Betti strata of height two ideals
Betti strata of height two ideals
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贝蒂层的高度两个理想
DOI：
10.1016/j.jalgebra.2004.12.012
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发表时间：
2004
影响因子：
0.9
通讯作者：
A. Iarrobino
中科院分区：
文献类型：
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作者：
A. Iarrobino
设R=k[x，y]表示无限域k上的二元多项式环.本文研究了R=k[x，y]的分次Artin多项式族G（H）在给定Hilbert函数H的情况下的参数化Betti层. Betti层Gβ（H）表示具有由H确定的分次Betti数和最小关联度β的所有行列式A，其中βi= dimkTor 1 R（I，k）i.我们恢复了Betti层是不可约的，并且我们计算了它们在族G（H）中的余维数。 这里μ=min{i| Hi<i+1}，理想的初始度。当k是代数闭的时，我们还证明了Betti层的闭包是更特殊层的并，并且是Cohen-Macaulay（定理2.18）。我们的方法是将Gβ（H）确定为行列式簇的乘积。我的话比他的话更有说服力。Iarrobino，V. Kanev，Power Sums，Gorenstein Algebras，and Determinantal Loci，Lecture Notes in Math.，第1721卷，Springer，Heidelberg，1999，345+xxvii页，定理5.63]，它得到了P2的准时子概型的一个假设层的Betti基的余维数; Boij确定高度为3的Gorenstein代数的Betti层的余维数[M. Boij，余维空间的贝蒂数层3 Gorenstein Artin代数，预印本，2001]。贯穿全文的关键工具包括不变量τ（V）的性质，祖先理想V <$R的生成元的数目，以及[A]中关于投射簇G（H）的先前结果。Iarrobino，Punctual Hilbert schemes，Mem. Amer. Math. Soc. 10（188）（1977）]。我们采用M. Boij，减少计算余维的贝蒂地层，以显示最特殊的地层有正确的余维。作为应用，我们确定哪些希尔伯特函数对于给定柱脚类型的R的Artin代数是可能的（定理3.2），并且我们还确定R中的t个一般足够水平理想的交集的希尔伯特函数，每个理想都有一个指定的希尔伯特函数（定理3.6）。
Let R=k[x,y] denote the polynomial ring in two variables over an infinite field k. We study the Betti strata of the family G(H) parametrizing graded Artinian quotients of R=k[x,y] having given Hilbert function H. The Betti stratum Gβ(H) parametrizes all quotients A of having the graded Betti numbers determined by H and the minimal relation degrees β, with βi=dimkTor1R(I,k)i. We recover that the Betti strata are irreducible, and we calculate their codimension in the family G(H). Here μ=min{i|Hi<i+1}, the initial degree of the ideals. When k is algebraically closed, we also show that the closure of a Betti stratum is the union of more special strata, and is Cohen–Macaulay (Theorem 2.18). Our method is to identify Gβ(H) as a product of determinantal varieties. Ours is a more direct argument than that of [A. Iarrobino, V. Kanev, Power Sums, Gorenstein Algebras, and Determinantal Loci, Lecture Notes in Math., vol. 1721, Springer, Heidelberg, 1999, 345+xxvii pp., Theorem 5.63], which obtains the codimension of Betti substrata of a postulation stratum for punctual subschemes of P2; there we relied on a result of M. Boij determining the codimension of Betti strata of height three Gorenstein algebras [M. Boij, Betti number strata of the space of codimension three Gorenstein Artin algebras, preprint, 2001]. Key tools throughout include properties of an invariant τ(V), the number of generators of the ancestor ideal V¯⊂R, and previous results concerning the projective variety G(H) in [A. Iarrobino, Punctual Hilbert schemes, Mem. Amer. Math. Soc. 10 (188) (1977)]. We adapt a method of M. Boij that reduces the calculation of the codimension of the Betti strata to showing that the most special stratum has the right codimension. As applications we determine which Hilbert functions are possible for Artinian quotients of R, given the socle type (Theorem 3.2), and we also determine the Hilbert function of the intersection of t general enough level ideals in R, each having a specified Hilbert function (Theorem 3.6).