Questions tagged [constructible-sheaves]
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62
questions
19
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0
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866
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Epsilon factors - a la Beilinson - What is it?
I understand, to some extent, Tate's thesis. Could somebody explain perhaps what are the epsilon factors in Beilinson's works, such as "$\epsilon$-factors for Gauss-Manin determinants", or "...
- 5,492
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votes
0
answers
405
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Constructible derived category and fundamental category
Introduction (may be skipped)
Given a nice topological space $X$, the category of local systems (say over a field $k$) on it is equivalent to the category of representations of its fundamental ...
- 12.9k
17
votes
0
answers
656
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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)...
- 355
14
votes
2
answers
1k
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"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 ...
- 7,549
10
votes
0
answers
835
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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 ...
- 546
8
votes
1
answer
982
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Analogues of D-modules and constructible sheaves
For a smooth complex variety, one can consider the category of say holonomic $\mathcal D$ modules on it. It is equipped with the deRham functor, which turns a $\cal D$-module into a constructible ...
- 12.9k
8
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0
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364
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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 ...
- 1,635
8
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0
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357
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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, ...
- 81
8
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0
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407
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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-...
- 633
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346
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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 ...
- 8,643
7
votes
2
answers
778
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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 ...
- 5,630
6
votes
1
answer
930
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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, ...
- 405
6
votes
1
answer
951
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Constructible sheaves and dg-modules
Let $M$ be a smooth manifold, $A_M$ the de Rham algebra of $M$, $D_{A_M}$ the derived category of the category of differential graded (dg) $A_M$-modules and $D^+_c(M)$ the bounded below constructible ...
- 23.2k
6
votes
1
answer
366
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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 ...
- 9,802
6
votes
1
answer
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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^\...
- 3,031
6
votes
1
answer
441
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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 ...
- 3,031
6
votes
1
answer
600
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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 ...
- 21.1k
6
votes
1
answer
365
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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 ...
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258
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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 (...
- 273
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0
answers
451
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Purity and six operations?
The six operations $f_!,f^!,f_*,f^*,\otimes,\mathcal Hom$ have the property that they preserve estimates on weights in one direction.
For $f_!,f^!,f_*,f^*$ I can see, that they don't preserve purity ...
- 12.9k
5
votes
3
answers
662
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 ...
- 8,726
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votes
1
answer
210
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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) ...
- 2,035
5
votes
1
answer
606
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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 ...
- 453
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votes
1
answer
652
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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 ...
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0
answers
229
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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
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0
answers
189
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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 ...
- 21.1k
5
votes
0
answers
618
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singular support of D-module smooth w.r.t. a stratification
(1) Suppose that $X$ is a smooth complex algebraic variety, stratified by some nice smooth stratification $S$. Let $M$ be a $D$-module on $X$, s.t. its shriek-pullback (or star... whatever is ...
- 5,492
4
votes
1
answer
249
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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 $...
- 41
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1
answer
489
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Nearby cycles and specialisation - properties
I am looking for reference for properties of nearby cycles - specifically, commutation with non-characteristic pull-back (good enough - commutation with pull-back to closed subvariety which is ...
- 5,492
4
votes
0
answers
133
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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 $\...
- 6,125
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0
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225
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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? ...
- 1,706
4
votes
0
answers
247
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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 ...
- 997
3
votes
1
answer
341
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Vanishing of !-restriction of constructible sheaves
If $\mathcal F$ is a constructible sheaf (say of $\mathbb C$-modules) on a (real) manifold concentrated in degree $0$ and $i\colon Z \hookrightarrow X$ is a submanifold, can I say anything about $H^j(...
3
votes
1
answer
201
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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 ...
- 163
3
votes
1
answer
476
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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....
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0
answers
255
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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 ...
- 5,851
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 $T^{*}X$ ...
- 3,432
2
votes
1
answer
279
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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 ...
- 153
2
votes
1
answer
143
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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 ...
- 23
2
votes
1
answer
168
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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{...
- 6,000
2
votes
1
answer
478
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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 $Y$ ...
- 155
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2
answers
421
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Reference wanted - preservation of constructible sheaves (in classical topology) by all functors
Hello,
Can anybody point to me a reference about the preservation of the derived bounded category of sheaves with constructible cohomology on the underlying classical (anayltic) space of a complex ...
- 5,492
2
votes
0
answers
209
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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 $U \to X$,...
- 6,000
2
votes
0
answers
133
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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 ...
- 1,021
2
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0
answers
130
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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 ...
- 21.1k
2
votes
0
answers
118
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 ...
- 373
2
votes
0
answers
190
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 ...
- 1,472
2
votes
1
answer
413
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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 ...
- 21
2
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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 ...
- 617
2
votes
0
answers
179
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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,
(...
- 927