Morita equivalence of DG algebras? (reference needed)

Question

A stupid question: whether ther exists a universally recognized definition of when two differential graded algebras should be Morita equivalent? I mean a sort of equivalence which would incorporate the usual Morita equivalence of algebras (when there is no differential) and the usual homotopy equivalence of differential graded algebras induced by an $A_\infty$/-quasi-isomorphism. In particular, I would like to believe, that the cyclic cohomology of such algebras should be canonically isomorphic via a generalised trace map.

I believe, this should exist, but a short search on Google rendered only papers on derived Morita equivalence, i.e. as much as I can judge, on equivalence between derived categories of algebras (non-differential). If I am mistaken and this is actually what I need, please, do explain it to me! Or if you know a good reference on the question I am asking, please, do share it with me!

Answer 1

Classical Morita equivalence is given by tensoring with a projective bimodule. The obvious generalization to dg algebras is to tensor with a dg bimodule that is some sense projective (the technical term is 'perfect'). But since complexes of modules over a classical algebra are the same thing as dg modules over that classical algebra thought of as a dg algebra with zero differential and concentrated in degree zero, one is led to the notion of derived Morita invariance by thinking of your classical algebra as a very simple kind of dg algebra.

Derived Morita equivalence is thus a generalization of classical Morita equivalence for algebras and enjoys many of the properties of the classical version. For example, every linear invariant that I know of (algebraic K-theory, cyclic (co)homology, Hochschild (co)homology,...) is not only invariant under Morita equivalence but also under derived Morita equivalence. Such invariance is well-known and can be found for example in papers of Bernhard Keller. So I would say that already for classical algebras (with zero differential and concentrated in degree zero), derived Morita invariance is a natural notion, and for more complicated dg algebras it is hard to imagine any other reasonable notion.

I can also very much recommend Stefan Schwede's Morita theory in abelian, derived and stable model categories, in Structured Ring Spectra, 33--86, London Mathematical Society Lecture Notes 315. You can also find this on Schwede's website.

Answer 2

Just a comment on "For example, every linear invariant that I know of (algebraic K-theory, cyclic (co)homology, Hochschild (co)homology,...) is not only invariant under Morita equivalence but also under derived Morita equivalence." :

Given an algebra $A$ (view it as trivially d.g.), the (say left) global dimension of $A$ is a Morita invariant, but it is not a derived-Morita invariant. On the positive, the global dimension to be finite or infinite is a derived Morita invariant.