Questions tagged [constructible-sheaves]

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Find stratification to decompose constructible sheaf to constant parts (example from Wikipedia)

I have a question about techniques used in determining the stratification over which a constructible sheaf falls into even constant pieces demonstrated on this example from Wikipedia. Let $f:X = \text{...
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1 answer
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Isomorphic IC sheaves induced from different locally closed subvarieties

Let us work with varieties over $\mathbb{C}$ and $D^{b}_{c}(X)$ the bounded constructible derived category of sheaves of $\mathbb{Q}$ vector spaces. Say $X$ and $Y$ are smooth locally closed ...
arczn's user avatar
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"Simple Limit Argument" in Freitag's and Kiehl's Etale Cohomology

I have a question about an argument used in Freitag's and Kiehl's Etale Cohomology and the Weil Conjecture in the proof of: 4.4 Lemma. (p 41) Every sheaf $F$ representable by an étale scheme $U \to X$,...
user267839's user avatar
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1 vote
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Cohomology of an intermediate extension (perverse) sheaf on the affine line

Let $\mathbb{A}^1$ be defined over a finite field or $\mathbb{C}$, $j: \mathbb{G}_m \rightarrow \mathbb{A}^1$ and $\mathcal{F}$ a local system on $\mathbb{G}_m$. I wonder what is known about the ...
BnPrs's user avatar
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Reference request: algebraic characteristic 0 version of microlocalization

I am trying to learn about microlocalization and singular supports with the end goal of understanding at least some form of the coherent-constructible correspondence. I am currently powering through ...
Sergey Guminov's user avatar
1 vote
1 answer
158 views

$\text{Ext}$-groups of perverse sheaves with a fixed stratification

Let $X$ be a complex variety with a good stratification $S$ and consider the category $Perv_S(X)$ of sheaves perverse with respect to the given stratification (with middle perversity) lying in $D^b_S(...
Sergey Guminov's user avatar
2 votes
0 answers
127 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 ...
asv's user avatar
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6 votes
1 answer
354 views

Exit path categories of regular CW complexes

Given a finite, regular CW complex $X$ (by regular, I mean that the gluing maps $D^n \to X$ from the closed unit ball to $X$ are homeomorphisms onto their image), denote by $S$ the finite partially ...
Markus Zetto's user avatar
2 votes
0 answers
115 views

Local systems as a Serre subcategory of the category of perverse sheaves

Let $X$ be an algebraic variety. Let $Perv(X)$ be the (abelian) category of perverse sheaves on $X$ and let $Loc^{ft}(X)$ be the subcategory of local systems with finitely-generated stalks. It is ...
Laurent Cote's user avatar
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1 answer
161 views

Fourier transform for constructible sheaves on spheres

Let $S_1 = S_2 = S^d$ be two copies of the $d$-dimensional sphere. Let $p_i : S_1 \times S_2 \to S_i$ be the projection, $j : U \to S_1 \times S_2$ the inclusion of the complement of the diagonal and $...
Nicolas Hemelsoet's user avatar
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0 answers
188 views

Stratified sites/topoi and constructible sheaves

Is it possible to define (possibly derived) categories of constructible sheaves over sites more general than those of open subsets of topological spaces while still retaining essential features, like ...
Dat Minh Ha's user avatar
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2 votes
1 answer
386 views

Conservativity of stalks of ind-constructible sheaves

I have a simple question about the conservativity of stalks of ind-constructible sheaves. Let $X$ be a topologically noetherian scheme, $S$ a set of geometric points of $X$ corresponding bijectively ...
Owen Barrett's user avatar
5 votes
3 answers
660 views

Deequivariantisation of indecomposable sheaves

Let G be a connected group acting on a space X. All spaces should be reasonable, so e.g. G is a complex affine algebraic group acting on an algebraic variety X, with everything done using the usual ...
Peter McNamara's user avatar
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0 answers
157 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 ...
Adrien MORIN's user avatar
5 votes
0 answers
229 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 $D_{\mathcal{S}}(X)$ ...
Laurent Cote's user avatar
6 votes
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258 views

The geometric "hands-on" vs. algebraic approach to nearby cycles

Feel free to skip to the question below; the following is just context and discussion: An interesting, but seemingly less used result in the theory of nearby cycles of constructible sheaves is (...
Mathmank's user avatar
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Constructible motivic sheaves

Motivic complexes are a complex of Zariski sheaves that their Zariski hypercohomology gives us the motivic cohomology groups. There are various constructions of these complexes. As far as I know they ...
user127776's user avatar
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Is analytification of regular holonomic D modules a fully faithful functor?

