Patching modules over commutative squares
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 it is shown that patching is always possible over a Milnor square. In Section 4 the authors give some general patching properties. In Section 5 the authors deal with patching of projective modules. In Section 6 the authors investigate patching for analytic isomorphisms when more assumptions are made about the rings. In Section 7 they give a number of examples.
scientific article; zbMATH DE number 4105190