Gorenstein algebras and the Cayley-Bacharach theorem
Gorenstein algebras and the Cayley-Bacharach theorem
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Gorenstein 代数和 Cayley-Bacharach 定理
DOI：
10.1090/s0002-9939-1985-0776185-6
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发表时间：
1985
影响因子：
0.9
通讯作者：
F. Orecchia
中科院分区：
文献类型：
--
作者：
E. Davis;A. Geramita;F. Orecchia
本文研究了p2中完全交点的经典Cayley-Bacharach定理与梯度Gorenstein代数性质之间的联系。介绍。众所周知，p2中完全交点的Cayley-Bacharach定理对于P'的0维算术Gorenstein子格式是有效的。更一般地，我们证明了该结果对具有最小Cohen-Macaulay型与Hilbert函数兼容的P'的0维子格式是有效的。Gorenstein情况下的Cayley-Bacharach定理是一个特殊的定理，它本身很有趣，在技术上也很有用，它与P'的关联子格式的Hilbert函数有关。最后，我们证明了Pn的0维，算术上的Gorenstein约简子格式具有Cayley-Bacharach定理的有效性和Hilbert函数的对称性。固定的符号。A表示标准n阶k代数，k为域：AO= k；A = k[AI]；X (A1) < ox。我们用X表示k-线性维数，保留“dim”表示环或方案的维数，8表示这类代数的多重性（或相应投影方案的度）。注意，如果dim A = 0，则8(A) = X(A)。我们用I表示a的非零，非单位，齐次理想，并且J = ann I（“ann”=湮灭子）。我们总是假设A和A/I是CM (Cohen-Macaulay)，而Ass(A/I) c Ass(A)。因此是Ass(A/J) c Ass(A)。（事实上，由于A的0理想不混合高度为0，所以A的任何非0理想的湮灭子也是如此）因此当A < 1时，A/J为CM。在任何情况下，如果A是Gorenstein [PS，命题1.3]，A/J是CM。在我们的应用程序中，由于这两个原因之一，A/J将是CM。回想一下，根据定义，环R是Gorenstein，条件是对于R的每一个素数a， RI是Gorenstein局部环，并且a是Gorenstein <* AA， a是Gorenstein局部环[AG]。下面使用的Gorenstein局部环的性质，我们参考[K]。1. 观察。假设人工智能是Gorenstein (A)。则：(a) 8(a) = S(a /I) + S（a /J）。1984年1月3日编辑收到。1980数学学科分类。Primary 13H10, 14N10。
This paper is an examination of the connection between the classical Cayley-Bacharach theorem for complete intersections in p2 and properties of graded Gorenstein algebras. Introduction. It is known, if not well known, that the Cayley-Bacharach theorem for complete intersections in p2 is valid for 0-dimensional arithmetically Gorenstein subschemes of P'. More generally, we show that the result is valid for 0-dimensional subschemes of P' having minimal Cohen-Macaulay type compatible with their Hilbert functions. The Cayley-Bacharach theorem in the Gorenstein case is a special instance of a theorem, interesting and technically useful in its own right, relating the Hilbert functions of linked subschemes of P'. Lastly we show that the 0-dimensional, arithmetically Gorenstein, reduced subschemes of Pn are characterized by the validity of the Cayley-Bacharach theorem and the symmetry of the Hilbert function. Fixed notation. A denotes a standard N-graded k-algebra, k a field: AO= k; A = k[AI]; X (A1) < ox. We use X to denote k-linear dimension, reserving "dim" for dimension of rings or schemes, and 8 to denote multiplicity for such algebras (or degree of the corresponding projective scheme). Note that 8(A) = X(A) if dim A = 0. We use I to denote a nonzero, nonunit, homogeneous ideal of A, and J = ann I ("ann" = annihilator). We assume always that A and A/I are CM (Cohen-Macaulay) and that Ass(A/I) c Ass(A). Hence Ass(A/J) c Ass(A). (Indeed, since the 0-ideal of A is unmixed of height 0, so is the annihilator of any nonzero ideal of A.) Therefore A/J is CM if dim A < 1. In any case, A/J is CM if A is Gorenstein [PS, Proposition 1.3]. In our applications A/J will be CM for one of these two reasons. Recall that, by definition, a ring R is Gorenstein provided that RI is a Gorenstein local ring for every prime idealA, of R, and A is Gorenstein <* AA,A is a Gorenstein local ring [AG]. We refer to [K] for those properties of Gorenstein local rings which are used below without specific reference. 1. OBSERVATIONS. Assume that Al is Gorenstein for all, EAss(A)). Then: (a) 8(A) = S(A/I) + S(A/J). Received by the editors January 3, 1984. 1980 Mathemnatics Subject Classification. Primary 13H10, 14N10.