Transactions of the American Mathematical Society

Abstract:

Let $M$ be a left module over a Weyl algebra in characteristic zero. Given natural weight vectors $\nu$ and $\omega$, we show that the characteristic varieties arising from filtrations with weight vector $\nu +s\omega$ stabilize to a certain variety determined by $M$, $\nu$, $\omega$ as soon as the natural number $s$ grows beyond a bound which depends only on $M$ and $\nu$ but not on $\omega$.

As a consequence, in the notable case when $\nu$ is the standard weight vector, these characteristic varieties deform to the critical cone of the $\omega$-characteristic variety of $M$ as soon as $s$ grows beyond an invariant of $M$.

As an application, we give a new, easy, non-homological proof of a classical result, namely, that the $\omega$-characteristic varieties of $M$ all have the same Krull dimension.

The set of all $\omega$-characteristic varieties of $M$ is finite. We provide an upper bound for its cardinality in terms of supports of universal Gröbner bases in the case when $M$ is cyclic. By the above stability result, we conjecture a second upper bound in terms of total degrees of universal Gröbner bases and of Fibonacci numbers in the case when $M$ is cyclic over the first Weyl algebra.