Journal of Noncommutative Geometry

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Volume 9, Issue 1, 2015, pp. 21–34
DOI: 10.4171/JNCG/186

Published online: 2015-04-13

Non-commutative resolutions and Grothendieck groups

Hailong Dao[1], Osamu Iyama[2], Ryo Takahashi[3] and Charles Vial[4]

(1) University of Kansas, Lawrence, USA
(2) Nagoya University, Japan
(3) Nagoya University, Japan
(4) University of Cambridge, UK

Let $R$ be a noetherian normal domain. We investigate when $R$ admits a faithful module whose endomorphism ring has finite global dimension. This can be viewed as a noncommutative desingularization of Spec($R$). We show that the existence of such modules forces stringent conditions on the Grothendieck group of finitely generated modules over $R$. In some cases those conditions are enough to imply that Spec($R$) has only rational singularities.

Keywords: Non-commutative resolutions, Grothendieck groups, rational singularities

Dao Hailong, Iyama Osamu, Takahashi Ryo, Vial Charles: Non-commutative resolutions and Grothendieck groups. J. Noncommut. Geom. 9 (2015), 21-34. doi: 10.4171/JNCG/186