Let A be an algebra (or dg algebra). Where can I find a proof of HH\_\*(A) = HH\_\*(Mod\_A) and HH^\*(A) = HH^\*(Mod\_A)? (And does this hold for any A?) Here Mod\_A is, e.g., the category of left A-modules.


One reason why this is interesting/important/useful is because many categories which arise "in nature" are of the form Mod\_A. For example, there is a theorem of Bondal and van den Bergh which states that derived categories of a large class of varieties (I forget their exact hypotheses) are equivalent to Mod\_A for some A. Dyckerhoff also proved that categories of matrix factorizations are of this form. By mirror symmetry, Fukaya-type categories should be of this form as well...


Anyway, so to compute HH of such a category, it suffices to find this A and then compute HH(A). I think that it generally(?) should be easier to compute HH of an algebra than HH of a category. (Of course finding this A can be a very nontrivial task.)



Hochschild (co)homology of A and of Mod_A
Basically this follows from the fact that the derived category of bimodules over two algebras is equivalent to the (suitably defined) functor category between the derived category of modules of each algebra. Say, Toen's paper on derived Morita equivalence. Then, the identity functor is given by the algebra itself interpreted as a bimodule, so the Hochschild cohomology is $\mathrm{Ext}^i\_{A-A}(A,A)$. You can compute this using the bar resolution and a quick calculation gives you the usual definition of Hochschild cohomology.


