Gr\"obner bases and equations of the multi-Rees algebras

TL;DR

This paper characterizes the defining equations of multi-Rees algebras over Noetherian rings with ideals generated by subsets of a weak regular sequence, advancing algebraic understanding of these structures.

Contribution

It provides a detailed description of the equations defining multi-Rees algebras in a specific algebraic setting, extending prior knowledge on Rees algebra equations.

Findings

Explicit equations for multi-Rees algebras are derived.

The structure of these equations depends on the properties of the generating ideals.

The results apply to ideals generated by subsets of weak regular sequences.

Abstract

In this paper we describe the equations defining the multi-Rees algebra $R[I_1^{a_1}t_1,\dots,I_r^{a_r}t_r]$, where $R$ is a Noetherian ring and the ideals are generated by subsets of a fixed weak regular sequence.