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

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

2012-03-21T21:36:01Z

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?

Answer by Chris Brav for Does the Riemann-Hilbert Correspondence work at the DG level?

2012-03-22T11:17:59Z

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.

Answer by Jan Weidner for Does the Riemann-Hilbert Correspondence work at the DG level?

2012-05-16T06:56:23Z

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.