Finitely generated saturated multi-Rees algebras

TL;DR

This paper investigates the finite generation of saturated multi-Rees algebras, establishing conditions under which they are finitely generated and analyzing the asymptotic behavior of related length functions.

Contribution

It provides new criteria for finite generation of saturated multi-Rees algebras in various algebraic settings and describes their asymptotic length functions.

Findings

Saturated multi-Rees algebras are finitely generated when the analytic spread is not maximal in excellent local domains.

Length functions associated with these algebras eventually agree with a polynomial.

In polynomial rings modulo monomial ideals, the length function exhibits piecewise quasi-polynomial behavior.

Abstract

We study the question of finite generation of saturated multi-Rees algebras and investigate the asymptotic behaviour of related length functions. In the setup of excellent local domains, we show that the saturated multi-Rees algebra of a finite collection of ideals is finitely generated when the analytic spread is not maximal and the associated length function eventually agrees with a polynomial. Similar results are obtained when we restrict to two-dimensional local UFDs with no restrictions on the analytic spread. We further prove that the saturated multi-Rees algebra of finitely many monomial ideals in a polynomial ring modulo an irreducible monomial ideal, is always finitely generated. In this case, the corresponding length function is shown to exhibit piecewise quasi-polynomial behaviour. We also produce multi-ideal versions of a theorem of Amao.