let $S$ be a base scheme. Let me write $\Vec(S)$ to denote the category of $\mathbb A_S$-vector space objects ...
- 63.1k
9
votes
0
answers
475
views
Using higher topos theory to study Cech cohomology
It seems to me that many classical statements about Cech cohomology should without much effort follow from Lurie's HTT, but I am struggling to fill out the details. I want to show that, given an ...
- 898
2
votes
1
answer
218
views
About the support of a holonomic D-module
Let $X$ be a smooth algebraic variety over $\mathbb{C}$ and let $M^\bullet\in\mathsf{D}^b_\text{h}(\mathcal{D}_X)$ be a complex of D-modules with holonomic cohomologies. We define the support of $M^\...
- 6,829
0
votes
0
answers
215
views
Singular cohomology to cohomology of quasi-coherent sheaf
Let $X$ be a projective nonsingular variety (integral Noetherian scheme of finite type that is proper over a field $k=\overline{k}$ such that $\Omega^1_X$ is locally free). Suppose one knows the ...
- 511
54
votes
3
answers
11k
views
Sheaves and bundles in differential geometry
Because the theory of sheaves is a functorial theory, it has been adopted in algebraic geometry (both using the functor of points approach and the locally ringed space approach) as the "main theory" ...
- 894
3
votes
0
answers
186
views
The site and the space
There is a (seemingly simple) statement in the literature on sheaf theory, namely,
If $E$ is the site of opens of a topological space $X$, the notion of sheaf over $X$ coincides with that of sheaf of ...
- 894
1
vote
0
answers
82
views
Do the covariant maps of a sheaf with transfer automatically satisfy a dual gluing axiom?
I'm interested in describing a notion of equivariant sheaves that uses Mackey functors on topoi. We can describe a genuine G-spectrum as a spectral Mackey functor on the topos of finite G-sets. If we ...
- 1,969