Can we use Mann's six-functor formalism with D-modules?

Question

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:

1. $f_+$: the usual direct image of D-modules, which is the analog of the functor $f_*$;
2. $f^!$: the exceptional inverse image, which plays the same role as the homonymous functor in other six-functor formalisms;
3. $\otimes^\mathsf{L}_{\mathcal{O}}$: the left derived functor of the O-module tensor product;
4. $\mathrm{D}$: 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?

Answer 1

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.

Answer 2

I have finally found some time to write up the $6$-functor formalisms in coherent cohomology (a la Gaitsgory--Rozenblyum) and for $D$-modules, see Lecture 8 and its appendix.

A short synopsis is that one can start with the functor taking any separated finite type $k$-scheme $X$ to its symmetric monoidal presentable stable $\infty$-category of crystals on the infinitesimal site, i.e. $$ \mathrm{Crys}(X) = \mathrm{lim}_{R,\mathrm{Spec}(R_{\mathrm{red}})\to X} D(R) $$ where $R$ runs over finite type $k$-algebras with a map from $\mathrm{Spec}(R_{\mathrm{red}})\to X$.

To construct a $6$-functor formalism, you need to specify in addition the class of morphisms $I$ for which $f_!$ should definitionally be a left adjoint of $f^\ast$, and the class of morphisms $P$ for which $f_!$ should definitionally be a right adjoint of $f^\ast$. Usually $I$ are the open immersions and $P$ the proper maps. Here, we do the opposite! I.e., $I$ are the proper maps, $P$ are the open immersions. The construction principle still works, yielding the desired $6$-functor formalisms, based on three functors $\otimes$, $f^\ast$ and $f_!$.

Now my notation is completely off with respect to the standard notation. This $6$-functor formalism $X\mapsto \mathrm{Crys}(X)$ with $\otimes,f^\ast,f_!$ is isomorphic to Gaitsgory--Rozenblyum's $X\mapsto \mathrm{IndCoh}(X_{\mathrm{dR}})$ with their $\otimes^!,f^!,f_\ast$. And this is isomorphic to the other $6$-functor formalisms on big categories of $D$-modules, with similarly sounding functors.

As I discuss in the notes, things are slightly less confusing if you work with opposite categories, i.e. with $$ \mathrm{Crys}(X)^{\mathrm{op}} = \mathrm{lim}_{R,\mathrm{Spec}(R_{\mathrm{red}})\to X} D(R)^{\mathrm{op}} $$ where $D(R)^{\mathrm{op}}=\mathrm{Pro}(\mathrm{Perf}(R))$ (using naive duality on $\mathrm{Perf}(R)$). Also, as $\mathrm{Crys}(X)$ is compactly generated with compact objects the coherent $D$-modules, and the compact objects admit a selfduality (basically, Verdier duality), one has a canonical identification $$ \mathrm{Crys}(X)^{\mathrm{op}} = \mathrm{Pro}(\mathrm{Crys}(X)^\omega). $$

The functors $\otimes,f^\ast,f_!$ of course induce similar functors on opposite categories, and when you restrict to the compact objects, you get the usual functors on $\mathrm{Crys}(X)^\omega$ i.e. coherent $D$-modules.

As you can see in the notes, I think the formalism really helps in the construction (or at least I found it helpful...), everything can be done by hand. In particular, Gaitsgory--Rozenblyum's $\mathrm{IndCoh}$ formalism can be defined without difficult properties of the Gray tensor product of $(\infty,2)$-categories, in fact without any $(\infty,2)$-categories at all.

Final comment: I am using here as pullback what is usually called "naive" pullback; the "true" pullback is usually a shift of the naive pullback. I think this shift shouldn't actually be introduced, and ultimately results from a "wrong" identification of left and right $D$-modules -- one should identify them by tensoring with $\omega_X = \Omega^d_X[d]$, but often one simply uses $\Omega^d_X$ (where things are defined on the abelian level).