It seems to me that by the algebraic Riemann Hilbert functor(which factors through the analytification map) and also the analytic Riemann Hilbert functor that the (derived) category of (algebraic) ...
davik's user avatar
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Sheaves with specified singular support at infinity coming from hyperplane arrangements

Given a manifold $M$, we consider its cotangent bundle $T^*M$, and its cocircle bundle $T^\infty M$, quotienting out by the scaling action of the positive reals. Given a Legendrian submanifold $\...
Hugh Thomas's user avatar
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10 votes
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825 views

intuition about perverse sheaves

firstly, I would know if my very basic intuition on perverse sheaves is correct . secondly, I would have some clarification in what perverse sheaves behaves better than regular sheaves . my intuition ...
Amos Kaminski's user avatar
1 vote
1 answer
235 views

Relation between characteristic cycle and singular support of constructible sheaf

Let $M$ be a real analytic manifold. Let $F$ be an object of the bounded derived category of sheaves on $M$ with real constructible cohomology sheaves. Let $CC(F)$ denote the characteristic cycle of $...
asv's user avatar
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3 votes
1 answer
474 views

Weak Lefschetz theorem for Lef line bundles

I'm studying M. A. A. de Cataldo, L. Migliorini - The Hard Lefschetz Theorem and the topology of semismall maps, Ann. sci. École Norm. Sup., Serie 4 35 (2002) 759-772. The premises are the following....
Armando j18eos's user avatar
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159 views

Galoisian perspective on local system tamely ramified along a smooth divisor

This question is about (1.7.8) and (1.7.11) in Deligne’s Weil II paper. Let $X$ be a regular scheme and $D\subset X$ a smooth principal divisor cut out by the function $t$. Let $\mathcal F$ be a ...
Tomo's user avatar
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8 votes
0 answers
357 views

Cohomology of constructible sheaves via exit paths

Let $X$ be a stratified space, with stratification $S$ (we will ignore technicalities). The category of exit paths $Ex(X,S)$ is a directed refinement of the path groupoid of $X$ accounting for the ...
Patrick Elliott's user avatar
2 votes
1 answer
276 views

Help with $\mathbf{Q}_{\ell}$ sheaves

Let $X\to S$ be a morphism of smooth connected varieties over an algebraically closed field $k$; let $j:\eta\to S$ be the inclusion of the generic point into $S$ (not a geometric generic point) and ...
delgato's user avatar
  • 153
1 vote
0 answers
74 views

Effaceability conditions in the derived category

In abelian categories effaceability of functors is often an interesting property. Is there any general equivalent condition on derived functors in the derived category? For example, for a functor ...
Patrick Elliott's user avatar
3 votes
0 answers
162 views

singular support in the singular case

For any constructible sheaf (or D-module) $\mathcal{F}$ over a smooth variety $X$ over $\mathbb{C}$, there is a notion of singular support $SS(\mathcal{F})$ that lives in the cotangent bundle $T^{*}X$ ...
prochet's user avatar
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5 votes
1 answer
597 views

Constructible étale sheaves on X are étale algebraic spaces over X

I saw the following statement in a paper of Bhatt-Mathew: Let $X$ be a quasicompact quasiseparated scheme. Then there is an equivalence of categories between constructible étale sheaves (of sets) on ...
Steve's user avatar
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1 vote
1 answer
129 views

Stratification along which a constructible complex is smooth

Let $X$ be a smooth complex algebraic variety. A constructible complex $F$ on $X$ has a singular support $SS(F)\subset T^*X$. Assume you are given a stratification of $X$ such that $SS(F)$ is the ...
hennlu's user avatar
  • 323
8 votes
0 answers
356 views

Proof of Kashiwara's constructibility theorem for algebraic D-modules

I am trying to understand the proof of Kashiwara's constructibility theorem for algebraic D-modules, following either the book "D-modules, Perverse sheaves, and representation theory" by Hotta, ...
sesame's user avatar
  • 81
6 votes
1 answer
926 views

Generalized Behrend version for Grothendieck-Lefschetz trace formula

[MOVED HERE FROM MSE.] The statement of the Grothendieck-Lefschetz fixed point theorem is well-known. For a proper algebraic variety $X$ over $\mathbb F_q$, $$\#X(\mathbb F_q) =\sum_i (−1)^i Tr(Fr_X, ...
W. Rether's user avatar
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2 votes
0 answers
177 views

Grothendieck group of constructible sets

Let $K_0$ be the Grothendieck group of complex algebraic varieties. This is the group generated by all complex algebraic varieties, subject to the relations: (i) $[X]=[Y]$ if $X,Y$ are isomorphic, (...
user142700's user avatar
4 votes
1 answer
249 views

Interesting (non) examples of singular support

I'm trying to better understand singular support of sheaves on smooth manifolds---to this end: What are examples of conical subsets of $T^*X$ that cannot arise as the singular support of a sheaf on $...
SS-SOS's user avatar
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17 votes
0 answers
654 views

Proof of MacPherson's result about set-valued constructible sheaves and exit paths

I'm looking for a proof of a theorem that is attributed to MacPherson. Treumann (Section 1.1 in Exit paths and constructible stacks, 2009) states the theorem as: Theorem 1.2 (MacPherson). Let $(X,S)...
Jānis Lazovskis's user avatar
6 votes
1 answer
664 views

