University Microfilms

INFORMATION TO USERS

This dissertation was produced from a microfilm copy of the original document. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependent upon the quality of the original submitted.

The following explanation of techniques is provided to help you understand markings or patterns which may appear on this reproduction.

1. The sign or "target" for pages apparently lacking from the document photographed is "Missing Page(s)". If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting thru an image and duplicating adjacent pages to insure you complete continuity.

2. When an image on the film is obliterated with a large round black mark, it is an indication that the photographer suspected that the copy may have moved during exposure and thus cause a blurred image. You will find a good image o f the page in the adjacent frame.

3. When a map, drawing or chart, etc., was part of the material being photographed the photographer followed a definite method in "sectioning" the material. It is customary to begin photoing at the upper left hand corner of a large sheet and to continue photoing from left to right in equal sections with a small overlap. If necessary, sectioning is continued again — beginning below the first row and continuing on until complete.

4. The majority of users indicate that the textual content is of greatest value, however, a somewhat higher quality reproduction could be made from "photographs" if essential to the understanding of the dissertation. Silver prints of "photographs" may be ordered at additional charge by writing the Order Department, giving the catalog number, title, author and specific pages you wish reproduced.

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

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 (.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-algebra 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(

is again a 2-cocycle equivalent to f which is obviously nor malized. Now, if A is a crossed product A(f,E,G) with nat ural E-basis we introduce the elements:

*0 = ”f(l,!)**

t;.t; = g(o,T).t;T

a.t .t, ~ a.t cf 1 o

. / ,. , t tn .a.t„ » a.t . 1 a a

This shows that any crossed product can be viewed as one with a

normalized cocycle and with a natural basis for which the ele

ment associated to the unit element of G is the identity of

the algebra.

The identity t .t , = f(o,o" ).f(l,l), which is a conse- a quence of the preceding discussion, shows that if A is a

crossed product A(f,E,G) each element of a natural E-basis

is a unit of A. 6

Proposition 1.2. If A is a crossed product A(f.E.G) with natural E-basis /t^ Q £ c> then G can be viewed as a sub group of Autj.(A), in the sense that for each 0 6 G the inner automorphism defined by t is an extension of cr from E to A .

Proof. If x is any element of E we have:

V*-*®-1 - V x- r-ii 1 “ f(0,0 > .fd,i)° 0

a ------1 ------.o(x).f(o,o‘1 ).f(l,l) a f(o,o“A ).f(l,l)

= o(x).

o • cL*

Proposition 1.3» Let A be a crossed product A(f.E.G) and let

s s E = ® Ei , 1 » JL e.. , i=l i=l be the decomposition of the semisimple algebra E into its sim« pie components 1 and the corresponding decomposition of the iden tity 1. Then, for each i=l,....s, E, is a Galois extension 1 ----- ^ of Fe^ whose Galois group is « I a € G / o(E^)=Ej^ and dim^A = n2 , where n = dim^E = [Gill.

Proof. Since E is commutative and semisimple, each E^ is a field, a field extension of F e ^ and G operates on the set of the E^'s. Let 0 be a G-orbit of this set;

0 = | e . E . \• 11 r

Then, if e = e. + ... + e. , it is clear that for all o £ G, X1 r q o(e) = e. Thus, e is an idempotent in E = F, whence e=l and 0 = •[ E^,... ,Egj . In other words, in the case of a crossed product, the group G operates transitively on the set of sim ple components of E, -^E^,...,Eq^ which is hence a homogene ous G-space isomorphic to any of the G/H^, where ELsStabgE^.

If s.6G is chosen in such a way that a. (E. ) = E., we have i i l l that H. = c.H,c7^ and also that i i l l

cf.Hi U ... ^ 0 1 1 s i

is a coset decomposition for G with respect to (for which we may take 0^=1 ). G H1 Let us consider now any xe^£ E^ = E^ = fixed subfield of

E^ under G (or ) and let us call y = Z» =

^ q^(xe^). Given any a £ G, there is an index j(i) and an

element such that oo^ s j(i) *be function j being bijective, we have:

« =2 «j ( i ) °1, J(i ) (xel ) * = y.

This implies that y £ F and therefore that x€F. Thus, E^ »

Fe^ and E^/Fe^ is a Galois extension with group H^. By con jugation, the analogous statement is true for every i.

Then, the F-dimension of each E^ is [H^:l^ and hence

dinipE = s.[H^:l] = [Gil].

