Pontryagin duality for topological modules
Pontryagin duality for topological modules
复制标题
拓扑模块的庞特里亚金对偶性
DOI：
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发表时间：
1979
期刊：
影响因子：
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通讯作者：
J. Flood
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文献类型：
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作者：
J. Flood
以环的Pontryagin对偶为特征模，刻划了任意局部紧环上拓扑模的Pontryagin对偶.一些作者试图将局部紧Abel群的Pontryagin对偶推广到拓扑环上的拓扑模类。G.C.Preston[6]得到了/>；-adi整数和域及其局部积上模的对偶性，M.F.Smith[7]得到了线性空间上模的对偶，M.D.Levin[4]得到了离散交换环上模的对偶.在每种情况下，特征模或表示逆变对偶函子的模都被认为是环R的Pontryagin对偶，它被证明是典型的R-模。以R的对偶R为特征标模，证明了局部紧环上的局部紧模的对偶性。二元性是有价值的，因为它增加了对对象进行结构性分解的可能性，并增强了所考虑的范畴的功能丰富性。部分结果已经在拓扑环的民间传说中得到了暗示，在拓扑环的民间传说中，Kaplansky[3]和其他人优雅地使用特征来获得结构定理。其中包括几个明显的应用。1.设A和Y是拓扑空间和连续函数范畴中的Top中的对象。给定紧开拓扑，用(A，Y)表示从A到Y的连续函数空间。然后由集合U(K，O)={/|/(A)c O}给出这种拓扑的子基，其中K是A的任何紧子集，O是Y的拓扑的任何子基的成员。映射X X Y-(A，Y)是函子从Topop X Top到Top[5，p.181]对对象的作用。紧-开拓扑的一个众所周知的性质给出了连续函数A-»(A，Y)与形式为A×K的集合上从A×X到Y连续的函数之间的1-1对应，其中A是A的局部紧子集。设IP：X X Y-？Z在顶层，A是一个拓扑空间。自然映射(I)Af.X X(A，Y)^(A，Z)，其中^(x，/)(A)=*(x，FIA))，由编辑于1978年5月10日收到，并以修订后的形式于1978年11月1日收到。AMS(MOS)主题分类(1970)。小学22D35，13J99。©1979美国数学学会OOO2-9939/79/00O0-0328/$02.2S 329许可证或版权限制可能适用于再分发；请参阅https://www.ams.org/journal-terms-of-use
A Pontryagin duality for topological modules over any locally compact ring is described, using the Pontryagin dual of the ring as character module. Several authors have attempted to extend the Pontryagin duality for locally compact Abelian groups to classes of topological modules over topological rings. G. C. Preston [6] obtained duality for modules over the />-adic integers and fields and their local products, M. F. Smith [7] for linear spaces, and M. D. Levin [4] for modules over discrete commutative rings. In each case, the character module, or module representing the contravariant dual functor, was taken to be the Pontryagin dual of the ring R, which was shown to be canonically an R-module. It is proven that duality holds for locally compact modules over locally compact rings, using the dual R of R as character module. A duality is valuable, in that it increases the possibilities for structural dismemberment of objects, and enhances the functional richness of the category under consideration. Parts of the result have been hinted at in the folklore of topological rings, where Kaplansky [3] and others have elegantly used characters to obtain structure theorems. A few obvious applications are included. 1. Suppose A and Y are objects in Top, the category of topological spaces and continuous functions. Denote by (A, Y) the space of continuous functions from A to Y given the compact-open topology. Then a subbasis for this topology is given by the sets U(K, O) = {/|/(A) c O}, where K is any compact subset of A and O a member of any subbasis for the topology of Y. The map X X Y -> (A, Y) is the action on objects of a functor from Topop X Top to Top [5, p. 181]. A well-known property of the compact-open topology gives a 1-1 correspondence between continuous functions A -» (A, Y) and functions from A x X to Y continuous on sets of the form A X K, where A is a locally compact subset of A. Another property is the following: 2. Lemma. Suppose ip: X X Y -» Z is in Top, and A is a topological space. The natural mappings (i) Af. X X (A, Y)^(A, Z), where ^(x,/)(a) = *(x,fia)) and Received by the editors May 10, 1978 and, in revised form, November 1, 1978. AMS (MOS) subject classifications (1970). Primary 22D35, 13J99. © 1979 American Mathematical Society OOO2-9939/79/00O0-0328/$02.2S 329 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use