Primary Decompositions of Regular Sequences

TL;DR

This paper proves the existence of uniform bounds for primary decompositions of powers of regular sequences in Noetherian rings, ensuring a finite set of associated primes and controlled containment relations.

Contribution

It establishes a uniform bound on primary decompositions of powers of regular sequences, linking algebraic structure to prime ideals and their powers.

Findings

Finite set of associated primes for all decompositions

Existence of a uniform exponent C controlling containment

Primary components have bounded complexity

Abstract

Let $R$ be a Noetherian ring and $x_1,\ldots,x_t$ a permutable regular sequence of elements in $R$. Then there exists a finite set of primes $\Lambda$ and natural number $C$ so that for all $n_1,\ldots,n_t$ there exists a primary decomposition $(x_1^{n_1},\ldots,x_t^{n_t})=Q_1\cap \cdots \cap Q_\ell$ so that $\sqrt{Q_i}\in \Lambda$ and $\sqrt{Q_i}^{C(n_1+\cdots + n_t)}\subseteq Q_i$ for all $1\leq i\leq \ell$.