Perturbations in Noetherian Local Rings m-adic Perturbations in Noetherian Local Rings Nick Cox-Steib
[email protected] University of Missouri Abstract We develop new methods to study m-adic stability in an arbitrary Noetherian local ring. These tech- niques are used to prove results about the behavior of Hilbert-Samuel and Hilbert-Kunz multiplicities under fine m-adic perturbations. 1 Introduction The topic of this paper involves comparing a fixed ideal, I =( f1,..., fc) = f , inside a Noetherian local ring, (R,m ), with ideals of the form ( f + ε , f + ε ,..., f + ε ) = f + ε , where ε ,...,ε are in R 1 1 2 2 c c 1 c some large power of the maximal ideal of R. When it is in the interest of clarity and space and the ideal I is understood, we often use the notation M M M M := = , and M := IM f M ( f +ε) f + ε M where M is an R-module. We will refer to the ideals f + ε , as mR-adic perturbations of I, and, more generally, we may also speak of the quotients M as perturbations of M. At the most basic level, we ( f +ε) want to understand the relationship between I and its perturbations. Recent interest in these problems stems from their relationship to the deformation of properties. This arXiv:2011.08044v2 [math.AC] 14 Dec 2020 relationship has been formalized in a recent paper by De Stefani and Smirnov with the concept of m-adic stability [SS20]. A property, P, of Noetherian local rings is m-adically stable if it is stable under sufficiently small perturbations of ideals generated by a regular element — that is, for every regular element f in a Noetherian local ring (R,m) such that R/( f ) has property P, there is an N ∈ N such that R/( f + ε) also has property P for all ε ∈ mN.