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arXiv · 1305.2256

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

Dmitry Kerner·Victor Vinnikov
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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.

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Dmitry Kerner, Victor Vinnikov. 2013-05-10. Block-diagonal reduction of matrices over commutative rings I. (Decomposition of modules vs decomposition of their support). https://arxiv.org/abs/1305.2256

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