10 votes
Accepted

Generalized Behrend version for Grothendieck-Lefschetz trace formula

This is Theorem 4.2 of Shenghao Sun's paper $L$-Series of Artin stacks over finite fields, Algebra & Number Theory 6 (2012) pp 47–122, doi:10.2140/ant.2012.6.47, arXiv:1008.3689. Let $f:\...
Will Sawin's user avatar
  • 135k
8 votes

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

Consult the book "Sheaves on manifolds" by Kashiwara and Schapira. It's a hard nut to crack, but it is the most efficient presentation I've seen. In Chapter 8, they work with stratifications ...
Liviu Nicolaescu's user avatar
6 votes

Deequivariantisation of indecomposable sheaves

Take $X=pt$ and $G=G_m$ and $k$ to have characteristic zero. The equivariant derived category in this case is equivalent to modules for the homology of the circle, ie exterior algebra on a generator ...
David Ben-Zvi's user avatar
6 votes

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

A bit more reader friendly than Kashiwara-Schapira is Borel's Intersection Cohomology. It doesn't treat perverse sheaves, but it has a good overview of stratified spaces in the first few chapter and ...
Greg Friedman's user avatar
6 votes
Accepted

Picard group of derived category of sheaves

Thanks to Drew Heard's comment I was able to find answers to my questions. In his paper "Picard groups of derived categories" H. Fausk proves the following theorem (see Theorem 4.2). Theorem: Let $(\...
David C's user avatar
  • 9,792
6 votes
Accepted

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

A necessary condition is that for each stratum $X_{\eta}^i$ of some stratification of $X_{\eta}$ on which $\mathcal F$ is lisse, the associated representation of $\pi_1(X_{\eta}^i)$ is unramified away ...
Will Sawin's user avatar
  • 135k
5 votes

Constructible étale sheaves on X are étale algebraic spaces over X

In case anyone wants to know a reference, I found it now: SGA4, Exp. IX, Prop. 2.7 Statement (Translated): Proposition 2.7 Let $X$ be a quasicompact and quasiseparated scheme, and let $F$ be a ...
Steve's user avatar
  • 453
5 votes
Accepted

Isomorphic IC sheaves induced from different locally closed subvarieties

It's not hard to see that the constructtion you give is the unique way to do it. It's not hard to check that the intermediate extension of an intermediate extension is an intermediate extension, and ...
Will Sawin's user avatar
  • 135k
5 votes
Accepted

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

The following answer was emailed to me by Claude Sabbah. I have received his permission to post it here. Whenever you need to get some object on $Y$ from an object existing on the normal cone (in ...
Avi Steiner's user avatar
  • 3,031
5 votes
Accepted

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

As noted by Chris Gerig in the comments, letting $cX$ denote the open cone on the compact space $X$ then $(cX)\times (cY)\cong c(X*Y)$, where $X*Y$ is the join. In the case at hand, Goresky and ...
Greg Friedman's user avatar
5 votes
Accepted

Is analytification of regular holonomic D modules a fully faithful functor?

Yes, for a smooth algebraic variety $X$, the analytification functor $D^b(\mathcal D_X)_{rh} \to D^b(\mathcal D_{X^{an}})_{rh}$ is fully faithful. As you note, it is a consequence of the usual ...
Sam Gunningham's user avatar
4 votes

Interesting (non) examples of singular support

The singular support of a sheaf is always coisotropic (Theorem 6.5.4 in Kashiwara and Schapira). In the example of the 2-dimensional conic subset of $T^\ast \mathbb R^2$ defined by a covector field ...
Sam Gunningham's user avatar
4 votes
Accepted

Comparing Frobenius weights with Mixed Hodge theory

For constant coefficients, the comparison statement, along with a sketch, appears in Deligne's ICM talk, Poids dans la cohomologie.... For things to work the way you seem to want in your second ...
Donu Arapura's user avatar
  • 34.2k
4 votes
Accepted

