# Questions tagged [constructible-sheaves]

The constructible-sheaves tag has no usage guidance.

44
questions

**4**

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104 views

### 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 $\...

**8**

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**0**answers

369 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 ...

**1**

vote

**1**answer

95 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 $...

**2**

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66 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....

**1**

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78 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 ...

**6**

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172 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 ...

**2**

votes

**1**answer

197 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 ...

**1**

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66 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 ...

**3**

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94 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 $T^{*}X$ ...

**5**

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**1**answer

226 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 ...

**1**

vote

**1**answer

102 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 ...

**7**

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175 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, ...

**4**

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**1**answer

737 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, ...

**2**

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153 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,
(...

**4**

votes

**1**answer

156 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 $...

**15**

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428 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)...

**5**

votes

**1**answer

377 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^\...

**6**

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**1**answer

365 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 ...

**8**

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**0**answers

266 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-...

**2**

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31 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 ...

**4**

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211 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? ...

**6**

votes

**2**answers

529 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 ...

**5**

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**1**answer

428 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 ...

**5**

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**1**answer

441 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 ...

**1**

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**1**answer

184 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: ...

**6**

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**1**answer

263 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 ...

**3**

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**1**answer

183 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 ...

**13**

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**2**answers

616 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 ...

**5**

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158 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 ...

**4**

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195 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 ...

**2**

votes

**1**answer

346 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 $Y$ ...

**8**

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310 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 ...

**0**

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125 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 ...

**2**

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**1**answer

364 views

### 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 ...

**17**

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737 views

### 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 "...

**2**

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98 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,...

**1**

vote

**2**answers

237 views

### on a characterisation of the intersection complex

Let $X$ be an integral scheme of finite type over a field $k$ of dimension $d$ and $U$ an open dense smooth subscheme.
Let $K\in D_{c}^{b}(X,\bar{\mathbb{Q}}_{l})$ be such that $K_{U}=\bar{\mathbb{Q}}...

**5**

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500 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 ...

**6**

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386 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 ...

**17**

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368 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 ...

**3**

votes

**1**answer

294 views

### 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(...

**7**

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**1**answer

735 views

### 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 ...

**2**

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**1**answer

343 views

### 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 ...

**6**

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**1**answer

841 views

### 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 ...