Date: Thursday 25th October 2012 br Speaker: Charles Vial (Cambridge) br Title: Non-commutative resolutions and Grothendieck groups br br Abstract: This is joint work with Hailong Dao, Osamu Iyama and Ryo Takahashi. A finitely generated module M over a commutative noetherian ring R is said to give a non-commutative resolution (NCR) of R if M is faithful and EndR(M) has finite global dimension. The aim of this talk is to discuss the relevance of such a definition and to give necessary conditions for the existence of NCRs. These conditions focus on the Grothendieck group of the category of finitely generated modules over R and its subcategories. This group is related, via Riemann-Roch, to the group of so-called algebraic cycles. I will explain how methods from the theory of algebraic cycles can be used in that setting and I will show that a standard graded Cohen-Macaulay algebra R over a field of zero characteristic with only rational singularities outside the irrelevant ideal has a NCR only if R has rational singularities.