Seminar in Real and Complex Geometry

Thursday, July 8, 2021, 16:00-17:30, online (via zoom)

Dmitry Kerner
(Ben-Gurion University)

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


Matrices over (algebraically closed) field can be block-diagonalized by conjugation, each block being a Jordan cell. Matrices over a principal ideal domain (e.g. power series in one variable) admit the diagonal reduction by left-right equivalence, A-> UAV. (The well known Smith normal form.) Over more general rings, e.g. power series in several variables, most matrices do not admit such diagonal reduction. The simplest obstruction is the irreducibility of the determinant. Suppose that the determinant factorizes, what are the necessary/sufficient conditions to ensure the corresponding block-diagonal reduction? Equivalently, assuming the support of a module is reducible, when is the module the direct sum of the corresponding components? We give the complete answer for square matrices, assuming co-primeness of the factorization. For rectangular matrices the criterion is more technical. As an immediate application we give criteria of simultaneous (block-)diagonal reduction for tuples of matrices over a field, the splitting of determinantal representations, the decomposability of sheaves on chains of curves, and so on. Joint work with V. Vinnikov.