Block-diagonal reduction of matrices over commutative rings I. (Decomposition of modules vs decomposition of their support)

TL;DR

This paper investigates conditions under which matrices over commutative rings can be reduced to block-diagonal form, linking module decomposition with matrix factorization and extending criteria across various ring types.

Contribution

It establishes criteria for matrix decomposability over rings, analyzes how decomposability persists under ring changes, and applies results to tuples of matrices and determinantal representations.

Findings

Decomposability persists under certain ring extensions.

Criteria for decomposability when det(A)=f_1*f_2 with coprime factors.

Application to linear determinantal representations.

Abstract

Consider rectangular matrices over a commutative ring R. Assume the ideal of maximal minors factorizes, I_m(A)=J_1*J_2. When is A left-right equivalent to a block-diagonal matrix? (When does the module/sheaf Coker(A) decompose as the corresponding direct sum?) If R is not an elementary divisor ring (i.e. not a close relative of a principal ideal ring) one needs additional assumptions on A. No necessary and sufficient criterion for such block-diagonal reduction is known. In this part we establish the following: * The persistence of (in)decomposability under the change of rings. For example, the passage to Noetherian/local/complete rings, the decomposability of A over a graded ring R vs the decomposability of Coker(A) locally at the points of Proj(R), the restriction to a subscheme in Spec(R). * The necessary and sufficient condition for decomposability of square matrices in the case: det(A)=f_1*f_2 is not a zero divisor and f_1,f_2 are co-prime. As an immediate application we give criteria of simultaneous (block-)diagonal reduction for tuples of matrices over a field, i.e. linear determinantal representations.