Filter regular sequence under small perturbations
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Mathematische Annalen
Filter regular sequence under small perturbations Linquan Ma1 · Pham Hung Quy2
· Ilya Smirnov3
Received: 31 July 2019 / Revised: 7 April 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We answer affirmatively a question of Srinivas–Trivedi (J Algebra 186(1):1–19, 1996): in a Noetherian local ring (R, m), if f 1 , . . . , fr is a filter-regular sequence and J is an ideal such that ( f 1 , . . . , fr ) + J is m-primary, then there exists N > 0 such that for any ε1 , . . . , εr ∈ m N , we have an equality of Hilbert functions: H (J , R/( f 1 , . . . , fr ))(n) = H (J , R/( f 1 + ε1 , . . . , fr + εr ))(n) for all n ≥ 0. We also prove that the dimension of the non Cohen–Macaulay locus does not increase under small perturbations, generalizing another result of [20].
1 Introduction Many fundamental questions in singularity theory arise from studying deformations. One particular way of deforming a singularity is by changing the defining equations by adding terms of high order. This problem often arises while working with analytic singularities. The first instance is the problem of finite determinacy which asks whether for a singularity defined analytically, e.g., as a quotient of a (convergent) power series ring, can be transformed into an equivalent algebraic singularity by truncating the defining equations. More generally, such truncation arise from Artin’s approximation that gives a way to descend finite structures, such as modules and finite complexes,
Communicated by Vasudevan Srinivas.
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Pham Hung Quy [email protected] Linquan Ma [email protected] Ilya Smirnov [email protected]
1
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA
2
Department of Mathematics, FPT University, Hanoi, Vietnam
3
Department of Mathematics, Stockholm University, 10691 Stockholm, Sweden
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L. Ma et al.
over the completion of a ring to finite structures over its henselization, which is a direct limit of essentially finite extensions with many nice properties. This problem was first considered by Samuel in 1956 [17], who showed for hypersurfaces f ∈ S = k[[x1 , . . . , xd ]] with isolated singularities that for large N if ε ∈ m N then there is an automorphism of S that maps f → f +ε. Samuel’s result was extended by Hironaka in 1965 [8], who showed that if S/I is an equidimensional reduced isolated singularity, and the ideal I obtained by changing the generators of I by elements of sufficiently larger order is such that S/I is still reduced, equidimensional, and same height as I , then there is an automorphism of S that maps I → I . Cutkosky and Srinivasan further extended Samuel and Hironaka’s result, we refer to [4,5] for more details. On the other hand, by results in [5,7], in order to get an isomorphism under small perturbation, it is essential to perturb by elements contained in the Jacobian ideal. So instead of requiring the deformation to give isomorphic rings, we consider a weaker question: what properties are preserved by a sufficiently
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