Topic
Topological module
About: Topological module is a research topic. Over the lifetime, 37 publications have been published within this topic receiving 191 citations.
Papers
TL;DR: In this article, it was shown that every linearly compact topology on a module is weaker than a unique, maximal topology over a ring with radical r. In particular, if 0 r " = (0) where r is the radical of a (not necessarily commutative) ring, then the topology of a module, for the discrete topology, is necessarily noetherian.
Abstract: We assume familiarity with the definitions and basic properties of linearly compact rings and modules as contained in [6] or [3, Exercises 14 20, pp. 108-111]. In § 1 we show that every linearly compact topology on a module is weaker than a unique, maximal linearly compact topology. In § 2 we apply the results of§ 1 to a discussion of the circumstances under which the following statements about a linearly compact topological module E over a linearly compact ring A with radical r are equivalent: (1) the topology of E is stronger than the r-adic topology; (2) rZE is open; (3) rE is open and finitely generated; (4) r--E is open and finitely generated. Our results, applied to linearly compact rings, yield improvements of known theorems, even in the compact case. In §3 we give conditions under which a module, linearly compact for the discrete topology, is necessarily noetherian. In particular, we show that if 0 r "= (0) where r is the radical of a (not necessarily commutative)
25 citations
TL;DR: A topological module over the ring of polynomials of vector-valued functions, holomorphic in a domain, is said to be stable and saturated if and only if it admits division by binomials as discussed by the authors.
Abstract: Let be a topological module (over the ring of polynomials) of vector-valued functions , holomorphic in a domain .A closed submodule is local (that is, uniquely determined by the collection , , of its localized submodules) if and only if is stable and saturated. A submodule is said to be stable if it admits division by binomials: , . Being saturated amounts to possessing sufficiently many elements.Bibliography: 26 titles.
24 citations
TL;DR: In this paper, a theory of Newton polygons for fractional power series is developed, which allows us to compute the Haar measure of the set of zeros of f of a given valuation, given the valuations of the coefficients.
Abstract: Let C be an algebraically closed field containing Fq which is com- plete with respect to an absolute value | |. We prove that under suitable constraints on the coefficients, the series f(z) = ∑ n∈Z anz q converges to a surjective, open, continuous Fq-linear homomorphism C → C whose kernel is locally compact. We characterize the locally compact sub-Fq-vector spaces G of C which occur as kernels of such series, and describe the extent to which G determines the series. We develop a theory of Newton polygons for these series which lets us compute the Haar measure of the set of zeros of f of a given valuation, given the valuations of the coefficients. The “adjoint” series f∗(z) = ∑ n∈Z a 1/qn n z1/q n converges everywhere if and only if f does, and in this case there is a natural bilinear pairing ker f × ker f∗ → Fq which exhibits ker f∗ as the Pontryagin dual of ker f . Many of these results ex- tend to non-linear fractional power series. We apply these results to construct a Drinfeld module analogue of the Weil pairing, and to describe the topological module structure of the kernel of the adjoint exponential of a Drinfeld module. Mathematical Sciences Research Institute, Berkeley, California 94720-5070 E-mail address: poonen@msri.org Current address: Department of Mathematics, Princeton University, Princeton, New Jersey 08544-1000 E-mail address: poonen@math.princeton.edu License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
14 citations
TL;DR: In this article, the notion of module homomorphisms in the general topological module setting is considered and the strict and uniform topologies on the modules of continuous linear homomorphism are studied.
12 citations
TL;DR: In this article, the concepts of open unit ball and closed unit ball in a real or complex normed space are naturally extended to the scope of topological rings with unity, and a type of open (closed) sets called open closed unit neighborhoods of 0 are defined.
Abstract: The concepts of open unit ball and closed unit ball in a real or complex normed space are naturally extended to the scope of topological rings with unity. We then define a type of open (closed) sets called open (closed) unit neighborhoods of 0. We show among other things that in R and C the only non-trivial open and closed unit neighborhoods of 0 are the open unit ball and the closed unit ball, respectively.
10 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 2 |
| 2020 | 3 |
| 2019 | 2 |
| 2018 | 1 |
| 2017 | 5 |
| 2016 | 1 |