Fulton's deformation to the normal cone vs Verdier's

Let $X$ be a smooth variety over a field $k$, and let $Y$ be a smooth subvariety. In the literature, I've seen two versions of the deformation to the normal cone: Verdier's version: $\tilde{X}_Y^\...
Avi Steiner's user avatar
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6 votes
1 answer
438 views

Confusion about a proof from Goresky and MacPherson's "Intersection Homology II"

Context My question is about the "proof of claim" on page 84 of Goresky and MacPherson's "Intersection Homology II". For ease of reading, here's the claim: Claim: Suppose $X$ is a topological ...
Avi Steiner's user avatar
  • 3,031
8 votes
0 answers
404 views

Category of representations of the path-groupoid

The path-groupoid $\mathcal{P}_1(X)$ of a (smooth) topological space $X$ is a refinement of the fundamental groupoid $\Pi_1(X)$ whose morphisms are given by (piecewise smooth) paths in $X$ up to thin-...
Carlos's user avatar
  • 633
2 votes
0 answers
35 views

If the set of non-0 stalks of F is relatively open, is the same true of its Verdier dual?

Let $X$ be a complex manifold, $F$ a bounded complex of $\Bbb C_X$-modules with constructible cohomology. If the set $\{x: F_x\neq0\}$ is relatively open (i.e. open in its closure), is the same true ...
Avi Steiner's user avatar
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4 votes
0 answers
225 views

Is there an analogue to the koszul complex for constructible sheaves?

Given a variety $X$ and a complete-intersection morphism $$ Y \to X $$ is there an analogue of the Koszul complex for $\mathcal{O}_Y \in \textbf{Coh}(X)$ in the setting of constructible sheaves? ...
54321user's user avatar
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7 votes
2 answers
774 views

What is the need for torsion in the definition of lisse sheaves?

I am studying the basics of constructible and lisse sheaves, and am trying to understand SGA 4, IX. As Grothendieck himself observes at the beginning of the chapter, one is forced to work with torsion ...
Filippo Alberto Edoardo's user avatar
5 votes
1 answer
646 views

Comparing Frobenius weights with Mixed Hodge theory

For a variety over a finite field Deligne define weights by looking at the eigenvalues of the Frobenius. On the other hand, if we take a variety over $\mathbb{Q}$, at least for its constant sheaf we ...
S. S.'s user avatar
  • 335
6 votes
1 answer
600 views

A property of nearby cycles functor

Let $f\colon X\to Y$ be a flat morphism of irreducible projective algebraic varieties over $\mathbb{C}$ (or any other algebraically closed field of characteristic 0). Assume that $Y$ is smooth, and ...
asv's user avatar
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1 vote
1 answer
273 views

condition for constructibility of direct images of constructible sheaves under open embedding

In $D$-Modules, Perverse Sheaves and Representation Theory from R. Hotta, K. Takeuchi and T. Tanisaki, I found the following statement (in section 8.2, the lines before Definition 8.2.2): Setting: ...
user103697's user avatar
6 votes
1 answer
365 views

Picard group of derived category of sheaves

Let $X$ be a topological space and $R$ be a commutative ring with unit, $D(X,R)$ is the derived category of unbounded complexes of sheaves of $R$-modules. Moreover we suppose that $X$ is a stratified ...
David C's user avatar
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3 votes
1 answer
201 views

Sheaves with $\mathbb{R}$-constructible proper direct image closed under dualizing?

I learned about $\mathbb{R}$-constructible sheaves very recently, so I hope this isn't a silly question: Let $M$ be a real analytic manifold, and $U\subset M$ an open subanalytic subspace. Denote the ...
user103697's user avatar
14 votes
2 answers
1k views

"Correct" definition of stratified spaces and reference for constructible sheaves?

It seems that the theory of constructible sheaves (in particular anything that goes into proving that they form an abelian category) requires some technical statements about existence of certain ...
Saal Hardali's user avatar
  • 7,549
5 votes
0 answers
188 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 ...
asv's user avatar
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4 votes
0 answers
246 views

How to compute the first etale cohomology of a constructible torsion-free sheaf?

I am interested in the following example! Let $k$ be a field, let $X_0$ be the scheme $\mathrm{Spec}R$ with $R_0=k[x,y]/(xy)$, let $R$ be the strict Hensilian localalisation of $R_0$ at the origin ...
Heer's user avatar
  • 1,007
2 votes
1 answer
473 views

Higher direct image of locally constant torsion sheaf (étale cohomology)

Let $\phi:X\rightarrow Y$ be a generically smooth projective surjective morphism of algebraic varieties over $k=\bar k.$ Is it possible for $R^1\phi_*(\mathbb Z/l)$ to be supported on a divisor of $Y$ ...
user3001's user avatar
  • 155
8 votes
0 answers
345 views

Why do Kashiwara and Schapira require a base ring of finite global dimension?

In the book "Sheaves on Manifolds" by Kashiwara and Schapira, they work always with sheaves of $R$-modules, where $R$ is a ring of finite global dimension. Why do they do this, what care ...
Vivek Shende's user avatar
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