Block-diagonal reduction of matrices over commutative rings I. (Decomposition of modules vs decomposition of their support)
Abstract
Consider rectangular matrices over a commutative ring R. Assume the ideal of maximal minors factorizes, Im(A)=J1*J2. 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)=f1*f2 is not a zero divisor and f1,f2 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.