VARIETIES FOR MODULES OVER ABLOCK OF A FINITE GROUP I

Abstract. Chapter Iis asurvey on the three notions introduoed by M. Linckelmann [10],[11] : the transfer maps in Hochschild cohomology of symmetric algebras, the cohomology rings of blocks, and the varieties in block theory. The transfer maps can be called “multiplicative transfer ” in asuitable situation. The definition of the block cohomology rings is based on Puig’s pointed group theory [14]. The block variety is defined as the subvariety of the maximal ideal spectrum of the block cohomology ring, and it is the block version of well-known Carlson’s module variety [3]. 1. $A\text {} $ algebra over acommtative ring $R, $ $U\text {} $ bounded complex of $A$-modules&, module left module. $\mathscr{P}(U) $ $U $ projective resolution (Le., right bounded complex of projective A-modules, $\mathscr{P}(U) $ $U$ quasi-isomorphism). $\mathrm{I} $ , $\infty \mathrm{h}\mathrm{o}\mathrm{m}\mathrm{o}1\mathrm{o}y $ element chain map homotopy class ([1, $\mathrm{I} $ , Section 27], [7, Chapte