Patching modules over commutative squares
type: Item
instance of: scholarly article
title: Patching modules over commutative squares
zbMATH Open document ID: 0675.18008
DOI: 10.1016/0021-8693(88)90161-5
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: All rings considered are unitary and modules are unital. The paper is divided into seven sections. In Section 1, the authors define the ``patch'' of modules and discuss general properties of commutative squares. A commutative diagram \[ S : \quad \begin{tikzcd} R \ar[r,"p_2"]\ar[d,"p_1" '] & R_2 \ar[d,"j_2"]\\ R_1 \ar[r,"j_1" '] & R_3 \end{tikzcd} \] of ring homomorphisms is said to be a square. Let \({\mathcal S}\) be a square. A patching data for \({\mathcal S}\) is said to be a triple \((M_ 1,M_ 2,\zeta)\), where \(M_ i\) is a left \(R_ i\)-module for \(i=1,2\) and \(\zeta\) : \(R_ 3\otimes_{R_ 1}M_ 1\to R_ 3\otimes_{R_ 2}M_ 2\) is an isomorphism of left \(R_ 3\)-modules. The \({\mathcal S}\)-patch of \(M_ 1\) and \(M_ 2\) (via \(\zeta)\) is said to be a triple \((P,\pi_ 1,\pi_ 2)\), where P is a left R-module and \(\pi_ i: P\to M_ i\) are R-module homomorphisms \((i=1,2)\) with the following properties: (1) For each \(p\in P\), \(\zeta (1\otimes \pi_ 1(p))=1\otimes \pi_ 2(p)\). (2) If \((P',\pi '_ 1,\pi '_ 2)\) is another triple satisfying (1), then there is a unique R-module map f: P\({}'\to P\) for which \(\pi_ i\circ f=\pi '_ i\), \(i=1,2\). (3) The induced maps \(\tau_ i: R_ i\otimes_ RP\to M_ i\), induced by \(\tau_ i(r_ i\otimes p)=r_ i\pi_ i(p)\) for \(r_ i\in R_ i\), \(p\in P\) \((i=1,2)\) are isomorphisms of left \(R_ i\)-modules. If (3) holds for all patching data \((M_ 1,M_ 2,\zeta)\), then \({\mathcal S}\) is called a patching square. There are two major results. The first one shows that a certain property of the square itself implies the corresponding property for ``modules over the square''. The second result is a general theorem on the existence of a patch. It says, roughly, that if one side of the square is under control, then so is the other side. In Section 2 the authors begin the study of analytic isomorphisms. The main result provides a necessary and sufficient condition on the analytic isomorphism for the patch to always exist. In Section 3 …
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 13C10
Mathematics Subject Classification ID: 16B50
zbMATH DE Number: 4105190
zbMATH Keywords: patch of modules
zbMATH Keywords: commutative squares
zbMATH Keywords: patching square
zbMATH Keywords: analytic isomorphisms
zbMATH Keywords: Milnor square
zbMATH Keywords: projective modules
author: Budh Singh Nashier
author: Warren D. Nichols
reviewed by: Iuliu Crivei
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90161-5
OpenAlex ID: W1994443997
cites work: Q5562621
cites work: La périodicité de Bott en $K$-théorie générale
cites work: Serre's conjecture
cites work: The Milnor patching property
cites work: Projektive Moduln über Polynomringen \(A[T_1,\dots,T_M\) mit einem regulären Grundring \(A\)
cites work: On the Bass-Quillen conjecture concerning projective modules over polynomial rings
cites work: Introduction to Algebraic K-Theory. (AM-72)
cites work: On the conormal bundle of ideals
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description: scientific article; zbMATH DE number 4105190