高度三水平代数的可约族
Reducible family of height three level algebras
摘要：
设$R=k[x_1,..., x_r]$是无限域$k$上$r$变量的多项式环，$M$是$R$的极大理想.这里，\emph{level algebra}是$R$的分级阿廷商$A$，其具有单次$j$中的柱脚$Soc(A)=0:M$。希尔伯特函数$H(A)=(h_0,h_1,... ,h_j)$给出了$0\le i\le j$的$A$的每个度-$i$分级片段的维度$h_i=\dim_k A_i$。$A$的嵌入维数是$h_1$，$A$的\emph{type}是$\dim_k \Soc (A)$，这里是$h_j$。具有Hilbert函数$H$的$R$的水平代数子族$\Levalg (H)$形成分次代数族的开子概型，或者通过Macaulay对偶形成格拉斯曼代数族的开子概型。我们证明了对于每一个Hilbert函数$H=H_1=(1,3,4,4)$和$H=H_2=(1,3,6,8,9,3)$，Hilbert函数$H$的$LevAlg (H)$族参数化水平的Artin代数有几个不可约的组件。我们还表明，这些例子每个电梯点。然而，在第一个例子中，阿廷代数的不可约贝蒂层在提升到点时变得可约。这些是我们在嵌入维3中获得$\Levalg(H)$多个组件的第一个例子。我们还表明，第二个例子是第一个在一个无限序列的例子类型三希尔伯特函数$H(c)$中，也有一些组件的LevAlg（H）得到任意大。第一种情况下，多个组件的现象可能会发生（即最低的嵌入维数，然后最低的类型）是三维和类型二。第一种情况的例子已经由作者和J. O.克莱珀
英文摘要：
Let $R=k[x_1,..., x_r]$ be the polynomial ring in $r$ variables over an infinite field $k$, and let $M$ be the maximal ideal of $R$. Here a \emph{level algebra} will be a graded Artinian quotient $A$ of $R$ having socle $Soc(A)=0:M$ in a single degree $j$. The Hilbert function $H(A)=(h_0,h_1,... ,h_j)$ gives the dimension $h_i=\dim_k A_i$ of each degree-$i$ graded piece of $A$ for $0\le i\le j$. The embedding dimension of $A$ is $h_1$, and the \emph{type} of $A$ is $\dim_k \Soc (A)$, here $h_j$. The family $\Levalg (H)$ of level algebra quotients of $R$ having Hilbert function $H$ forms an open subscheme of the family of graded algebras or, via Macaulay duality, of a Grassmannian. We show that for each of the Hilbert functions $H=H_1=(1,3,4,4)$ and $H=H_2=(1,3,6,8,9,3)$ the family $LevAlg (H)$ parametrizing level Artinian algebras of Hilbert function $H$ has several irreducible components. We show also that these examples each lift to points. However, in the first example, an irreducible Betti stratum for Artinian algebras becomes reducible when lifted to points. These were the first examples we obtained of multiple components for $\Levalg(H)$ in embedding dimension three. We also show that the second example is the first in an infinite sequence of examples of type three Hilbert functions $H(c)$ in which also the number of components of LevAlg(H) gets arbitrarily large. The first case where the phenomenon of multiple components can occur (i.e. the lowest embedding dimension and then the lowest type) is that of dimension three and type two. Examples of this first case have been obtained by the authors and also by J.-O. Kleppe.
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