Cohomology Theory of Finite Groups

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Many problems of fundamental importance are left unsolved in the theory of cohomology of block ideals of finite groups. Let G be a finite groups and k an algebraically closed field of prime characteristic dividing the order of G. Let B be a block ideal of kG and let D be a defect group. Let P be a subgroup of D and let H be a subgroup of G containing DC _G (D) and N_G (P). Assume that a block ideal C of kH and the block B are in Brauer correspondence and D is also a defect group of C. It is significantly important to investigate relationships between the block cohomologies H^* (G, B) and H^* (H,C). For example, when H^* (G, B) ⊆ H^* (H,C) does hold, this inclusion map should be understood through transfer maps between Hochshild cohomology rings of the blocks B and C. We showed that the (B, C) -bimodule L which is the Green correspondent of C to G x H has many nice properties which are useful not only for applications for the cohomology theory but also for modular representation theory of finite groups. Under some additional conditions the module L defines the transfer map L:HH^* (B)→ HH^* (C) which induce the inclusion map l:H^* (G,B)→ H^* (H,C) through embeddings of H^* (G, B) into HH^* (B) and of H^* (H,C) into HH^* (c). It is very interesting from a view point of modular representation theory to determine the blocks of kH in which the Green correspondent V of an indecomposable module U lying in the block B lies. We showed that, using the module L above that under some condition the module V lies in the Brauer correspondent C and when H^* (G,B)⊆ H^* (H,C) the block varieties of the modules coincides in the sense that V _(G,B)(U)= l^* V _(H,C)(V).