Serre's vanishing conjecture for Ext-groups
Serre's vanishing conjecture for Ext-groups
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塞尔关于 Ext-groups 的消失猜想
DOI：
10.1016/j.jpaa.2003.07.007
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发表时间：
2004
影响因子：
0.8
通讯作者：
I. Mori
中科院分区：
文献类型：
--
作者：
I. Mori
设R是Noether交换局部环，M，N是N-生成R-模.然后一个广义形式的塞尔的消失猜想可以陈述如下：如果然后[公式：见正文]这是已知的塞尔的消失猜想成立一个完整的交叉环R，但不知道一个Gorenstein环R。我们可以做出类似的猜想，用Ext代替Tor，即，如果M和N满足上述三个条件，则[公式：在本文中，我们将证明，在Gorenstein环R上，Tor-version的Vanishing猜想，Ext-version的Vanishing猜想，Mori和Smith（J. Pure Appl. Algebra 157（2001）279）中定义的交叉重数的交换性都是等价的。此外，我们将证明，一个特定的Ext版本的消失猜想成立的一大类非交换投影计划，通常包括所有的交换投影计划在一个领域，通过扩展Bézout的定理。
Let R be a noetherian commutative local ring, and M,N be finitely generated R-modules. Then a generalized form of Serre's Vanishing conjecture can be stated as follows: if then [Formula: see text] It is known that Serre's Vanishing conjecture holds for a complete intersection ring R, but is not known for a Gorenstein ring R. We can make a similar conjecture replacing Tor by Ext, namely, if M and N satisfy the above three conditions, then [Formula: see text] In this paper, we will prove that, over a Gorenstein ring R, the Tor-version of the Vanishing conjecture, the Ext-version of the Vanishing conjecture, and the commutativity of the intersection multiplicity defined in Mori and Smith (J. Pure Appl. Algebra 157 (2001) 279) are all equivalent. Further, we will prove that a certain Ext-version of the Vanishing conjecture holds for a large class of noncommutative projective schemes, typically including all commutative projective schemes over a field, by extending Bézout's Theorem.