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Title:  Generalized Clifford Theory (Group Ring, Frobenius Extensions) 
Author(s):  Uno, Katsuhiro 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
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 Smodule W can be regarded as an Rmodule W(,R). Also, for any right Rmodule V, we can form the "induced" SModule V('S) = V (CRTIMES)(,R) S. Fixing a right Rmodule V, we let A = End(,R)(V) and B = End(,S)(V('S)). Then there is a natural ring homomorphism l' : A (>) B. The Rmodule V is said to be weakly Sinvariant if V('S) is isomorphic to a direct summand of a direct sum of a finite number of copies of V as Rmodules. (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 Smodules W such that W(,R) weakly divides V and whose morphisms are all Shomomorphisms 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 Sinvariant, 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 Tinvariant and that V('S) is weakly Tinvariant. Assuming that l(,1) : S (>) T is Frobenius, we prove that an object W of Mod(S(VBAR)V) is weakly Tinvariant if and only if the object Y of Mod(B(VBAR)A) corresponding to W is weakly Cinvariant. We then establish the compounding of Clifford correspondences constructed in Chapter 3. This is done in Chapter 5. 
Issue Date:  1985 
Type:  Text 
Description:  70 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 1985. 
URI:  http://hdl.handle.net/2142/71232 
Other Identifier(s):  (UMI)AAI8511681 
Date Available in IDEALS:  20141216 
Date Deposited:  1985 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois