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Finitely generated saturated multi-Rees algebras

Suprajo Das, Sudeshna Roy

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.

Finitely generated saturated multi-Rees algebras

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.
Paper Structure (9 sections, 18 theorems, 88 equations)

This paper contains 9 sections, 18 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Multigraded Rees algebras and Hilbert functions
  4. Multigraded generalisations of a theorem of Amao
  5. Saturated multi-Rees algebras of ideals with low analytic spread
  6. Saturated multi-Rees algebras of monomial ideals
  7. Main result
  8. Saturated multi-Rees algebras in dimension two
  9. Acknowledgements

Key Result

Theorem 1.1

Suppose that $(R,m_R)$ is an excellent local domain of dimension $d$ and $I_1,\ldots,I_r,J$ are ideals in $R$. Assume that the analytic spread $\ell({I_1}_P\cdots {I_r}_P) < \dim R_P$ for all $P\in V(J)$. Then the saturated multi-Rees algebra $S_J(I_1,\ldots,I_r)$ of $I_1,\ldots,I_r$ with respect to for all $n_1,\ldots,n_r\gg0$.

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 3.1
  • proof
  • Proposition 3.1
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof
  • Lemma 4.1
  • ...and 25 more