nLab derived Morita equivalence

Idea

Derived Morita equivalence is a generalization of Morita equivalence to the “derived” context (homotopy theory of dg-algebras). Just as two $k$-algebras are Morita equivalent if and only if their categories of left modules are equivalent, the coarser equivalence relation of derived Morita equivalence holds whenever for two differential graded algebras their (bounded) derived categories of modules, along with their triangulated category structure, are equivalent.

The existence of a tilting complex is necessary and sufficient for an equivalence between the unbounded derived categories of two rings. A tilting complex is a special small generator of the derived category. It is a bounded complex $T$ of finitely generated projective $R$-modules which generates the derived category $\mathcal{D}(R)$ and whose graded ring of self maps $\mathcal{D}(R)(T, T)_{\ast}$ is concentrated in dimension zero.

Examples

A derived Morita equivalence in the context of homological mirror symmetry appears in (Okada 09)

References

Related notions include Fourier-Mukai transform, mirror symmetry

The classical work in algebra concerning the Morita for derived categories of modules is due Rickard (also called Rickard equivalence)

• Jeremy Rickard, Morita theory for derived categories, J. London Math. Soc. 2-39:3 (1989) 436-456 doi

Contemporary generality is outlined in

• Stefan Schwede, Morita theory in abelian, derived and stable model categories, In: Structured Ring Spectra (eds. A. Baker, B. Richter) arXiv:math/0310146
• B. Shipley, Morita theory in stable homotopy theory, Handbook of Tilting Theory, vol. 13 (L. A. Hügel, D. Happel)

In the setting of dg-categories:

• B. Toen, The homotopy theory of dg-categories and derived Morita theory, arXiv:math/0408337.

In the setting of stable (infinity,1)-categories (section 4):

• Andrew J. Blumberg, David Gepner, Goncalo Tabuada, A universal characterization of higher algebraic K-theory, arXiv:1001.2282.

• So Okada, Homological mirror symmetry of Fermat polynomials (arXiv:0910.2014)

For a treatment in terms of bicategories:

• Niles Johnson, Morita theory For derived categories: a bicategorical perspective (arXiv:0805.3673)