Torsion in class groups of modules over certain one-dimensional rings

Abstract

Let k be a field and let A be an affine, semiprime noetherian λ-algebra of pi degree d and Krull dimension 1. We extend results of [G3] and [GJRW] on the torsion subgroup of Cl(λ), the genus class group of λ. Let Γ be a maximal order containing λ. Then we prove that, if there exists a (λ,Γ) -conductor C which is a semiprime ideal of Γ, there is a positive integer m (depending on A) such that Cl(λ) _n is finite for all n prime to m, where the subscript denotes the n torsion subgroup. If k is a number field or has positive characteristic, then the assumption on the conductor can be dropped. Moreover, we give examples to show that our result is sharp, in the sense that every potential source of infinite torsion identified in the proof actually arises for some λ. Using a change of rings functor of [G2], we can partially extend this result to arbitrary finitely-generated λ-lattices. We prove that, for each positive integer r, there exists an integer m(r) depending on r and λ such that G(M) _n is finite for all n prime to m(r) for any λ-lattice of rank r, where G(M) is the genus class group of the lattice M. We also prove sharper versions of this result when k is a number field or has positive characteristic.

Keywords:

• genus class group
• order
• lattice
• noncommutative curve

1991 Mathematics Subject Classification:

• Primary: 16D70
• Secondary: 16G30
• Secondary: 16R20