This completes the proof.

Proposition 1.4. If A is a crossed product A(f«E,G)« E is a maximal commutative subalgebra of A .

Proof. We keep the notations introduced above. Let B be a commutative subalgebra of A containing E, and b = ^

€ B. If x is any element of E, from xb = bx follows that

b X° = xb ( V a e G). 0 0

Now, if x6Ee^, x^O, the relation

a. x x 1 a xb _ ( V x €H_, Vi) O.X 0.X 1 implies that b is 0 for all i^l and for all t£H. • If 0_^T X /£ 1, we can choose an element x in Ee^ such that for a 9 given tCH^, t / 1 , we have t(x)/x. Then, the relation bTxx a xb^ implies that b^ a 0. Eventually, we obtain that b a ^1*1 £ E#

(}• e.d.

Proposition 1.5. If A is a crossed product, it is a central simple F-algebra.

Proof. This is a particular case of Prop.6.1. However, we can give a simple direct proof as follows:

For each i=l,...,s, we can consider the F-algebra A^ = e.Ae. = E.t which, being a classical crossed product, 1 1 t£H. 1 T is a central simple F-algebra with identity e^. If B is a non-zero two-sided ideal of A, = BAA^ is a two-sided ideal of A. and it is non-zero too because given x = a t 1 o 6 G 6 B, x £ 0, we can get, by proper multiplication with an ele ment of A, another non-zero element y = a V f B for oe G which a'c / 0 for some 0 6H i ., and then the element e.ye. x x would be a non-zero member of BAA.. It follows that each B. JL 1 is equal to A^ and hence that eaoh e^£B. This forces 16B and B must be equal to A.

q. e. d. 10

2. Crossed product orders.

Definition 2.1. Let R be an integral domain with quotient field F« E a semisimple commutative F-algebra. G a finite subgroup of AutpE whose fixed subalgebra is equal to F. A an R-order in E which is G-invariant;

o ( A 0> = A 0 ( V o € G) and let f be a 2-cocycle of the G-module E with values in

A^. Then we call the crossed product A(f > A fl,G) the ring A defined by;

A = z A 0t0£ A

A can be defined directly in the same way as crossed pro duct algebras. It is a free A o-module of the form:

with the multiplication defined by:

xt0ytT = x.y°.f(atT).t0X (Vx,y e Ao*Vo»t£0). 11

It is clear that such a A is a ring and a finitely generated

R-module which generates a crossed product A(f,E,G) over F, in the sense that:

F(2RA =A(f,E,G).

If it happens that a crossed product A(f, A C»G) does contain an identity element 1 we say that A ( f , A fl«G) is a crossed product order (over R).

It should be observed that A ( f , A c ,G) is a crossed product order if and only if f(l,l) is a unit of A c * This forces also each f(o,l), f(l,d) to be a unit and we see at once that any crossed product order A ( f , A Q ,G) can also be written in the form A ( g , A q ,G), where g is a normalized 2-cocycle in the same class as f.

In what follows we will always assume that f is normalized.

The family t„ will be called a natural A -basis for the o o

crossed product order A(^» A q *G)« Because of our convention, we always have that t^=l.

Keeping the notations already introduced, let us call, for

each o € G, the projection of A onto E defined by: 12

n X T x t ) = x. . a r x x a T 6 u

Each is E-linear. in the sense that o

n^Cx.a) = x.7i0 (a) (1/x^Ej^aeA).

Let now M be a A-right ideal in A (i.e. a right A-module which generates A over F) and let us call M = it (M) (

o — the M ' s are equal to some A -ideal in E. If M is finitely generated over A (or over R), the same is true for M ... If, furthermore, M is a two-sided A -ideal, then M is invar- - x—

------riant under G. And also, in any case, M ^ /. M x t_.x

Proof. m £ M means that there exists an element Z a t_€M —— x o o such that m^ = m. If we multiply this element by t on the right we see that m^.f^T** ) 6 M^. This proves the first assertion. If M is two-sided, the relation t0M s M implies cf(M^)

e. d. 13

2.3. Remark on the study of orders whose factor system is not composed by units.

It seems that crossed product orders, even in the classical

case, have never appearod in the literature with the same degree

of generality we have given them here. All authors introduce

the restriction that the factor system f : GxG ■■■■- > A Q has

its values in the group of units of A q:

f: GxG ----> U( A q).

