Frobenius categories, Gorenstein algebras and rational surface singularities

Abstract

We give sufficient conditions for a Frobenius category to be equivalent to the category of Gorenstein projective modules over an Iwanaga-Gorenstein ring. We then apply this result to the Frobenius category of special Cohen-Macaulay modules over a rational surface singularity, where we show that the associated stable category is triangle equivalent to the singularity category of a certain discrepant partial resolution of the given rational singularity. In particular, this produces uncountably many Iwanaga-Gorenstein rings of finite GP type. We also apply our method to representation theory, obtaining Auslander-Solberg and Kong type results.

Graded quiver varieties and derived categories

“…section 3.3. In the philosophy of [21], the Frobenius category E = proj(R C ) 'admits a resolution', namely itself, and so one expects its category of projectives proj(S C ) to be Gorenstein and the category itself to be equivalent to the category Gorensteinprojective modules over proj(S C ). Technically, the categories we consider do not quite fit into the framework of [loc.…”