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
leftright equivalence, A\to UAV, where U,V are invertible matrices over R.
When such a matrix is equivalent to a blockdiagonal matrix? To an
upperblocktriangular?
In the numerical case (i.e. matrices over a field) the question is trivial
(every matrix is leftright diagonalizable).
For matrices over a local ring the situation is more involved.
In commutative algebra, such a matrix A is the presentation of the Rmodule
coker(A). Blockdiagonal matrices correspond to decomposable modules, while
upperblocktriangular 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 blockdiagonal form.
Equivalence to an upperblocktriangular form is a more delicate story, we
give various necessary conditions.
As an application we prove ThomSebastiani results on the decomposability of
maps/systems of vector fields(or PDE's).
(Joint work with V. Vinnikov)
