Γ-群的（余）同调与Γ-同调代数
(Co)homology of Γ-groups and Γ-homological algebra
摘要：
本文是对同调代数中的群作用在Γ-单群的同调，特别是Γ-等变同调和上同调的情形下的进一步研究.这种方法可称为Γ-同调代数。将群的非交换扩张的抽象核、它与非交换扩张存在性的障碍以及与群的第二上同调的关系推广到了Γ-群的非交换Γ-扩张的情形。计算了有限循环Γ-群的有理Γ-等变（余）同调群.证明了一个Γ-群G的n重Γ-等变扩张群与一个Go Γ-模A的同构，其中G的第n+1阶Γ-等变群上同调的系数在A中.定义了当基环含有有理数时，包含循环同调的Γ- Hochschild复形的同调为Γ-等变霍赫希尔德同调，推广了Γ-等变（上）同调的性质.当群Γ在Hochschild复形上的作用由它在基环上的作用引起时，建立了与Kahler微分、Morita等价和导出函子有关的Γ-等变霍赫希尔德同调的重要性质.引入并利用相关导出函子研究了交叉Γ-模的群（上）同调和Γ-等变群（上）同调，最后给出了它们在代数K理论、交换环的伽罗瓦论和群的上同调维数中的应用.
英文摘要：
This is a further investigation of our approach to group actions in homological algebra in the settings of homology of Γ-simplicial groups, particularly of Γ-equivariant homology and cohomology of Γ-groups. This approach could be called Γ-homological algebra. The abstract kernel of non-abelian extensions of groups, its relation with the obstruction to the existence of non-abelian extensions and with the second group cohomology are extended to the case of non-abelian Γ-extensions of Γ-groups. We compute the rational Γ-equivariant (co)homology groups of finite cyclic Γ-groups. The isomorphism of the group of n-fold Γ-equivariant extensions of a Γ-group G by a G o Γ-module A with the (n+1)th Γ-equivariant group cohomology of G with coefficients in A is proven.We define the Γ-equivariant Hochschild homology as the homology of the Γ- Hochschild complex involving the cyclic homology when the basic ring contains rational numbers and generalizing the Γequivariant(co)homology of Γ-groups when the action of the group Γ on the Hochschild complex is induced by its action on the basic ring. Important properties of the Γ-equivariant Hochschild homology related to Kahler differentials, Morita equivalence and derived functors are established. Group (co)homology and Γ-equivariant group (co)homology of crossed Γ-modules are introduced and investigated by using relevant derived functors Finally, applications to algebraic K-theory, Galois theory of commutative rings and cohomological dimension of groups are given.
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