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On the boundary operator of differential modules

On the boundary operator of differential modules EMANUEL SPERNER zum 65. Geburtstag gewidmet By R. J. WILL~ (University of South Africa) Introduction This paper deals with the following problem: Let (X, d) be a differential module, find other boundary operators d' for the same module X, such that (X, d) and (X, d') have the same homology group. We shall give some existence theorems, which we obtained in our research to get a better knowledge of this problem. Section 1. concerns factorization of d in the acyclic case. In Section 2. we discuss graded modules. In Section 3. non graded modules are treated. For definitions we refer to [1], [2] and [3]. w 1. Factorization of d in the acyelic case Theorem 1. Let X be a module over a ring A, (X, d) acyclic, q~l and q~ : X ---> X A-linear and such that then (X, ~x) is a di]erentlal module, which is also acycllc and q~ is a module isomorphism in (X, d) and (X, q~l). Proof. Since ~| = 0 it follows that (X, ~1) is a differential module. (i) First we shall prove that (X, ~) is aeyelie, i. e. its homology group H (X, r = 0. Let x e http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Springer Journals

On the boundary operator of differential modules

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Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Mathematics, general; Algebra; Differential Geometry; Number Theory; Topology; Geometry
ISSN
0025-5858
eISSN
1865-8784
DOI
10.1007/BF02993622
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Abstract

EMANUEL SPERNER zum 65. Geburtstag gewidmet By R. J. WILL~ (University of South Africa) Introduction This paper deals with the following problem: Let (X, d) be a differential module, find other boundary operators d' for the same module X, such that (X, d) and (X, d') have the same homology group. We shall give some existence theorems, which we obtained in our research to get a better knowledge of this problem. Section 1. concerns factorization of d in the acyclic case. In Section 2. we discuss graded modules. In Section 3. non graded modules are treated. For definitions we refer to [1], [2] and [3]. w 1. Factorization of d in the acyelic case Theorem 1. Let X be a module over a ring A, (X, d) acyclic, q~l and q~ : X ---> X A-linear and such that then (X, ~x) is a di]erentlal module, which is also acycllc and q~ is a module isomorphism in (X, d) and (X, q~l). Proof. Since ~| = 0 it follows that (X, ~1) is a differential module. (i) First we shall prove that (X, ~) is aeyelie, i. e. its homology group H (X, r = 0. Let x e

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Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg – Springer Journals

Published: Nov 18, 2008

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