Small subalgebras of polynomial rings and Stillman’s Conjecture

Abstract

Let n , d , η n, d, \eta be positive integers. We show that in a polynomial ring R R in N N variables over an algebraically closed field K K of arbitrary characteristic, any K K -subalgebra of R R generated over K K by at most n n forms of degree at most d d is contained in a K K -subalgebra of R R generated by B ≤ η B ( n , d ) B \leq {}^\eta \mathcal {B}(n,d) forms G 1 , … , G B {G}_1,\,\ldots ,\,{G}_{B} of degree ≤ d \leq d , where η B ( n , d ) {}^\eta \mathcal {B}(n,d) does not depend on N N or K K , such that these forms are a regular sequence and such that for any ideal J J generated by forms that are in the K K -span of G 1 , … , G B {G}_1,\,\ldots ,\,{G}_{B} , the ring R / J R/J satisfies the Serre condition R η \mathrm {R}_\eta . These results imply a conjecture of M. Stillman asserting that the projective dimension of an n n -generator ideal

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多项式环的小子代数与Stillman猜想

设n，d，ηn，d为正整数。我们证明了在任意特征的代数闭域K K上N个变量的多项式环R R中，在K K上生成至多n个次数至多d d形式的R R的任何K K-子代数都包含在由B≤ηB（n，d）B\leq｛｝^\eta\mathcal｛B｝G 1，…，G B{G}_1，\，\ldots，\，{G}_{B} 阶≤d\leq d，其中ηB（n，d）｛｝^\eta\mathcal｛B｝（n，d）不依赖于n n或K K，使得这些形式是正则序列，并且对于由在G1，…，GB的K K跨度中的形式生成的任何理想J J{G}_1，\，\ldots，\，{G}_{B} ，环R/J R/J满足Serre条件Rη\mathrm{R}_\eta。这些结果暗示了M.Stillman的一个猜想，即n-生成元理想的投影维数<内联公式

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