Frobenius categories, Gorenstein algebras and rational surface singularities

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 Gorenstein projective type. We also apply our method to representation theory, obtaining Auslander–Solberg and Kong type results.

RELATIONS BETWEEN MULTIPLICITY AND DIVISOR CLASS GROUP FOR RATIONAL SURFACE SINGULARITIES

In this paper we prove that a rational surface singularity with divisor class group ℤ/(2) is a rational double point. This generalizes a result by Brieskorn: if the divisor class group of a rational singularity is trivial then it is the E8 singularity [3]. We also prove several inequalities involving the integers e, δ, mi, [Formula: see text], where [Formula: see text] is the fundamental cycle. The proof of this result uses ideas from Minkowski's theory of reduction of positive-definite quadratic forms. We also give some interesting counterexamples to some of the related questions in this context.

On symplectic fillings of links of rational surface singularities with reduced fundamental cycle

We prove that every symplectic filling of the link of a rational surface singularity with reduced fundamental cycle admits a rational compactification, possibly after a modification of the filling in a collar neighbourhood of the link.

A Note on the Multiplicity of Cohen-Macaulay Algebras with Pure Resolutions

Let R = k[X1, …, Xn] with k a field, and let I ⊂ R be a homogeneous ideal. The algebra R/I is said to have a pure resolution if its homogeneous minimal resolution has the formSome of the known examples of pure resolutions include the coordinate rings of: the tangent cone of a minimally elliptic singularity or a rational surface singularity [15], a variety defined by generic maximal Pfaffians [2], a variety defined by maximal minors of a generic matrix [3], a variety defined by the submaximal minors of a generic square matrix [6], and certain of the Segre-Veronese varieties [1].If I is in addition Cohen-Macaulay, then Herzog and Kühl have shown that the betti numbers bi are completely determined by the twists di.