Questions tagged [six-operations]

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3
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0answers
239 views

Most general form of Poincaré duality in étale cohomology

I am interested in Poincaré duality from the point of view of Grothendieck's 6-functor formalism. I am predominantly interested in the proof that Poincaré duality holds in étale cohomology from this ...
5
votes
0answers
296 views

Intuition behind exceptional inverse image?

The story is probably well-known: given a map $f:X\to Y$ of spaces (say schemes, but there are many other contexts), we have two classical operations between sheaves on $X$ and those on $Y$: the ...
4
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0answers
217 views

Universal six-functor formalism on an $\infty$-category

In the article The Universal Six-Functor Formalism by Brad Drew and Martin Gallauer it is proved that for an ordinary category $S$ with a wide subcategory $P$ of 'smooth morphisms' containing all ...
2
votes
1answer
232 views

$f^!=f^*[d]$ for quasismooth maps?

Given a smooth map of schemes $f:X\to Y$ of relative dimension $d$, then there is a natural isomorphism $f^!\simeq f^*[d](2d)$ (in any context where the six operations are defined; see Cesinski-...
3
votes
1answer
118 views

Subspace inclusion with non-vanishing higher direct images

I'm looking for concrete topological intuition for the derived pushforward. Let $f:X\to Y$ be a continuous map. The derived pushforward $\mathbf Rf_\ast$ takes a sheaf $F$ to the sheafification of ...
1
vote
0answers
120 views

Six operations passage from $X_0$ to $X$ reference request

Let $X_0$ be a variety over $\mathbb F_q$ and denote by $X$ its basechange to the algebraic closure. Consider the constructible derived categories $D^b_c(X_0,\mathbb E)$ and $D^b_c(X,\mathbb E)$, ...
5
votes
0answers
528 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
votes
0answers
414 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 ...
7
votes
1answer
657 views

Overconvergent/infinitesimal site, base change and six operations

This question is about 6 operations formalism for 'crystalline' cohomology theories - more specifically the infinitesimal cohomology of smooth $\mathbb{C}$-varieties, and the overconvergent cohomology ...
8
votes
1answer
881 views

Tensor product of $\mathcal{D}$-modules and constructible sheaves

The Riemann-Hilbert correspondence, as proved by Kashiwara and Mebkhout, says that for X a smooth algebraic variety over $\mathbb{C}$ there is an equivalence of triangulated categories $D^b_c(X,\...
2
votes
1answer
354 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 ...
11
votes
1answer
1k views

When do six operations work?

This question comes (heavily edited) from my notes, thus slightly unusual structure. We know that algebraic maps have very strict structure, and in many settings the operations ...
11
votes
4answers
4k views

When does the sheaf direct image functor f_* have a right adjoint?

Say f: X → Y is a morphism of schemes. The sheaf direct image functor f★ always has a left adjoint, namely the sheaf inverse image functor f★ (with tensoring). Under what (...