Recall that a noetherian ring R is regular if every finitely generated R-module has finite projective dimension. In a paper from 2009, Iacob and Iyengar characterize the regularity of R in terms of properties of (unbounded) R-complexes. Their proofs build on results of Jorgensen, Krause, and Neeman on compact generation of the homotopy categories of complexes of projective/injective/flat modules. In the commutative case, these results can be obtained with derived category methods in local algebra. I will illustrate how this is done by proving that the following conditions are equivalent for a commutative noetherian ring R:
1) R is regular.
2) Every complex of finitely generated projective R-modules is semi-projective.
3) Every complex of projective R-modules is semi-projective.
4) Every acyclic complex of projective R-modules is contractible.
The second condition is new, compared to the 2009 results, and relating it to the regularity of R is the novel part of the proof. This argument also plays a central role in the new proof of the corresponding results for complexes of injective modules and complexes of flat modules.
This video is part of the New Directions in Group Theory and Triangulated Categories seminar series.
