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

Take a rectangular matrix over a commutative ring A∈Mat_m×n(R). Suppose that the ideal of maximal minors factorizes, I_m(A)=J₁⋅J₂⊂R. When is A left-right equivalent to a block-diagonal matrix? (When does the module/sheaf Coker(A) decompose as Coker(A)|_V(J1)⊕Coker(A)|_V(J2)?) 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 paper we establish the following results. • The persistence of (in)decomposability under the change of rings. For example: - the passage to Noetherian/local/Henselian/complete rings; - the decomposability of the module Coker(A) over a graded ring R vs the decomposability of the sheaf Coker(A) locally at the points of Proj(R), (this is the matrix decomposability version of M. Noether's AF+BG); - the restriction to a subscheme in Spec(R). • The necessary and sufficient condition for decomposability of square matrices in the case: det⁡(A)=f₁⋅f₂ is not a zero divisor and f₁,f₂ are co-prime in R . As an immediate application we give new criteria of simultaneous (block-)diagonal reduction for tuples of matrices over a field, i.e., linear determinantal representations.