Algebras defined by powers of determinantal ideals
Algebras defined by powers of determinantal ideals
复制标题
由行列理想的幂定义的代数
DOI：
10.1016/0021-8693(91)90222-t
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发表时间：
1991
影响因子：
0.9
通讯作者：
W. Bruns
中科院分区：
文献类型：
--
作者：
W. Bruns
设X为域K上的一个m × n的不定式矩阵，a = K [X]， I= Z,(X)是由X的T次元生成的理想矩阵。本文主要研究的对象是Rees代数2= &Y,(a)=@ p ' ' = 0 I ' 'c a [T]， T是一个新的不定式，以及相关联的梯度环B= $(a)=@ T ~ ozi/zi+ l= 6%/Z&%。&？和3在（简单情况t= 1和）极大次元的理想中是很容易理解的：t= min (m, n)，详细讨论参见[BV]，第91节。这里我们主要对更复杂的情况1< t< min （m, n）感兴趣。我们的结果的关键是由conccini， Eisenbud和Procesi [DEP]确定的对于char K= 0的幂次I的初等分解，并由Bruns和Vetter [BV]推广到char K> min （m-t, n-t, t）的情况；这些特征将被称为非异常。我们证明了这个结果如何推广到一个任意的系数积分域：初等理想的交点给出I ‘的非例外特征，总是I ’的积分闭包。从初等分解可以立即得出幂函数I在非例外特性上是全封闭的。所以& !是一个法域，并且i的初等分解一个有趣的观察：如果事先知道所有这些理想都是全封闭的，那么理想I '的初等分解可以很快计算出来。在特征0处得到了最好的结果，因为在K [x]上存在线性约化群GL (m， K) x GL （n， K）的无多重作用，在此作用下Z是稳定的，参见[DEP]或[BV]，第111节。应用u不变量理论（Kraft [Kr]）可以证明，！G美元!有理性奇点吗，特别是&？因此，有9个是科恩-麦考利环。我们毫不怀疑%！和3是科恩-麦考利任意非异常特征。然而，在特殊特征中，它们似乎尽可能远离这个性质：t= 2的情况表明，人们必须期望Y的深度为0，w的深度为1
Let X be an m xn matrix of indeterminates over a field K, A= K [X], and I= Z,(X) the ideal generated by the t-minors of X. The main objects of this article are the Rees algebra 2= &Y,(A)=@ p”= 0 I’T’c A [T], T a new indeterminate, and the associated graded ring B= $(A)=@ t~ ozi/zi+ l= 6%/Z&%. The structures of &? and 3 are well-understood in (the simple case t= 1 and) for ideals of maximal minors: t= min (m, n), cf.[BV, Sect. 91 for a detailed discussion. Here we are mainly interested in the much more complicated situation 1< t< min (m, n). The key to our results is the primary decomposition of the powers I’determined by DeConcini, Eisenbud, and Procesi [DEP] for char K= 0 and extended to the situation char K> min (m-t, n-t, t) by Bruns and Vetter [BV]; these characteristics will be called non-exceptional. We show how this result generalizes to an arbitrary integral domain of coefficients: The intersection of primary ideals which gives I’in non-exceptional characteristics, always is the integral closure of I’. It follows immediately from the primary decomposition that the powers I’are integrally closed in non-exceptional characteristics. Therefore &! is a normal domain, and the primary decomposition of I.!% turns out easy, giving some insight into the structure of Y. An interesting observation: The primary decomposition of the ideals I’can be computed very quickly if one knows in advance that all these ideals are integrally closed. The best results are obtained in characteristic 0 since one has a multiplicity free action of the linearly reductive group GL (m, K) x GL (n, K) on K [X] under which Z is stable, cf.[DEP] or [BV, Sect. 111. Applying the theory of U-invariants (Kraft [Kr]) one shows that,! G $! has rational singularities, in particular &? and, hence, 9 are Cohen-Macaulay rings. We have no doubt that%! and 3 are Cohen-Macaulay in arbitrary non-exceptional characteristic. It seems however that in exceptional characteristic they are as far as possible from this property: The case t= 2 indicates that one has to expect depth 0 for Y and depth 1 for W. 150