Does the Riemann-Hilbert Correspondence work at the DG level? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T23:35:08Z http://mathoverflow.net/feeds/question/91865 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/91865/does-the-riemann-hilbert-correspondence-work-at-the-dg-level Does the Riemann-Hilbert Correspondence work at the DG level? deltaphi 2012-03-21T21:36:01Z 2012-07-24T13:21:36Z <p>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:</p> <p>$$Sol: D_{rh}(X) \rightarrow Mod_c(X^{an})$$ 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:</p> <p>$$RSol: D_{rh}^b(D(X)) \cong D^b_c(X^{an}, \mathbb{C}).$$</p> <p>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:</p> <p>$$Sol_{DG}: K_{rh}^b(D(X)) \rightarrow K^b(X^{an},\mathbb{C}).$$</p> <p>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?</p> http://mathoverflow.net/questions/91865/does-the-riemann-hilbert-correspondence-work-at-the-dg-level/91900#91900 Answer by Chris Brav for Does the Riemann-Hilbert Correspondence work at the DG level? Chris Brav 2012-03-22T11:17:59Z 2012-03-22T11:17:59Z <p>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 <em>complexes</em> $F_{x,y}:D_{1}(x,y) \rightarrow D_{2}(F(x),F(y))$ 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.)</p> <p>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. </p> http://mathoverflow.net/questions/91865/does-the-riemann-hilbert-correspondence-work-at-the-dg-level/97090#97090 Answer by Jan Weidner for Does the Riemann-Hilbert Correspondence work at the DG level? Jan Weidner 2012-05-16T06:56:23Z 2012-07-24T13:21:36Z <p>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:</p> <p>$$\mathcal F \rightarrow \prod_{x\in X} {i_x}_* {i_x}^* \cal F$$ 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</p> <p>$$I: C^+(X)\rightarrow C^+(\mathbb C-inj)$$</p> <p>which maps each complex to a quasi-isomorphic complex of injectives.</p> <p>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 $$\tilde{DR}:C^+(\mathcal D_X-inj) \rightarrow C^+(\mathbb C-inj)$$</p> <p>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:</p> <p>$$\tilde{DR}(\mathcal M):=I(\tilde{\Omega}\otimes_{\mathcal D_{X^{an}} \mathcal M^{an}})$$</p> <p>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:</p> <p>$$\tilde{DR}:C^b_{rh}(\mathcal D_X-inj) \rightarrow C^b_c(\mathbb C-inj)$$</p> <p>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.</p> <p>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.</p>