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

No, this is not a purely $\ell$-adic phenomenon. Let $X = \mathbb P^1$, $S$ the stratification with one stratum so sheaves constructible with respect to this stratification are lisse and complexes ...
Will Sawin's user avatar
  • 135k
4 votes

Deequivariantisation of indecomposable sheaves

Take $G_m$ acting on itself via $z\mapsto z^2$, and take $k$ to be of characteristic $2$. The equivariant derived category here is the derived category of $\mathbb{Z}/2\mathbb{Z}$-modules. Now take ...
rvk's user avatar
  • 553
3 votes
Accepted

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

I'm going to use $g$ for the equation of the curve since using $f$ for both the equation of the curve and the map to $\operatorname{Spec}(\mathbb C[s,t])$ leads to ambiguity. The situation just using ...
Will Sawin's user avatar
  • 135k
3 votes
Accepted

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

For a discussion of Whitney stratifications, see for example Stratified Morse theory Goresky and Macpherson, Notes on topological stability by Mather, or Stratifications de Whitney et théorème de ...
Donu Arapura's user avatar
  • 34.2k
3 votes

Deequivariantisation of indecomposable sheaves

This is essentially a "down to earth" version of what David Ben-Zvi is saying. The object/example produced below essentially matches with what David is suggesting. I am just producing it &...
rvk's user avatar
  • 553
2 votes

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

I just came across my own question again and think meanwhile I can give an answer: By definition (e.g. Definition 8.3.4 in sheaves on manifolds by M. Kashiwara and P. Schapira), a sheaf $G\in D^b(M)$ ...
user103697's user avatar
2 votes
Accepted

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

I think you are misinterpreting things slightly. Lemma 2.1 says nothing about abelian groups. Lemma 2.1(i) is about sets, and Lemma 2.1(ii) is about A-modules. For $A = \mathbb Z$, $\mathbb Z$-...
Will Sawin's user avatar
  • 135k
2 votes

A property of nearby cycles functor

The answer is no! Start with any $g:Z \to S$, with smooth generic fiber. Write $Z_0$ for the closed fiber; in order for your conditions to be eventually satisfied this should be assumed smooth as ...
Gus's user avatar
  • 21
2 votes
Accepted

Relation between characteristic cycle and singular support of constructible sheaf

No. Consider $M = \mathbb R$, $F$ the direct sums of the constant sheaves on the positive real numbers, negative real numbers, and $0$, extended by zero to the whole space. Then $F$ is the associated ...
Will Sawin's user avatar
  • 135k
2 votes
Accepted

Fourier transform for constructible sheaves on spheres

We have $q_1 = p_1 \circ j$ and $q_2 = p_2 \circ j$ so $$ {q_2}_! q_1^*(\mathcal F) \cong {p_2}_! j_! j^* p_1^*(\mathcal F) \cong {p_2}_! j_! ( j^* p_1^*(\mathcal F) \otimes \mathbb Q_U) \ \cong {...
Will Sawin's user avatar
  • 135k
1 vote

Exit path categories of regular CW complexes

It seems to me like this statement is folklore, since e.g. the paper Stellar Stratifications on Classifying Spaces tries to show a generalization of it and at least hints that my simpler claim is true ...
Markus Zetto's user avatar
1 vote

Stratification along which a constructible complex is smooth

I think the answer is yes for the Bruhat stratification, and moreover for the orbit stratification of any space $X$ with an action of a group $G$ with finitely many orbits, such that any irreducible ...
Will Sawin's user avatar
  • 135k
1 vote
Accepted

Weak Lefschetz theorem for Lef line bundles

It is based on certain vanishing property of $U= X\backslash Y$. First you have a long exact sequence (a derived categorical version is given in the end) $$H^k(X,Y;\mathbb{Q})\rightarrow H^k(X,\mathbb{...
XT Chen's user avatar
  • 1,064
1 vote

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

There are two different issues here which I think you are confusing: 1) that the right coefficients are $l$-adic instead of archimedean (these are the "moral" reasons you mention), and 2) the issue ...
Nicolás's user avatar
  • 2,802

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