In a recent course in Bonn, P. Scholze explains a formalization of a six-functor formalism due to L. Mann. In this axiomatization, three of the functors $f_!,f^*,\otimes$ are "constructed" (in the form of a lax monoidal functor) and then the other three are defined as right adjoints.
Now, in the derived category of holonomic D-modules over an algebraic variety in characteristic zero, we have four functors which are simple to construct:
- $f_+$: the usual direct image of D-modules, which is the analog of the functor $f_*$;
- $f^!$: the exceptional inverse image, which plays the same role as the homonymous functor in other six-functor formalisms;
- $\otimes^\mathsf{L}_{\mathcal{O}}$: the left derived functor of the O-module tensor product;
- $\mathrm{D}$: the "Verdier duality", which also works as in other six-functor formalisms.
Let me be precise: the functor $\otimes^\mathsf{L}_{\mathcal{O}}$ is not one of the six functors. (For example, in a six-functor formalism one would expect that the inverse image functor is monoidal with respect to the tensor product. This does not happen here.) But we can make it work!
Since $f^!$ is defined as a shift of $\mathsf{L}f^*$ (the O-module functor, which is also not one of the six-functors), and $\mathsf{L}f^*$ is monoidal with respect to $\otimes^\mathsf{L}_{\mathcal{O}}$, we can put $\otimes^!:=\otimes^\mathsf{L}_{\mathcal{O}}[-\dim]$. Then $f^!$ is monoidal with respect to $\otimes^!$. Finally, as the usual inverse image of D-modules $f^+$ is defined as the Verdier dual of $f^!$, we can define $\otimes$ as the Verdier dual of $\otimes^!$.
This tensor product $\otimes$ has a right adjoint $\underline{\operatorname{Hom}}(-,-)=\mathrm{D}(-\otimes\mathrm{D}(-))$ and the functors $f_+,f^!$ have left adjoints $f_!,f^+$; constituting a full six-functor formalism. (Which satisfies every nice property imaginable. And makes the analogy between holonomic D-modules and wildly ramified perverse sheaves in positive characteristic very clear.)
My question is: how could we use (or modify) L. Mann's foundations to establish this six-functor formalism? If we already know everything about this six-functor formalism, I guess it should not be too bad to prove that it enters in Mann's axiomatization. But those foundations should give simpler proofs to known (or not) statements; not repose on them, right?
Best Answer
The six functor formalism applies to $D$-modules, but you need to extend the theory to possibly non-smooth schemes. For this, we see that hypersheaves on the site of pairs $(X,Z)$, with $Z$ a closed subscheme of a smooth scheme $X$, with maps the obvious commutative squares, with coverings those maps $(X,Z)\to (X',Z')$ inducing an étale covering $Z\to Z'$ form an $\infty$-category equivalent to the $\infty$-category of hypersheaves on the usual big étale site of our ground field of characteristic zero (but we could work with arithmetic $D$-modules à la Berthelot for fields of positive characteristic, using the work of Caro). Sending $(X,Z)$ to $D$-modules over $X$ that are supported on $Z$, we get a hypersheaf of symmetric monioidal stable $\infty$-categories. What precedes says that there is a unique (hyper)sheaf of symmetric monoidal stable $\infty$-categories on the big étale that coincides with usual $D$-modules on smooth schemes (the proof is an interpretation of Kashiwara's theorem). This says how to define $\otimes$ and $f^*$ for any map of schemes of finite type over the ground field $f:Z\to Z'$. To define $f_!$, we reduce by descent as above to the case where $f$ is induced by a map of pairs $(X,Z)\to (X',Z')$, in which case we use the $g_!$ induced by $g:X\to X'$ restricted to $D$-modules supported on $Z$ and $Z'$ respectively.