Co-Localization, Co-Support and Local Homology
Co-Localization, Co-Support and Local Homology
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共定位、共支持和局部同源
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发表时间：
2006
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通讯作者：
A. S. Richardson
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作者：
A. S. Richardson
我们提出了一个定义的共同支持的模块交换，诺特环，我们希望将涉及到当地的同调函子的Greenlees和5月以同样的方式，普通的支持涉及到当地的上同调。通过表达局部同调的Koszul复形，我们证明了一些消失定理，涉及这个共同支持。我们还研究了共同本地化函子，从而产生我们的共同支持的定义。这个函子对待Artinian型构造的方式与普通的局部化函子对待Noether型构造的方式相同，并且这种对偶性扩展到共支撑和普通支撑之间的对偶性。导论.给定一个交换诺特环A上的模M和A的理想I，如果我们试图将M的支集限制在簇V（I）上，那就等于取M的I-adic挠。这个过程的右导函子是局部上同调函子H· I。因此，支集的概念从一开始就是局部上同调的一部分。它也出现在某些著名的消失结果中：如果M开始是挠的，也就是说如果Supp M <$V（I），则M的所有高局部上同调模消失。此外，任何模M的局部上同调在M的支集维数之后总是为零。Matlis在[8]中定义了局部同调函子为I-adic完备函子的左导函子。由于挠和完备化是对偶的，人们期望这些函子名副其实，并且以与局部上同调对偶的方式表现。特别是，它是自然的期望有消失定理的局部同调对偶的那些联系的局部上同调模的支持。然而，在我们能够证明甚至陈述这样的定理之前，我们必须决定如何对偶化支撑的概念。这是本文的主要目的。AMS数学学科分类。小学13 D 07，13 E10。编辑于2003年6月20日收到，并于2004年1月6日修订。版权所有©2006落基山数学联盟
We propose a definition of co-support for modules over commutative, Noetherian rings that we hope will relate to the local homology functors of Greenlees and May in the same way ordinary support relates to local cohomology. By expressing local homology in terms of the Koszul complex, we prove some vanishing theorems involving this co-support. We also investigate the co-localization functor which gives rise to our definition of co-support. This functor treats Artiniantype constructions the same way the ordinary localization functor treats Noetherian-type constructions, and this duality extends to one between co-support and ordinary support. Introduction. Given a module M over a commutative, Noetherian ring A and an ideal I of A, if we attempt to restrict the support of M to the variety V (I), that amounts to taking the I-adic torsion of M . The right derived functors of this process are the local cohomology functors H• I . The concept of support is thus part of local cohomology from the beginning. It also shows up in certain well-known vanishing results: If M is torsion to begin with, which is to say if Supp M ⊂ V (I), then all the higher local cohomology modules of M vanish. Furthermore, the local cohomology of any module M will always vanish past the dimension of the support of M . In [8], Matlis defined the local homology functors to be the left derived functors of the I-adic completion functor. Since torsion and completion are dual, one expects these functors to live up to their name and behave in a manner dual to local cohomology. In particular, it is natural to expect there to be vanishing theorems for local homology dual to the ones that relate the local cohomology of a module to the module’s support. Before we can prove or even state such theorems, however, we have to decide how to dualize the notion of support. That is the primary objective of this paper. AMS Mathematics Subject Classification. Primary 13D07, 13E10. Received by the editors on June 20, 2003, and in revised form on January 6, 2004. Copyright c ©2006 Rocky Mountain Mathematics Consortium