Singularity categories, Leavitt path algebras and Hochschild homology.

The singularity category of a noetherian (non commutative) algebra is the quotient of its bounded by its perfect derived category. This construction goes back to Buchweitz (1986) in this setting and, independently, to Orlov (2003) in a geometric setting. We will recall the description of the singularity category of a radical-square zero quiver algebra using a graded Leavitt path algebra following work of Paul Smith, Xiao-Wu Chen, Dong Yang and others. We will then combine this with a localization theorem for Hochschild homology to obtain a simple description of the Hochschild homology of these singularity categories (with their canonical differential graded enhancement) and of the corresponding Leavitt path algebras. Finally, we will report on recent work of Xiao-Wu Chen and Zhengfang Wang which yields a generalization from radical-square zero to arbitrary finite-dimensional algebras (over an algebraically closed field).

This is mainly a survey talk. The original parts are based on joint work with Umamaheswaran Arunachalam and Yu Wang.