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University Microfilms 300 North Zeeb Road Ann Arbor, Michigan 48106 A Xerox Education Company 73-2071 MERKLEN, Hector Alfredo, 1936- ON CROSSED PRODUCT ORDERS. The Ohio State University, Ph.D., 1972 Mathematics ; University Microfilms, A XEROX Company, Ann Arbor, Michigan THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED. ON CROSSED PRODUCT ORDERS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Hector Alfredo Merklen, Licenoiado + + + + + + The Ohio State University 1972 .pproved by Advisor Department of Mathematics PLEASE NOTE: Some pages may have indistinct print. Filmed as received. Microfilms, A Xerox Education Company ACKNOWLEDGMENTS I want to thank my advisor, Prof. Hans Zassenhaus, who never felt tired of listening to me or of making valuable suggestions and remarks, and to my wife Ana, whose spiritual help made all of this possible. Hector A. Merklen VITA July 19* 1936 . Born - Montevideo, Uruguay. 1933-1958 ..... High school teacher, Durazno, Uruguay. 1938-1966 ..... High school teacher (by competitive con­ test), Montevideo, Uruguay. r 1962-1966 ..... Profesor Adjunto, Facultad de Ingenieria y Agrimensura, Universidad de la Republics Oriental del Uruguay, Montevideo, Uruguay. 1962 . ........ Licenciado en Ciencias Matematicas, Facul­ tad de Ciencias Ex&ctas y Naturales, Uni­ versidad de Buenos Aires, Buenos Aires, Argentina. 1963-1965 ........ Lecturer in several Summer-courses for the improvement of high school teachers of Mathematics, in several countries: Argen­ tina, Peru, Uruguay, Venezuela. 1965-1966 ..... Field Assistant, Interamerican Program for the Improvement of Science Teaching, Orga­ nization of the American States, Montevi­ deo, Uruguay. 1967-1968 ..... Encargado de Investigacion, Facultad de Ciencias, Universidad de Chile, Santiago, Chile. 1969 - ..... Profesor Titular, Instituto de Matematicas, iii Universidad Catolica de Valparaiso, Valpa­ raiso, Chile. 1970-1972 ..... Teaching Associate, Department of Mathema­ tics, The Ohio State University, Columbus, Ohio. PUBLICATIONS " Introduccion al estudio de las funciones de una variable real" Mimeo. Notes, Durazno, Uruguay. (1958) " Algebra lineal" , Mimeo. Notes, Buenos Aires, Argentina (1963) " Geometria" , Mimeo. Notes, Ministerio de Educacion, Caracas, Venezuela (1963) " Geometria" , IPEM, Lima, Peru (1963 and 1965 ). Introduccion a la Matematica Formal" , Mimeo. Notes, IDAL, Montevideo, Uruguay (1963 ). " Geometria" , Mimeo. Notes, Buenos Aires, Argentina (196*0 Introduccion de la Geometria en el primer ano de la Ensenanza Secundaria" , Mimeo. Notes, IPEM, Lima, Peru (196*0 " Algunas ideas para la ensenanza moderna de la Matematica" , Mimeo. Notes, Ministerio de Educacion, Buenos Aires, Ar­ gentina (1964). " Lecciones para el Seminario Pedagogico", Mimeo. Notes, IPEM, Lima, Peru (1965 ) " La ensenanza moderna de la Geometria al nivel universitario basico" , Mimeo. Notes, PIMEC, Montevideo, Uruguay (1966 ). " Lecciones de Analisis" , Mimeo. Notes, PIMEC, Montevideo, Uru­ guay (1966). iv ** Curso Cero" , Universidad Catolica de Valparaiso, Valparaiso( Chile (1970 and 1972). FIELDS OF STUDY Major Field: Mathematics Studies in Measure Theory. Professor Mischa Cotlar. Studies in Functional Analysis. Professor Mischa Cotlar and Professor Jean Dieudonne. Studies in Algebra. Professor Hans Zassenhaus. v TABLE OF CONTENTS Page ACKNOWLEDGEMENTS . .................................... ii VITA ..................................................... iii Chapter I. CROSSED PRODUCT ORDERS ...................... 1 1. Generalized crossed products .......... • . 1 2. Crossed product orders ..... 10 3. Hereditary crossed product orders ..... 