The Spectrums Of The Character Rings And Basic Morita Equivalences

This dissertation is concerned with the connected components of the spectrums of three classical rings of a finite group and the basic Morita equivalence between the block algebras of two finite groups.Firstly, we focus on the spectrums of three kinds of rings of characters of a finite group, and show that the numbers of their connected components are equal to the numbers of special kinds of conjugacy classes of the group.Let G be a finite group, and K be a subfield of the complex field C. Let R(G) be the ring consisting of all Z-combinations of the irreducible characters of G over C, RK(G) be the ring consisting of all Z-combinations of the irreducible characters of G over K, and RK(G) be the ring consisting of all the functions in R(G) which take values in K.Letπbe a set of rational primes. Denote by Zπthe localized ring of the integer ring Z determined by the setπ. Let h be a multiple of the exponent eG of G andωh be a primitive h-th root of unity. Then all the rational primes in the setπare not invertible and other rational primes are invertible in the ring generated byωh over the localized ring Zπ. Let S be a subring of C such that SnQ(ωh)(?)K=Zπ[ωh](?)K.We construct a special function of G over the field K under a special condi-tion; since the connected components of the spectrum correspond one-to-one to the primitive idempotents,we prove that the number of connected components of the spectrum of the commutative ring Zπ[ωh](?)Z RK(G) is equal to the number ofπ-regular K-classes of G. For the ring S(?)Z RK(G), we can prove that its primitive idempotents are exactly the primitive idempotents of the ring Zπ[ωh](?)Z RK(G),so the number of connected components of the spectrum of the ring Zπ[ωh](?)Z RK(G) is equal to the number ofπ-regular K-classes of G.Further,we prove that the ring Zπ[ωh](?)Z RK(G)is the subring of the ring Zπ[ωh](?)Z R(G)consisting of the fixed elements under the action of some Galois group,and by means of the action of the Galois group we can get all the idempotents of the ring Zπ[ωh](?)ZRK(G) and show that the number of connected components of the spectrum of the ring Zπ[ωh](?)ZRK)(G) is equal to the number ofπ-regular K- classes of G. The ring S(?)ZRK(G) and Zπ[ωh](?)ZRK(G) have the same idempotents, so we can prove that the number of connected components of the spectrum of the ring S(?)Z RK(G) is also equal to the number ofπ-regular K-classes of G.The other problem which is concerned in this paper is that we give a precise description of the local property of basic Morita equivalences in module-theoretic methods.We prove that the modules inducing the basic Morita equivalences between the corresponding block algebras of these local subgroups can be associated by the induction and restriction.Let G and G'be two finite groups, and p be a rational prime. k is an alge-braically closed field of characteristic p. We denote by b and b' the block idempotents of G and G'over k, respectively. We assume that the block algebras kGb and kG'b' are basic Morita equivalent. In [29], it is proved that the corresponding block al-gebras of some special subgroups of G and G' are also basic Morita equivalent. In this paper, we investigate the relationships between these basic Morita equivalences; using the relationship between the graded module and its 1-component and some properties of the source algebras,we find a module such that its induced module and its restricted module induce the basic Morita equivalences respectively, hence give a precise description of these basic Morita equivalences.