Non-surjective Milnor patching diagrams

TL;DR

This paper investigates the conditions under which certain commutative squares of rings, called Milnor patching diagrams, allow for module gluing, providing new examples and a classification approach for these structures.

Contribution

It constructs non-surjective Milnor patching diagrams using determinant factorization and generalizes the concept to subcategories of modules, advancing the classification of such diagrams.

Findings

Constructed non-surjective Milnor patching diagrams

Related patching conditions to determinant factorization

Classified generalized Milnor patching diagrams in subcategories

Abstract

Milnor patching diagram is essentially the commutative square of rings, over which gluing of finitely generated projective modules is possible in the strongest sense. Necessary and sufficient conditions for a square to be Milnor patching diagram were studied by Milnor, Beauville-Laszlo and Landsburg. We relate this question to determinant-induced factorization in matrix rings to construct a series of non-surjective Milnor patching diagrams, settling the question of Landsburg, and make a step towards the classification of such examples. Also we consider a possible generalization of the notion of Milnor patching diagram to arbitrary subcategories of modules and obtain a classification result for this setting.