Back to Commutative Algebra Seminar: Fall 2015
Commutative Algebra Seminar
Date: Friday, Oct 2, 2015
Time: 3:10PM - 4:00PM
Location: LCB 222
Roger Wiegand
University of Nebraska-Lincoln
Title |
Big projective modules over slightly non-commutative rings |
Abstract |
Let H be an arbitrary positive normal affine semigroup (aka “Diophantine semigroup”, aka “reduced finitely generated Krull semigroup”). Early in the millennium I proved that one can find a one-dimensional local domain R (commutative and Noetherian) and a finitely generated torsion-free R-module M such that add M is isomorphic to H. Now let S be the endomorphism ring of R. It is well-known and easy to prove that add M is isomorphic to proj-S, the semigroup of isomorphism classes of finitely generated projective right S-modules. Thus, strange direct-sum behavior over R leads to strange direct-sum behavior of finitely generated projective modules over the slightly non-commutative ring S. What is surprising is that this leads also to strange behavior of countably generated projective modules. In this talk I will indicated how to build a countably generated projective S-module with no non-zero finitely generated direct summands. Put another way, there is a direct summand N of M^{(\omega)} (the direct sum of countably many copies of M) such that N has no finitely generated direct summands. This is joint work with Dolors Herbera and Pavel Prihoda. |
Commutative Algebra