Complete cohomological functors on groups
Complete cohomological functors on groups
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DOI：
10.1016/0166-8641(87)90015-0
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发表时间：
1987-03
影响因子：
0.6
通讯作者：
T. Gedrich;K. Gruenberg
中科院分区：
文献类型：
--
作者：
T. Gedrich;K. Gruenberg
如果Λ是环，a是Λ-module，则当且仅当Ext j Λ （a， P）对所有射影Λ-modules P和所有足够大的j = 0时，证明存在Ext * Λ （a,）的端点补全。当且仅当所有射影Λ-modules silp Λ的内射长度的上限时，存在每个a的端点补全。类似的结果也适用于Ext * Λ （， A），并且涉及spli Λ，这是内射Λ-modules的投影长度的上值。当Λ是一个整群环zg时，分裂zg是有限的，则表明zg是有限的。在群扩展下，也证明了spli的有限性。若G是可数可溶群，则当且仅当G的赫希数有限时，分划zg是有限的。
If Λ is a ring and A is a Λ-module, then a terminal completion of Ext∗ Λ (A,) is shown to exist if, and only if, Ext j Λ (A, P)= 0 for all projective Λ-modules P and all sufficiently large j. Such a terminal completion exists for every A if, and only if, the supremum of the injective lengths of all projective Λ-modules, silp Λ, is finite. Analogous results hold for Ext∗ Λ (, A) and involve spli Λ, the supremum of the projective lengths of the injective Λ-modules. When Λ is an integral group ring Z G, spli Z G is finite implies silp Z G is finite. Also the finiteness of spli is preserved under group extensions. If G is a countable soluble group, the spli Z G is finite if, and only if, the Hirsch number of G is finite.