## Abstract

Let (S, m) be a local ring over a field (the simplest example is analytic/formal

power series). Consider matrices with entries in S, up to left-right equivalence, A → UAV , where U, V are invertible matrices over S. When such a matrix is equivalent to a block-diagonal matrix? Alternatively, when the S-module coker(A) is decomposable?

An obvious necessary condition (for square matrices) is that the determinant of the matrixis reducible (as an element of S). This condition is very far from being sufficient. We prove a very simple necessary and sufficient condition for equivalence to block-diagonal form.

As immediate applications we prove several results: the Ulrich modules over local rings tend to be decomposable; an obstruction to Thom-Sebastiani decomposability of maps (or of systems of vector fields/PDE’s); decomposable representations of algebras are avoidable in projective families; decomposable matrices are highly unstable (they are far from being finitely

determined); etc..

power series). Consider matrices with entries in S, up to left-right equivalence, A → UAV , where U, V are invertible matrices over S. When such a matrix is equivalent to a block-diagonal matrix? Alternatively, when the S-module coker(A) is decomposable?

An obvious necessary condition (for square matrices) is that the determinant of the matrixis reducible (as an element of S). This condition is very far from being sufficient. We prove a very simple necessary and sufficient condition for equivalence to block-diagonal form.

As immediate applications we prove several results: the Ulrich modules over local rings tend to be decomposable; an obstruction to Thom-Sebastiani decomposability of maps (or of systems of vector fields/PDE’s); decomposable representations of algebras are avoidable in projective families; decomposable matrices are highly unstable (they are far from being finitely

determined); etc..

Original language | English |
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State | Published - 10 May 2013 |

### Publication series

Name | arXiv preprint |
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