Dualizing complexes of affine semigroup rings
Dualizing complexes of affine semigroup rings
复制标题
仿射半群环的对偶复形
DOI：
10.1090/s0002-9947-1990-1076179-6
复制
发表时间：
1990
影响因子：
1.3
通讯作者：
P. Schenzel
中科院分区：
文献类型：
--
作者：
U. Schäfer;P. Schenzel
对于仿射半群环，利用半群及其所张成的多面体锥的面格的同调构造了对偶复形。因此，有局部科恩-麦考利环，Buchsbaum环和科恩-麦考利环以及塞尔条件9的特征。设S表示加法幺半群N '的一个n-生成子幺半群，n为正整数。通过域X上S的仿射半群环I [S]，我们记X[xl，.，xn]由所有单项式xs，s E S生成。因此，直到同构，仿射半群环和由单项式参数化给出的仿射簇之间存在一一对应。现在，它是一些感兴趣的特点环的理论性质，科恩麦考利，Gorenstein等，关于半群S. Kempf等人[KKMS]和Hochster [Hol]在这个方向上首次取得了突破，他们证明了X[S]是一个Cohen-Macaulay环，只要它是正规的。此外，X[S]的正规性是用S来描述的（见[Hol]）。另一种特殊情况，当S是单纯半群时，分别由后藤，铃木和渡边[GSW]讨论. Stanley [St 2]证明了X[S]是Cohen-Macaulay环，如果它满足Serre的22 -条件。注意，正态性等同于5 Y2和R1。此外，还有关于S的5 '-条件的描述（见定理6.3）。一般来说，52-条件不足以使I[S]是Cohen-Macaulay环，如[Hol]中的例子所示。[TH，1.2]。另一方面，Grobner [G]提出了Cohen-Macaulay仿射半群环的分类问题。这个问题的解决方案是由Hoa和Trung [TH]完成的。分次环的结构性质是对偶理论的一个技术工具。这就是Grothendieck引入的对偶复形（见[Ha]）。近年来，它成为交换代数的有用工具。特别是，它的知识可以刻画科恩-麦考利环，局部科恩-麦考利环，Buchsbaum环，Gorenstein环，和5-条件接收编辑1988年12月18日。1980年数学学科分类（1985年修订）。小学13 H10;中学14 M05，06 A99。（1990年美国数学学会0002-9947/90 $1.Q0 +每页$0.25
For an affine semigroup ring we construct the dualizing complex in terms of the semigroup and the homology of the face lattice of the polyhedral cone spanned by the semigroup. As a consequence there are characterizations of locally Cohen-Macaulay rings, Buchsbaum rings, and Cohen-Macaulay rings as well as Serre's condition 9 . INTRODUCTION Let S denote a finitely generated submonoid of the additive monoid N', n being a positive integer. By the affine semigroup ring I [S] of S over a field X let us denote the subring of X[xl, ... , xn] generated by all monomials xs, s E S. So one has, up to isomorphisms, a one-to-one correspondence between affine semigroup rings and affine varieties given parametrically by monomials. Now it is of some interest to characterize ring theoretic properties as CohenMacaulay, Gorenstein, etc., in terms of the semigroup S. A first breakthrough in this direction was done by Kempf et al. [KKMS] and Hochster [Hol] who showed that X[S] is a Cohen-Macaulay ring provided it is normal. Moreover, the normality of X[S] is described in terms of S (see [Hol]). Another particular case, if S is a simplicial semigroup, is treated by Goto, Suzuki, and Watanabe [GSW], resp. Stanley [St2], who showed that X[S] is a Cohen-Macaulay ring if it satisfies the 22 -condition of Serre. Note that normality is equivalent to 5Y2 and R1. Moreover, there is a description of the 5'-condition in terms of S (see Theorem 6.3). In general the 52-condition is not sufficient for I[S] to be a Cohen-Macaulay ring, as follows by examples in [Hol] resp. [TH, 1.2]. On the other side Grobner [G] posed the problem to classify Cohen-Macaulay affine semigroup rings. The solution of this problem was done by Hoa and Trung [TH]. Related to structural properties of graded rings is a technical tool of the duality theory. This is the dualizing complex introduced by Grothendieck (see [Ha]). In recent years it became a helpful tool in commutative algebra. In particular, its knowledge allows characterizations of Cohen-Macaulay rings, locally CohenMacaulay rings, Buchsbaum rings, Gorenstein rings, and the 5-conditions of Received by the editors December 18, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 13H10; Secondary 14M05, 06A99. ( 1990 American Mathematical Society 0002-9947/90 $1.Q0 + $.25 per page