Holmstrom Grothendieck's six ops

Something about a similar setup in algebraic topology is here

Possibly “Hartshorne: Residues and duality” could be a good reference

Joseph Ayoub thesis

The six functors formalism give rise to (“all”?) spectral sequences… ???

http://mathoverflow.net/questions/21051/why-does-the-naive-definition-of-compactly-supported-etale-cohomology-give-the-wr

http://mathoverflow.net/questions/108614/intermediate-extension-functor-exact

In Voevodsky (and Deligne): Lectures on cross functors, there is a discussion of the formalism of the four Grothendieck operations. This is maybe superseded by Ayoub’s work. However, the lectures have some nice brief background info on the formalism (for etale sheaves), including PD and the four kind of (co)homology, indicating where there are difficulties. There is not really any material on motivic homotopy theory, it just says that such applications will be given “later”.

arXiv:1209.4020 The formalism of Grothendieck’s six operations in p-adic cohomologies from arXiv Front: math.AG by Daniel Caro Let $\mathcal{V}$ be a complete discrete valued ring of mixed characteristic $(0,p)$, $K$ its field of fractions, $k$ its residue field which is supposed to be perfect. Let $X$ be a separated $k$-scheme of finite type and $Y$ be an open subscheme of $X$. We construct the category $F\text{-}D ^\mathrm{b}_\mathrm{ovhol} (\mathcal{D} ^\dag_{(Y,X)/K})$ of overholonomy type over $(Y,X)/K$. We check that these categories satisfy a formalism of Grothendieck’s six operations.

arXiv:0908.2799 Natural Commuting of Vanishing Cycles and the Verdier Dual from arXiv Front: math.AG by David B. Massey We prove that the shifted vanishing cycles and nearby cycles commute with Verdier dualizing up to a {\bf natural} isomorphism, even when the coefficients are not in a field.

arXiv:1211.5294 Gluing restricted nerves of infinity-categories fra arXiv Front: math.AG av Yifeng Liu, Weizhe Zheng In this article, we develop a general technique for gluing subcategories of infinity-categories and prove that certain maps of simplicial sets are categorical equivalences. This applies in particular to maps naturally arising from the study of algebraic geometry, and allows us to upgrade the theory of derived categories of étale sheaves on schemes to the infinity-categorical level. In a subsequent article, we will use this technique to establish a general theory of Grothendieck’s six operations for Artin stacks.

arXiv:1211.5948 Enhanced six operations and base change theorem for sheaves on Artin stacks fra arXiv Front: math.AG av Yifeng Liu, Weizhe Zheng In this article, we develop a theory of Grothendieck’s six operations for lisse-étale sheaves on Artin stacks and prove all expected properties including the Base Change theorem. This extends all previous theories on this subject, including the recent one developed by Laszlo and Olsson. In particular, if we restrict ourselves to constructible sheaves, we obtain the same six operations as Laszlo and Olsson but for more general Artin stacks, with the Base Change isomorphism constructed in the derived category, and without their technical assumptions on the base scheme or on the coefficient rings. Moreover, our theory works for higher Artin stacks as well

Our method differs from all previous approaches, as we exploit the theory of stable infinity-categories developed by Lurie. These higher categories are viewed as enhancement of usual derived categories. We introduce the infinity-categorical (co)homological descent for Cartesian sheaves and develop several infinity-categorical techniques, which, together with those in our previous article arXiv:1211.5294, allow us to handle the “homotopy coherence”.

nLab page on Grothendieck's six ops