When does a perturbation of the equations preserve the normal cone

TL;DR

This paper proves that small algebraic perturbations of generators of an ideal preserve the associated graded ring with respect to a given ideal, confirming conjectures on invariance of algebraic invariants under such perturbations.

Contribution

It establishes conditions under which perturbations of ideal generators do not change the normal cone, solving longstanding conjectures and providing explicit bounds for the perturbation size.

Findings

Invariance of the normal cone under small perturbations of ideal generators.

Equivalence of Artin-Rees numbers for original and perturbed ideals.

Preservation of algebraic properties like Cohen-Macaulayness under perturbation.

Abstract

Let $(R,\mathfrak m)$ be a local ring and $I, J$ two arbitrary ideals of $R$. Let $\operatorname{gr}_J(R/I)$ denote the associated ring of $R/I$ with respect to $J$, which corresponds to the normal cone in geometry. The main result of this paper shows that if $I = (f_1,...,f_r)$, where $f_1,...,f_r$ is a $J$-filter regular sequence, there exists a number $N$ such that if $f_i' \equiv f_i \mod J^N$ and $I' = (f_1',...,f_r')$, then $\operatorname{gr}_J(R/I) \cong \operatorname{gr}_J(R/I')$. If $J$ is an $\mathfrak m$-primary ideal, this result implies a long standing conjecture of Srinivas and Trivedi on the invariance of the Hilbert-Samuel function under small perturbations, which has been solved recently by Ma, Quy and Smirnov. As a byproduct, the Artin-Rees number of $I$ and $I'$ with respect to $J$ are the same. Furthermore, we give explicit upper bounds for the smallest number $N$ with the above property. These results solve two problems raised by Ma, Quy and Smirnov. There are other interesting consequences on the invariance of the Achilles-Manaresi function, the relation type, the Castelnuovo-Mumford regularity, the Cohen-Macaulayness and the Gorensteiness of the Rees algebra of $R/I$ with respect to $J$ under small perturbation of $I$. We also prove a converse of the main result showing that the condition $I$ being generated by a $J$-filter regular sequence is the best possible for its validity. The main result can be also extended to perturbations with respect to filtrations of ideals. As a consequence, if $R$ is a power series ring, $f_1,...,f_r$ is a filter regular sequence, and $f_i'$ is the $n$-jet of $f_i$ for $n \gg 0$, then $I$ and $I'$ have the same initial ideal with respect to any Noetherian monomial order. A special case of this consequence was a conjecture of Adamus and Seyedinejad on approximations of analytic complete intersection singularities.