8

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 $f:\mathscr X_0\to\mathscr Y_0$ be a morphism of $\mathbb F_q$-algebraic stacks, and let $K_0\in W^{-,stra}_m(\mathscr X_0,\overline{\mathbb Q}_{\ell})$ be a ...


8

I'm not sure what you mean by "rings which look like D" but here's one point of view: the de Rham functor is just derived Hom from the structure sheaf $O$, i.e. you're testing all D-modules against your favorite one. One can imagine an analog in any context where you have a favorite module. For D-modules with an integral twist, you can just use the ...


7

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 whose strata satisfy the $\mu$-regularity condition. This is equivalent to the Verdier regularity condition which in turn is a bit stronger than the Whitney ...


7

I think the vanishing you want holds. For simplicity consider the case where $i$ is the inclusion of $Z = \{ z \}$ a point. (One should be able to reduce to this case by taking a normal slice.) The question is local so we can replace $X$ by a small neighbourhood $U$ of $z$. Let $j$ denote the inclusion of $U - \{ z \}$. Consider the distinguished triangle $...


6

This is not a helpful answer to your main question, but merely a negative answer to your "In particular..what happens.." question. But, the general idea may be helpful in figuring out what more precise things (weaker than Keller's result) would be reasonable to ask for. $\newcommand{\RR}{\mathbb{R}}\newcommand{\RHom}{\mathrm{RHom}}\newcommand{\pt}{\mathrm{...


6

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 $(\mathcal{E},\mathcal{O})$ be a commutative unital ringed Grothendieck topos with enough points such that for all points $p$ of $\mathcal{E}$ the ring $\mathcal{O}...


6

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 develops constructible sheaves and the six functors in Chapter V. You might also try Dimca's Sheaves in Topology, which does talk about perverse sheaves. Neither ...


5

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 MacPherson are treating $\mathbb{R}^i$ as $cS^{i-1}$. When $X$ and $Y$ are stratified, there is a natural stratification of the join. I discuss this in Section 2.11 ...


4

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 paragraph, you're going to need a lot more structure for $F$ than what you've given. At the very least, when $F$ is a local system (lisse), you need the weights ...


3

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 along a line, such a subset will not define a coisotropic submanifold unless the covector is conormal to the line. Thus it cannot be the singular support of a ...


3

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 fact normal bundle in your case), you need to proceed by pushforward. As it is better to use a proper pushforward, the version $\tilde{X}_Y^{\mathrm{Ful}}$ is best ...


3

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 Bertini-Sard by Verdier. The point is that the conditions imply that everything is topologically locally trivial along strata in a suitably strong sense; ...


2

Let's work over $\mathbf{C}$. Take a K3 surface $S$ with a fix point free involution $\sigma$ and let $X = S \times \mathbf{C}/\langle (\sigma, -1)\rangle$ and $Y = \mathbf{C}/\langle -1 \rangle \cong \mathbf{C}$. Take $l = 2$ and compute using proper base change that your $R^1$ is not zero and supported in $0$.


2

For the first item, section 3 of M. Saito's Modules de Hodge Polarizables is pretty thorough. He works in the filtered $D$-module setting, but I suppose you can ignore that aspect. For the constructible setting, perhaps you can look at Dimca's Sheaves in Topology.


2

I do this all over $\mathbb{C}$. By [BBD] this should not be a problem. Assume $X=\mathbb{C}P^1$, $U=\mathbb{C}$, $S_1= U$, $S_0=X-U$. Let further $j_i:S_i\hookrightarrow X$ be the inclusion maps. Then $K:=j_{1!} \underline{\mathbb{C}}_{S_1}[1]\oplus j_{0!} \underline{\mathbb{C}}_{S_0}$ satisfies your condition above and is obviously not isomorphic to $\...


2

This should fail already in the simplest case when $X = \mathbb A^1$ and $U$ is the complement of a point. Namely, the trivial local system on $U$ and the point admits several different gluings to a perverse sheaf on $X$. The precise gluing data you need is explained in Beilinson's "Gluing perverse sheaves", see also http://arxiv.org/abs/1002.1686


2

You could try Sheaves in Topology by Alexandru Dimca. There are no prerequisites other than basic sheaf theory, so you don't have to worry about microlocal troubles or anything else.


2

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 $G\in D^b(M)$ is $\mathbb{R}$-constructible if there is a locally finite covering $M=\cup_\alpha M_\alpha$ of $M$ by subanalytic subsets such that $H^i(G)\vert_{...


2

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 well. Then set $X = Z\times S$, $Y = S\times S$, $f = g\times id.$ Taking the base change of $f$ with the two coordinate-axis embeddings $S \to S\times S$ gives ...


2

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 $A = \mathbb Z$, $\mathbb Z$-modules are equivalent to abelian groups and so Lemma 2.1(ii) applies. It is only 2.1(i) that is used in Remark 2.3(i). That is used to show that a sheaf of groups ...


1

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 about building the theory first with torsion coefficients and then passing to the limit to get to characteristic $0$ (this is where the issue arises that the ...


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