Injectivity and flatness of semitopological modules

TL;DR

This paper explores the relationship between flatness and injectivity of modules over Noetherian algebras, especially in the context of topological vector spaces, revealing that injectivity of duals is a stronger condition than flatness.

Contribution

It establishes a link between flatness and injectivity of modules over Noetherian algebras, introducing the notion of (K) dual pairs and analyzing their properties.

Findings

Injectivity of dual modules is stronger than flatness of the original module.

The paper introduces the concept of (K) dual pairs over A.

It extends known results from spaces like D, S, E' to a broader algebraic context.

Abstract

The spaces D, S and E' over \mathbb{R}^(n) are known to be flat modules over A=\mathbb{C}[\partial_{1},...,\partial_{n}], whereas their duals D', S' and E are known to be injective modules over the same ring. Let A be a Noetherian k-algebra (k=\mathbb{R} or \mathbb{C}). The above observation leads us to study in this paper the link existing between the flatness of an A-module E which is a locally convex topological k-vector space and the injectivity of its dual. We show that, for dual pairs (E,E') which are (K) over A--a notion which is explained in the paper--, injectivity of E' is a stronger condition than flatness of E. A preprint of this paper (dated September 2009) has been quoted and discussed by Shankar.