Singularities, a bridge between commutative and noncommutative algebra
For simplicity assume that k is an algebraically closed field. Let R be a quasihomogenous
isolated (commutative) Gorenstein singularity of Krull dimension two.
As we are going to point out such singularities are best studied through
attached categories which live in the world of non-commutative algebras. These
categories have quite diferent descriptions but turn out to be interrelated:
The quotient category H of the category of all finitely generated graded
R-modules modulo its Serre subcategory of finite dimensional graded modules.
The category H has an interpretation as the category of coherent
sheaves on a non-commutative curve X, arising from a smooth projective
(commutative) curve X by insertion of a finite number of weighted points.
The triangulated category of the graded singularites of R, defined as the
quotient of the derived category of finitely generated graded R-modules
modulo the subcategory of perfect complexes.
The stable category of graded (maximal) Cohen-Macaulay modules over
R, defined as the stable category associated to the Frobenius category of
graded maximal Cohen-Macaulay modules over R.
The stable category of vector bundles on X depending on the selection of
a distinguished class of line bundles, giving the category of vector bundles
the structure of a Frobenius category.
The relationship between the above concepts will be discussed and illustrated
by specific examples. The talk will report on joint work with J. A. de la Peņa,
D. Kussin and H. Meltzer.
Helmut Lenzing, Paderborn