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!


2 Answers 2


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.

  • $\begingroup$ Thank you! I have actually run across Schwede's paper, but couldn't realize, if it deals with the subject I need, or something different, albeit close to what I am intersted in. Now I will look more attentively in it. $\endgroup$
    – gshar
    Dec 12, 2011 at 19:41
  • $\begingroup$ If you are specifically interested in Morita invariance of cyclic homology, then you can find the following on Bernhard Keller's webpage: Invariance and Localization for Cyclic Homology of DG algebras, Journal of Pure and Applied Algebra, 123 (1998), 223-273. $\endgroup$
    – Chris Brav
    Dec 12, 2011 at 22:40

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.