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A generalization of the Pontryagin-Hill theorems to projective modules over Pr\"ufer domains
Motivated by the Pontryagin-Hill criteria of freeness for abelian groups, we
investigate conditions under which unions of ascending chains of projective
modules are again projective. Several extensions of these criteria are proved
for modules over arbitrary rings and domains, including a genuine
generalization of Hill's theorem for projective modules over Pr\"{u}fer domains
with a countable number of maximal ideals. More precisely, we prove that, over
such domains, modules which are unions of countable ascending chains of
projective, pure submodules are likewise projective
The Frobenius Structure of Local Cohomology
Given a local ring of positive prime characteristic there is a natural
Frobenius action on its local cohomology modules with support at its maximal
ideal. In this paper we study the local rings for which the local cohomology
modules have only finitely many submodules invariant under the Frobenius
action. In particular we prove that F-pure Gorenstein local rings as well as
the face ring of a finite simplicial complex localized or completed at its
homogeneous maximal ideal have this property. We also introduce the notion of
an anti-nilpotent Frobenius action on an Artinian module over a local ring and
use it to study those rings for which the lattice of submodules of the local
cohomology that are invariant under Frobenius satisfies the Ascending Chain
Condition.Comment: 35 pages. Section 3 was revised to emphasize Theorem 3.1, and some
minor corrections/changes were performed. To appear in Algebra and Number
Theor
On the Linear Weak Topology and Dual Pairings over Rings
In this note we study the weak topology on paired modules over a (not
necessarily commutative) ground ring. Over QF rings we are able to recover most
of the well known properties of this topology in the case of commutative base
fields. The properties of the linear weak topology and the dense pairings are
then used to characterize pairings satisfying the so called -condition.Comment: 16 pages, to appear in "Topologu and its Applications
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