arXiv subjects

J. T. Stafford

Publications and source records attributed to J. T. Stafford.

26 records · Page 2Linked to original sources

arXiv2004

Sklyanin algebras and Hilbert schemes of points

We construct projective moduli spaces for torsion-free sheaves on noncommutative projective planes. These moduli spaces vary smoothly in the parameters describing the noncommutative plane and have good properties analogous to those of moduli spaces of sheaves over the usual (commutative) projective plane P^2. The generic noncommutative plane corresponds to the Sklyanin algebra S constructed from an automorphism sigma of infinite order on an elliptic curve E < P^2. In this case, the fine moduli space of line bundles over S with first Chern class zero and Euler characteristic (1-n) provides a symplectic variety that is a deformation of the Hilbert scheme of n points on P^2 - E.

math.AG↗

arXiv2003

Simplicity of noncommutative Dedekind domains

The following dichotomy is established: A finitely generated, complex Dedekind domain that is not commutative is simple. Weaker versions of this dichotomy are proved for Dedekind prime rings and hereditary noetherian prime rings.

math.RA↗

arXiv2003

Naive Noncommutative Blowing Up

Let B(X,L,s) be the twisted homogeneous coordinate ring of an irreducible variety X over an algebraically closed field k with dim X > 1. Assume that c in X and s in Aut(X) are in sufficiently general position. We show that if one follows the commutative prescription for blowing up X at c, but in this noncommutative setting, one obtains a noncommutative ring R=R(X,c,L,s) with surprising properties. In particular: (1) R is always noetherian but never strongly noetherian. (2) If R is generated in degree one then the images of the R-point modules in qgr(R) are naturally in (1-1) correspondence with the closed points of X. However, both in qgr(R) and in gr(R), the R-point modules are not parametrized by a projective scheme. (3) qgr R has finite cohomological dimension yet H^1(R) is infinite dimensional. This gives a more geometric approach to results of the second author who proved similar results for X=P^n by algebraic methods.

math.RA↗

arXiv2003

Noncommutative projective geometry

This article describes recent applications of algebraic geometry to noncommutative algebra. These techniques have been particularly successful in describing graded algebras of small dimension.

math.RA↗

arXiv2000

O_e(G) is a free module over O(G)

We show that the quantised function algebra O_e(G) of a simply-connected semisimple algebraic group G at a root of unity is a free module over the subring isomorphic to O(G).

math.QA↗

arXiv2000

Noncommutative curves and noncommutative surfaces

In this survey article we describe some geometric results in the theory of noncommutative rings and, more generally, in the theory of abelian categories. Roughly speaking and by analogy with the commutative situation, the category of graded modules modulo torsion over a noncommutative graded ring of quadratic, respectively cubic growth should be thought of as the noncommutative analogue of a projective curve, respectively surface. This intuition has lead to a remarkable number of nontrivial insights and results in noncommutative algebra. Indeed, the problem of classifying noncommutative curves (and noncommutative graded rings of quadratic growth) can be regarded as settled. Despite the fact that no classification of noncommutative surfaces is in sight, a rich body of nontrivial examples and techniques, including blowing up and down, has been developed.

math.RA↗

arXiv2000

Algebras without noetherian filtrations

We provide examples of finitely generated noetherian PI algebras for which there is no finite dimensional filtration with a noetherian associated graded ring; thus we answer negatively a question raised by M. Lorenz.

math.RA↗

arXiv1999

The graded version of Goldie's Theorem

The analogue of Goldie's Theorem for prime rings is proved for rings graded by abelian groups, eliminating unnecessary additional hypotheses used in earlier versions.

math.RA↗