论文标题
Betti序列在局部环上的多项式生长
Polynomial growth of Betti sequences over local rings
论文作者
论文摘要
这是对贝蒂数字数量有限生成的模块的序列的研究，该模块在完整的交点本地环（$ r $）上进行了研究。 $ \ {β^r_ {i}（m）\} _ {i \ geq 0} $偶数，偶数奇数$ i $最终是由i的i $最终在i中以同样的指导术语给出的。我们表明，如果$ i^\ square $，这些多项式是重合的，那是由$ r $的二次级别关系产生的理想，满足$ {\ rm height} \ i^\ i^\ square \ ge {\ ge {\ rm codim} \ rm codim} \ r -1 $，以及$ r $ $ $ $ $ $ $ $ $} 4 $。随后，Avramov，Packauskas和Walker证明了$ J> {\ rm codim} \ r - {\ rm高度} \ i^\ square $ ext and Betti polynomials的square $相等。我们基于C.I的残基环的内在表征给出了新的证明。本文获得的最小多重性的局部环。我们还表明该界限是最佳的。
This is a study of the sequences of Betti numbers of finitely generated modules over a complete intersection local ring, $R$. The subsequences $\{β^R_{i}(M)\}_{i\geq 0}$ with even, respectively, odd $i$ are known to be eventually given by polynomials in i with equal leading terms. We show that these polynomials coincide if $I^\square$, the ideal generated by the quadratic relations of the associated graded ring of $R$, satisfies ${\rm height}\ I^\square \ge {\rm codim}\ R -1$, and that the converse holds if $R$ is homogeneous or ${\rm codim}\ R \le 4$. Subsequently Avramov, Packauskas, and Walker proved that the terms of degree $j > {\rm codim}\ R - {\rm height}\ I^\square$ of the even and odd Betti polynomials are equal. We give a new proof of that result, based on an intrinsic characterization of residue rings of c.i. local rings of minimal multiplicity obtained in this paper. We also show that that bound is optimal.
