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BIRS Workshop Lecture Videos

Singularity categories and Morita equivalence Greenlees, John


(joint with Greg Stevenson) The singularity category of a ring R is defined to be the quotient of the triangulated category of complexes with finitely generated homology modulo the perfect complexes: D_{sg}(R)=D_{fg}(R)/D_{perf}(R). To define the analogue more generally (eg when R is a DGA or a ring spectrum satisfying mild finiteness hypotheses), we need to find analogues of the triangulated categories D_{fg}(R) and D_{perf}(R). The perfect complexes are precisely the small objects, so the denominator is easy. One way to make sense of the finitely generated R-modules is to suppose given a regular ring S and a map S->R making R a small S-module, and take D_{fg}(R) (which may depend on S) to consist of R-modules small over S. This covers interesting cases, and the talk will discuss the use of Morita theory to understand the resulting singularity category.

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