# Does the Riemann-Hilbert Correspondence work at the DG level?

let $X$ denote a smooth complex algebraic variety. Let $D_{rh}(X)$ denote the category of regular holonomic $D$-modules on $X$ and $D_{rh}^b(D(X))$ denote the bounded derived category of $D$-modules on $X$ with regular holonomic homology. Let $X^{an}$ denote the associated complex manifold and let $Mod_c(X^{an})$ denote the category of (algebraically) constructible sheaves on $X^{an}$. Also let $D^b_c(X^{an}, \mathbb{C})$ denote the bounded derived category of complex sheaves on $X$ with constructible homology. There is a (left exact, contravariant) solution functor:

$$Sol: D_{rh}(X) \rightarrow Mod_c(X^{an})$$ given by $Sol(M):= Hom_{D(X^{an})}(M^{an}, \mathcal{O}_{X^{an}}).$ The Riemann-Hilbert Correspondence asserts that this induces an anti-equivalence of categories:

$$RSol: D_{rh}^b(D(X)) \cong D^b_c(X^{an}, \mathbb{C}).$$

Now because the categories of $D$-modules on $X$ and complex sheaves on $X^{an}$ have enough injectives we can think about these derived categories as the homotopy categories of the DG-categories of complexes whose objects are injective with bounded homology. Thus $D_{rh}^b(D(X))$ is the homotopy category of the DG-category $K_{rh}^b(D(X))$, whose objects are injective chain complexes with bounded, regular holonomic homology. Similarly $D^b_c(X^{an}, \mathbb{C})$ is the homotopy category of the DG-category $K^b(X^{an},\mathbb{C})$, whose objects are injective chain complexes with bounded, constructible homology. The solution function naturally gives a functor:

$$Sol_{DG}: K_{rh}^b(D(X)) \rightarrow K^b(X^{an},\mathbb{C}).$$

Passing to the homotopy categories gives the Riemann-Hilbert Correspondence. My question is the following: Can the Riemann-Hilbert Correspondence be lifted to the DG setting? In other words, is $Sol_{DG}$ an equivalence of DG-categories?

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Maybe I'll let someone who's more of an expert answer, but I'm pretty sure the answer is yes. I shouldn't say there will never be an equivalence of derived categories that doesn't lift to the DG level, but you should be really shocked if you encounter one. If you can define the functors on the DG level, they can't suddenly stop being equivalences (after all, for free resolutions, quasi-isomorphism is the same as homotopy equivalence). The thing that could go wrong (though it very rarely does) is that the functors don't lift. –  Ben Webster Mar 22 '12 at 2:52
Most theorems that derived categories are equivalent are actually theorems that DG categories are the same, with precisely the same proof. It's just that derived categories appeared on the scene first, and it took people a long time to realize that DG categories are a better notion. –  Ben Webster Mar 22 '12 at 2:54
In this case, the equivalence exists on more than the dg-level. According to Riemann-Hilbert $Mod_{rh}(D_X) \simeq Perv(x)$. So you get an equivalence of the corresponding dg-categories. And a famous theorem of Beilinson "On the derived category of perverse sheaves" asserts that for algebraic $X$, $D^b_c(X) = D^b(Perv(X))$ (and also $D^b_{rh}(D_X) = D^bMod_{rh}(D_X)$). –  YBL Mar 22 '12 at 19:52
@ YBL on the other hand it is not clear, that $real:D^b(Perv) \rightarrow D^b_c(X)$ can be upgraded to the dg-level, is it? –  Jan Weidner May 10 '12 at 14:07

The answer is yes, if 'equivalence of dg categories' means the usual thing: given dg categories $D_{1}$, $D_{2}$, a dg equivalence between them is a dg functor $F: D_{1} \rightarrow D_{2}$ such that 1) the induced map on complexes $F_{x,y}:D_{1}(x,y) \rightarrow D_{2}(F(x),F(y))$ is a quasi-isomorphism for every $x,y \in D_{1}$ and 2) the induced functor on homotopy categories $[F]: [D_{1}] \rightarrow [D_{2}]$ is an equivalence. The first condition is the homotopical version of fully faithful and the second condition ensures essential surjectivity up to equivalence. The standard statement of Riemann-Hilbert gives 2), but in fact the proof usually verifies 1) along the way. See for instance 7.2.2 in D-modules, Perverse Sheaves, and Representation Theory by Hotta, Takeuchi, and Tanisaki. (They actually treat the covariant Riemann-Hilbert correspondence, using the de Rham functor, but you can get the contravariant version, involving the solution functor, by duality.)

About Ben's comment. There exists an example in positive characteristic of two dgas whose triangulated module categories are equivalent but this equivalence is not induced by a Quillen equivalence of model categories. See Dugger-Shipley, A curious example of triangulated-equivalent model categories which are not Quillen equivalent. Actually, what they show is that the algebraic K-theory of the two dgas is different, and so is not invariant under triangulated equivalence. I take this as convincing evidence that the notion of triangulated category is deficient.

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I like your answer, but I don't see yet how to lift the deRham functor to dg-level. So first of all the D_i are the dg categories of complexes of injective D-modules/sheaves with rh/constructible cohomology and usual Hom complex as morphisms right? So if you are given a complex of injective $D$-modules (with rh cohomology), how do you construct a complex of injective sheaves (with constructible cohomology) from it? Simply tensoring with a resolution of $\Omega$ won't preserve injectivity, right? –  Jan Weidner May 10 '12 at 14:47
In your reference it is shown that $DR$ commutes with $RHom$, but this seems a priori weaker then 1) if no dg lift is constructed before. Do you claim that "derived equivalences" which "preserve $RHom$" can be lifted to dg-equivalences? –  Jan Weidner May 10 '12 at 14:47

If I make no mistake, one can construct a dg-lift as follows: The key point is that any sheaf of vectorspaces embedds canonical into an injective sheaf of vectorspaces:

$$\mathcal F \rightarrow \prod_{x\in X} {i_x}_* {i_x}^* \cal F$$ By standard constructions this allows to construct natural injective resolutions of sheaves and even of bounded below complexes of sheaves. One can check that his actually yields a canonical dg-functor from the category of bounded below complexes of sheaves, to bounded below complexes of injective sheaves

$$I: C^+(X)\rightarrow C^+(\mathbb C\-inj)$$

which maps each complex to a quasi-isomorphic complex of injectives.

Now let $\tilde \Omega$ be a finite flat resolution of the top forms. For example the usual $\mathcal D_{X^{an}}$ valued differential forms will do. We can now define $$\tilde{DR}:C^+(\mathcal D_X\-inj) \rightarrow C^+(\mathbb C\-inj)$$

from the dg-category of bounded below complexes of injective $\mathcal D_X$-modules with to the dg-category of bounded below complexes of injective sheaves by the formula:

$$\tilde{DR}(\mathcal M):=I(\tilde{\Omega}\otimes_{\mathcal D_{X^{an}} \mathcal M^{an}})$$

It is clear by construction that $\tilde{DR}$ induces the usual $DR$ on homotopy categories, hence $\tilde{DR}$ actually restricts to a dg-equivalence in the sense of Chris Brav's answer:

$$\tilde{DR}:C^b_{rh}(\mathcal D_X\-inj) \rightarrow C^b_c(\mathbb C\-inj)$$

from the dg-category of finte complexes of injective $\mathcal D_X$-modules with regular holonomic cohomology to the dg-category of complexes of injective sheaves with bounded constructible cohomology.

In fact there are functorial injective embeddings in many abelian categories and by the same recipe this should allow to construct dg-lifts of many functors. For example the duality functor, the solution functor etc.

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