19 k. Reduction to the classical case ...... 31 5. Reduction to the totally ramified case . 4^ 6. Non-commutative crossed products ...... 39 7. The wildly ramified case ........... 75 II. ON THE HULL OF REPRESENTATION OR D E R S ......... 84 8. Introduction................. 84 9. Block Orders ..... 86 10. Representation o r d e r s ................... 113 BIBLIOGRAPHY ............................................ 13^ vi I. CROSSED PRODUCT ORDERS 1. Generalized crossed products. The terminology and general notations used in this paper are the same as the ones introduced in (7)« (2*f) and (25)* We begin with a generalization of the classical idea of a crossed product, for which we refer to (7)s Definition 1.1. Let F be a field, E a semisimple, commuta­ tive E-algebra of finite dimension and G a finite subgroup of Autp(E) with the property that its fixed subalgebra coin­ cides with F: E6 = jx6E/k/cf€.G: c(x)=x} = F; in addition, let f be a 2-cocycle of the G-module E, i.e.: f: GxG --- * U(E) f (<J,T>.f (ot,6) = f (T,6)°.f(atT6) (Vc,t,6€G). (we use freely the notation x° instead of o(x) for x € E and o€G, so that if t:A — »E is any function with values 1 in E, then ttf represents the composite function o*t. It should be emphasized that we think of functions as written on the left, so that for example: means o(x(x)) or (.x*)0 OX and, if t is a function as before, t means the composite: o.ft.) Then we call the crossed product A(f,E,G) the F- algebra defined as follows: A(f,E,G) is the F-vector space of all functions t: G --- > E , together with the multiplication defined thus: given t,t': G ■ — ■ —> E, tt' is the function defined by — , - 1 tt'(o) * f(tf*-1 ,T)t(OT~1 )t'OX C*> ( V o e G ) . T 6 G The proof that this definition makes A(f,E,G) an F-al- gebra would be clear after the following considerations. Among the elements of A(f,E,G) we distinguish the family ■ft r n> where t^(x) = 6 (Kronecker delta) (Vo,t £ G). *• o; a €. g a ox It is clear that {t0}0 g g is an E-basis for the E-module A(f,E,G) and, in fact, for each element t of the crossed product we have: t a Z 2 t(o).t . O €G ° According to the definition, the multiplication table for this basis is: t t (vO «= f(pe”\ e ) . t (®) » G T o T e =» Z f ^ " 1 ,.).# ,«6_ „ a e ^,pc T »* = f(pT_1,T).6 - => f(0,T).6 a o.px"1 aXt)l = f(c,x).taT. Conversely, if we start with the free E-module over the set ^t^ / o £ g } indexed by G and define: xt^.yt^ = x.y°.f (o,x) .t0T ( V x , y £ E ; \/o ,t £G) the associative property of f assures us, as in the classi­ cal case, that we obtain a well defined structure of F-alge- bras over that module. Since this structure is clearly isomor­ phic to what we called A(f,E,G)t this is also an F-algebra. More generally, we will say that an F-algebra A is a crossed product A(f,E,G) if it is isomorphic to the one constructed above. Up to isomorphism, as in the classical case, a crossed prod­ uct A(f»E,G) does not depend on the 2-cocycle f but on the class [f] of the second cohomology group which is repre­ sented by f. If an F-algebra A is a crossed product A(i»E,G), there is a basis -{t } e G of the E-module A with the properties: A = © E.t a £G 0 xt^.yt^ = x.ya .f(o,T).tci; ( Vx,y 6 E;\f o ,t 6 G) We will say that such a basis is a natural E-basis for the crossed product A. We say that a 2-cocycle g: GxG --- > U(E) is normalized when it happens that g(l,o) = 1 = g(o,l), for all a in G. From the associative property of cocycles it follows in general that f(l,0) » f(l,l) and f(o,l) = fCl,!)** (Vc€G). Hence, a 2-cocycle f is normalized if and only if f(l,l)=l. It is immediate to check that, given the cocycle f, the func­ tion: 5 / \ f(o.T) g(<J,T) = --- is again a 2-cocycle equivalent to f which is obviously nor­ malized.