Hereditary noetherian categories with a commutative function field
by
Helmut Lenzing
Fachbereich Mathematik-Informatik, Universität Paderborn, D-33100 Paderborn

Let k be a field, not necessarily algebraically closed. The talk deals with the classification of (connected) hereditary noetherian categories H having finite dimensional morphism and extension spaces (over k) and satisfying Serre duality in the form

D Ext1(X, Y) = Hom(Y, \tauX)

for some automorphism \tau of H, where D refers to the formation of the k-dual. We call these categories hereditary noetherian for short. A typical example is the category of coherent sheaves over a smooth projective curve C defined over k; moreover the process of inserting (positive integral) weights in a finite number of points of C, yields a hereditary noetherian category as well, to be interpreted as the category of coherent sheaves on a weighted smooth projective curve.

Let H be hereditary noetherian, and let H₀ denote the Serre subcategory of objects of finite length. Then the quotient category (sense of Serre-Grothendieck) is equivalent to the category of finite dimensional vector spaces over a skew field F, which is either a finite extension of k or a finite extension of a function field K in one variable over k.

If k is algebraically closed F is known to be commutative. Hence the following assertions extend recent related results of I. Reiten and M. van den Bergh.

Theorem. Let H be hereditary noetherian, and assume that F is a commutative function field in one variable over k. Then H arises from the category of coherent sheaves on a smooth projective curve C by insertion of weights.

Corollary 1. The Grothendieck group K₀(H) of H has the form K₀(C)\oplusZⁿ. In particular, K₀(H) is finitely generated if and only if C has genus zero.

Corollary 2. H has a tilting object if and only if C has genus zero.

Date received: April 26, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahh-11.