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Let $X$ be a quasi-projective variety over a field $k$. Let $D_{qcoh}$ be a dg enhancement of the unbounded derived category of quasi-coherent sheaves over $X$, and $D_{perf}$ its full subcategory of perfect complexes.

This question is about Hochschild cohomology in the dg category sense. The question in the title gives the gist, but a more precise question is the following. There is a restriction homomorphism $$ r\colon HH^\ast(D_{qcoh},D_{qcoh}) \to HH^\ast(D_{perf},D_{perf}). $$ This is clear if one computes Hochschild cohomology using the bar complex.

Question: Is $r$ an isomorphism?

The left-hand side makes me nervous because of the unbounded complexes, but I think this is true in the case of smooth $X$. I'm interested in singular varieties (or schemes). What about the affine case? To be concrete, what about the case where $X$ is a local complete intersection?

Comments: The category $D_{qcoh}$ looks much more ferocious than $D_{perf}$. The reason for bringing it into the picture is that its Hochschild cohomology seems to be better understood. One possible reference (which also discusses similar results for perfect complexes, but only in the smooth case) is Toen's article on derived Morita equivalence. If I understood correctly (did I?), $HH^\ast(D_{qcoh},D_{qcoh})$ is isomorphic to $$HH^\ast(X):= Ext^\ast_{\mathcal{O}_{X\times X}}(\delta_{\ast}\mathcal{O}_X,\delta_{\ast}\mathcal{O}_X)$$ where $\delta\colon X\to X\times X$ is the diagonal. And this is the thing I really want to compare to $HH^\ast(D_{perf},D_{perf})$, for the reason that I know how to compute it in examples. There is a local-to-global Ext spectral sequence converging to $HH^\ast(X)$, and for local complete intersections one can use the Hochschild-Kostant-Rosenberg isomorphism to understand the Ext-sheaf.

Motivation: I've been looking at manifestations of homological mirror symmetry in which one has an embedding of the Fukaya category of a symplectic manifold into $D_{per}$ for a mirror variety. I'd like to compute Hochschild cohomology of the Fukaya category via algebraic geometry.

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up vote 9 down vote accepted

The answer is yes - at least if you take for the definition of $HH^*$ the self-ext of the identity functor. For any quasicompact quasiseparated scheme we know (thanks to Thomason-Trobaugh) that $D_{qc}(X)$ is compactly generated by the perfect complexes. This means that $D_{qc}(X)=Ind(D_{perf}(X))$ -- the quasicompact [dg enhanced throughout!!] derived category is the ind category (in the $\infty$-categorical sense) of the perfect one. Now it's an easy consequence of various results in Lurie's Higher Topos Theory [and maybe by now appears in there explicitly?] that

$\bf Proposition$: Passing to ind-categories induces an equivalence of [symmetric monoidal] $\infty$-categories between small, idempotent-complete stable $\infty$-categories (with exact functors) and compactly generated presentable stable $\infty$-categories (with proper functors -- continuous functors that preserve compact objects).

(This is spelled out in my paper with Francis and Nadler.) This is very useful -- it means you can go back and forth freely between small concrete categories that you like and big flexible objects which have all (small) limits and colimits, where the adjoint functor theorem and many other things apply. So the passage from perfect to quasicoherent is part of an equivalence of categories. It also means in particular that all exact functors on perfect complexes are representable by quasicoherent sheaves on the square --- though unless you're smooth and proper these will not be precisely the perfect complexes on the square..

Since Hochschild cohomology is given by self-Ext of the identity functor (which is certainly a proper functor), we can choose to calculate it either with the small categories or with the large categories. (For a reference for the equivalence between dg and stable $\infty$-categories I suggest this paper by Blumberg, Gepner and Tabuada.)

So now you might want to check that the cyclic bar construction for a small dg category does indeed calculate self-Ext of the identity functor. (Maybe you want to take the dg category to be pretriangulated, or first prove that both sides are invariant under Morita equivalence.) This is pretty clear I think - at least notationally it's easier if we assume our category has one generator (equivalently finitely many -- and this is always the case for [q-c,q-s] schemes), hence is just modules over a dg algebra. Then we observe the standard bar construction is a free resolution of the algebra $A$ considered as a bimodule (i.e. of the identity functor), and the cyclic bar construction computes the Ext. For the multiobject version I think there's an MO discussion already somewhere..

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Thank you, David, that looks great (I have some reading to do). [Hm. My own colleagues have a good track record at answering my MO questions. I should say that I really do also talk to David and others in person...] –  Tim Perutz Feb 12 '11 at 19:53

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