Let (R,m) be a local ring (the simplest example is analytic/formal power
series), consider matrices with entries in R. They are studied up to
left-right equivalence, A\to UAV, where U,V are invertible matrices over R.
When such a matrix is equivalent to a block-diagonal matrix? To an
upper-block-triangular?
In the numerical case (i.e. matrices over a field) the question is trivial
(every matrix is left-right diagonalizable).
For matrices over a local ring the situation is more involved.
In commutative algebra, such a matrix A is the presentation of the R-module
coker(A). Block-diagonal matrices correspond to decomposable modules, while
upper-block-triangular matrices define modules that are extensions.
An obvious necessary condition (for square matrices) is that the determinant
of the matrix is reducible (as an element of R). This condition is very far
from being sufficient. We prove a very simple necessary and sufficient
condition for equivalence to block-diagonal form.
Equivalence to an upper-block-triangular form is a more delicate story, we
give various necessary conditions.
As an application we prove Thom-Sebastiani results on the decomposability of
maps/systems of vector fields(or PDE's).
(Joint work with V. Vinnikov)
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