On the Inverse Syzygy Problem

Abstract

We present a constructive solution of the inverse syzygy problem over arbitrary coherent rings. By relating the existence of a kernel representation to torsionlessness instead of the more common torsionfreeness, we do not need to assume the existence of a quotient field. As a by-product, we obtain an algorithm to compute the extension groups of finitely presented modules.

Key Words:

• Coherent rings
• Extension groups
• Syzygies
• Torsionfree modules
• Torsionless modules

2000 Mathematics Subject Classification:

• Primary 16E05, 16D10, 93B25
• Secondary 16D50, 16U20

ACKNOWLEDGMENT

This article is cordially dedicated to Ulrich Oberst on the occasion of his retirement in 2009.

Notes

Note that Auslander and Bridger speak of k-torsionfree modules, but their notion of 1-torsionfree is equivalent to what we call torsionless.

In the case of a commutative ring 𝒟, we may of course use the more common realisation β(m) = Bm and obtain then β*(x) = B ^t x, where B ^t denotes the transposed matrix.

Communicated by M. Cohen.