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|Title:||Generalized Clifford Theory (Group Ring, Frobenius Extensions)|
|Department / Program:||Mathematics|
|Degree Granting Institution:||University of Illinois at Urbana-Champaign|
|Abstract:||Let R and S be two rings with identity elements and let l : R (--->) S be a ring homomorphism preserving their identity elements. Then any S-module W can be regarded as an R-module W(,R). Also, for any right R-module V, we can form the "induced" S-Module V('S) = V (CRTIMES)(,R) S. Fixing a right R-module V, we let A = End(,R)(V) and B = End(,S)(V('S)). Then there is a natural ring homomorphism l' : A (--->) B. The R-module V is said to be weakly S-invariant if V('S) is isomorphic to a direct summand of a direct sum of a finite number of copies of V as R-modules. (We say that (V('S))(,R) weakly divides V in this case.)
The notion of weak invariance is introduced in Chapter 2. In Chapter 3, we prove a correspondence theorem, which generalizes the classical Clifford theorem studied by several authors such as Clifford, Cline, Conlon, Dade and Tucker. Let Mod(S(VBAR)V) denote the category whose objects are all right S-modules W such that W(,R) weakly divides V and whose morphisms are all S-homomorphisms among those modules. Likewise, we define Mod(B(VBAR)A). Then the theorem says that the two additive functors (.) (CRTIMES)(,R) S and Hom(,S)(V('S),(.)) form an equivalence between Mod(S(VBAR)V) and Mod(B(VBAR)A).
The map l : R (--->) S is called Frobenius if S(,R) is finitely generated projective and S (TURNEQ) Hom(,R)(S(,R),R(,R)) as (R,S)-bimodules. Such ring homomorphisms are studied in Chapter 4. They satisfy the "other" Frobenius Reciprocity Law. Also, it is shown that if l : R (--->) S is Frobenius and V is weakly S-invariant, then l' : A (--->) B is Frobenius.
Let l(,1) : S (--->) T be another homomorphism of rings. Let C = End(,T)(V('T)) with A and B being the same as before. Suppose that V is weakly S- and T-invariant and that V('S) is weakly T-invariant. Assuming that l(,1) : S (--->) T is Frobenius, we prove that an object W of Mod(S(VBAR)V) is weakly T-invariant if and only if the object Y of Mod(B(VBAR)A) corresponding to W is weakly C-invariant. We then establish the compounding of Clifford correspondences constructed in Chapter 3. This is done in Chapter 5.
Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1985.
|Date Available in IDEALS:||2014-12-16|