We answer affirmatively a question of Srinivas–Trivedi (J Algebra 186(1):1–19, 1996): in a Noetherian local ring $$(R,{{\,\mathrm{\mathfrak {m}}\,}})$$ ( R , m ) , if $$f_1,\dots ,f_r$$ f 1 , ⋯ , f r is a filter-regular sequence and J is an ideal such that $$(f_1, \ldots , f_r)+J$$ ( f 1 , … , f r ) + J is $${{\,\mathrm{\mathfrak {m}}\,}}$$ m -primary, then there exists $$N>0$$ N > 0 such that for any $$\varepsilon _1,\dots ,\varepsilon _r \in {{\,\mathrm{\mathfrak {m}}\,}}^N$$ ε 1 , ⋯ , ε r ∈ m N , we have an equality of Hilbert functions: $$H(J, R/(f_1,\dots ,f_r))(n)=H(J, R/(f_1+\varepsilon _1,\dots , f_r+\varepsilon _r))(n)$$ H ( J , R / ( f 1 , ⋯ , f r ) ) ( n ) = H ( J , R / ( f 1 + ε 1 , ⋯ , f r + ε r ) ) ( n ) for all $$n\ge 0$$ 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 ].

中文翻译：

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在小扰动下过滤规则序列

我们肯定地回答了 Srinivas–Trivedi (J Algebra 186(1):1–19, 1996) 的一个问题：在 Noetherian 局部环 $$(R,{{\,\mathrm{\mathfrak {m}}\,} })$$ ( R , m ) , 如果 $$f_1,\dots ,f_r$$ f 1 , ⋯ , fr 是一个过滤规则序列，J 是一个理想的，使得 $$(f_1, \ldots , f_r) +J$$ ( f 1 , … , fr ) + J 是 $${{\,\mathrm{\mathfrak {m}}\,}}$$ m -primary，则存在 $$N>0$$ N > 0 使得对于任何 $$\varepsilon _1,\dots ,\varepsilon _r \in {{\,\mathrm{\mathfrak {m}}\,}}^N$$ ε 1 , ⋯ , ε r ∈ m N ，我们有希尔伯特函数的等式： $$H(J, R/(f_1,\dots ,f_r))(n)=H(J, R/(f_1+\varepsilon _1,\dots , f_r+\varepsilon _r))(n)$$ H ( J , R / ( f 1 , ⋯ , fr ) ) ( n ) = H ( J , R / ( f 1 + ε 1 , ⋯ , fr + ε r ) ) ( n ) 对于所有 $$n\ge 0$$ n ≥ 0 。我们还证明了非 Cohen-Macaulay 轨迹的维数在小扰动下不会增加，概括了 [20] 的另一个结果。