# Questions tagged [six-operations]

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19
questions

8
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0
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### In Mann's six-functor formalism, do diagrams with the forget-supports map commute?

One of the main goals in formalizing six-functor formalisms is to obtain some sort of "coherence theorem", affirming that "every diagram that should commute, commutes". In these ...

4
votes

0
answers

162
views

### Every stable homotopical functor factors through $\mathbf{SH}$

In this nlab page, it says that the fact that every stable homotopical functor factors through $\mathbf{SH}$ (the motivic stable homotopy category of Morel-Voevodsky) is proven in Ayoub's thesis. ...

2
votes

1
answer

368
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### Hypersheaves vs derived category of sheaves

This question arose from Peter Scholze's notes on six functor formalisms, specifically lecture VII in the proof of proposition 7.1.
We fix a LCH space $X$ and consider the functor $D(\mathrm{Ab}(X)) \...

2
votes

1
answer

200
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### Removing quasi-projective assumption in the formalism of four operations

In Ayoub's thesis, Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique (I), Ayoub proved that given a stable homotopical $2$-functor (Definition 1.4.1) $\...

10
votes

1
answer

691
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### How Mayer-Vietoris follows from a six-functor formalism

In many reasonable six-functor formalisms, open and closed immersions satisfy the so-called recollement conditions. (This holds in all the "constructible" formalisms. For example, in the ...

6
votes

0
answers

210
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### Is the right adjoint to presheaf direct image interesting?

Let $X\overset{f}{\to}Y$
be a continuous map. It induces on presheaves a classical adjunction inverse image ⊣ direct image. However, the direct image functor has a further right adjoint, defined by ...

3
votes

0
answers

548
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### 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 ...

6
votes

0
answers

938
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### 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
votes

0
answers

316
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### 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

1
answer

247
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### $f^!=f^*[d]$ 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

1
answer

202
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### 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

0
answers

123
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

0
answers

625
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### 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

0
answers

451
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### 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

1
answer

775
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### 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

1
answer

1k
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### 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

2
answers

426
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### 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

1
answer

1k
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### 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

4
answers

5k
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 (...