Rationality of equivariant Hilbert series and asymptotic properties

Abstract

AnFI\operatorname {FI}- or anOI\operatorname {OI}-moduleM\mathbf {M}over a corresponding noetherian polynomial algebraP\mathbf {P}may be thought of as a sequence of compatible modulesMn\mathbf {M}_nover a polynomial ringPn\mathbf {P}_nwhose number of variables depends linearly onnn. In order to study invariants of the modulesMn\mathbf {M}_nin dependence ofnn, an equivariant Hilbert series is introduced ifM\mathbf {M}is graded. IfM\mathbf {M}is also finitely generated, it is shown that this series is a rational function. Moreover, if this function is written in reduced form rather precise information about the irreducible factors of the denominator is obtained. This is key for applications. It follows that the Krull dimension of the modulesMn\mathbf {M}_ngrows eventually linearly innn, whereas the multiplicity ofMn\mathbf {M}_ngrows eventually exponentially innn. Moreover, for any fixed degreejj, the vector space dimensions of the degreejjcomponents ofMn\mathbf {M}_ngrow eventually polynomially innn. As a consequence, any graded Betti number ofMn\mathbf {M}_nin a fixed homological degree and a fixed internal degree grows eventually polynomially innn. Furthermore, evidence is obtained to support a conjecture that the Castelnuovo-Mumford regularity and the projective dimension ofMn\mathbf {M}_nboth grow eventually linearly innn. It is also shown that modulesM\mathbf {M}whose widthnncomponentsMn\mathbf {M}_nare eventually Artinian can be characterized by their equivariant Hilbert series. Using regular languages and finite automata, an algorithm for computing equivariant Hilbert series is presented.