Questions tagged [constructible-sheaves]

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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 "...
Sasha's user avatar
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18 votes
0 answers
403 views

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 ...
Jan Weidner's user avatar
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17 votes
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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
10 votes
0 answers
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
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
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
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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
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8 votes
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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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6 votes
0 answers
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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6 votes
0 answers
451 views

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 ...
Jan Weidner's user avatar
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5 votes
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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 $D_{\mathcal{S}}(X)$ ...
Laurent Cote's user avatar
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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5 votes
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615 views

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 ...
Sasha's user avatar
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4 votes
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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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4 votes
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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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4 votes
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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 ...
Heer's user avatar
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3 votes
0 answers
251 views

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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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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2 votes
0 answers
208 views

"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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2 votes
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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
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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2 votes
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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
2 votes
0 answers
187 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
2 votes
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
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
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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2 votes
0 answers
102 views

Continuity of constructible derived category

Let $X_0$ be a variety over $\mathbb F_q$. Denote by $X_n$ its basechange to $\mathbb F_{q^n}$ and let $X=\lim X_n$ be its basechange to the algebraic closure $\overline{\mathbb F}_q$. Let $D^b_c(X_n,...
Jan Weidner's user avatar
  • 12.8k
1 vote
0 answers
48 views

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
  • 195
1 vote
0 answers
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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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
0 votes
0 answers
168 views

constructibility for pushforward

Let consider a quasicompact open $j:U\rightarrow\mathbb{A}^{\mathbb{N}}$ over a field $k$, Is there an example where $Rj_{*}\mathbb{Z}/n\mathbb{Z}$ is not constructible, where $n$ is prime to the ...
prochet's user avatar
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