Along our study we have found that most of the theorems we can

establish for crossed product orders (cf., for example, Prop.

2.2) are false or cannot be proved without this natural restric

tion. (We consider the restriction as natural because otherwise

f could not be viewed as a representative of a cohomological

class unless one leaves the ” arithmetical environment ” to go

to the ’’algebraic background” E of A Q.)

This is why we are going to follow suit by making the

Convention. In what follows, any 2-cocycle f appearing in the

construction of a crossed product order A(f, A 0»G) is sup

posed to have its values in the unit group U( A Q)»

However, we would like to say a few words as for the kind

of ideas one might think of when trying to study the more gen

eral crossed product orders. We are not going to enter into

much detail but only to show the kind of difficulties one en- lA counters.

The most natural way of approach should be to take profit of the known fact that, by enlarging the ground field F, one can assume that all values f(o,t) are roots of unity and there

fore fall into U(/\o ) (cf. (7), § 7» Satz 1). Of course, this would involve the introduction of some ramification which would make difficult to recover the pattern that follow hereditary or maximal orders containing the given crossed product.

Anyhow, let us suggest a possible way to the end of intro

ducing ground ring extensions in the theory of crossed product

orders. For the sake of simplicity we will work in the classi

cal case and for complete local Dedekind rings as groun rings.

As it will follow from our later results this does not seem to

be a significant lost of generality.

Let A = A(f , A 0 ,G), where A 0 is the maximal order in E

and E is a Galois extension of the quotient field F of the

complete local Dedekind domain R, with Galois group G. Let

A = F A = A(f,E,G).

Let L be a Galois extension of F containing E with

maximal R-order S and Galois group H. Let H' be the Ga

lois group of L/E:

1 — > H' — »■ H G — > 1 (exact).

Then f induces, through the inflation map, the 2-cocycle: 15

f : HxII — ► S as follows:

f HxH ------*• S

GxG

f(h,hO = f(h,h') (h = x(h) 6G, h£H).

Then we would call the order:

A S = A(f,S,H) = ® St

in AS =: A(f,L,H) the crossed product order obtained from A by extension from

A 0 to S. (If the extension L is properly chosen, it would be possible to replace f by an equivalent cocycle in such a way that all its values are in U(S).)

What we would like to have now would be a two-way passage to g go from orders in A containing A to orders in A contain- q ing A • To go in one direction, we can consider the mapping

(p defined thus: 16 g Given an order P in A, «<>( D 3 T is the Bet which is generated as an R-raodule by all expressions of the form:

g where the 60 #s are any elements in A but each family {xff}

(o£G) is chosen in E in such a way that e P • s s Then

To find the way in the reverse direction, we notice first

that there is a natural mapping of the E~module Et in- x £ K T to A, p, defined by: p(x.t^) = x.t^^j (xtE, t £H). Then

we consider the mapping y which associates to each order p*

3> A S in AS the order lf'( P') - P( T # A ( ® Et )) C. A. x £ H T The two correspondences (p, 4*, have the property that for

every order T containing A* y( P» so that

injective over the set of maximal orders containing A •

2.4. Examples.

Crossed product algebras and hence crossed product orders oc

cur very often in nature (cf. (7))» We will content ourselves

with the following illustrating examples:

1.- If E/F is a Galois extension with group G, then EndpE 17

is naturally isomorphic to A(1,E,G) (cf. (2), Appendix)*

Sitting inside is the crossed product order A(1,I1,G), where II is the maximal R-order in E.

2.- The ordinary integral quaternions, ^[i»j]» form a cross

ed product order A ( f , Z[i] ,G) where G = < > is the Gal

ois group of Q(i)/Q and f is the factor system:

f(“ ,l) = f(l,l) = f(l,“ ) * 1

f C D = -1.

A natural 2”£i3-basis is given by t^=l, t = k = ij:

^ [i v ^3 = + ^Ci]*k .

3.- Lot G be a finite group with an abelian normal subgroup

H and let 6 be sui absolutely irreducible faithfull repre

sentation of G over the complex numbers. Let A q be the

■^-module generated by 6(G). Then J\Q is an order in a sim

ple (Q-algebra which can be viewed as an epimorphic image of a

crossed product order A ( f » Ajj|G/H), where A g is the /£-mod

ule generated by 6(H) and f is the factor system correspond

ing to the group extension:

1 f H — > G ► G/H — ► 1 .

In some cases A q is actually a crossed product order. For

example, let G = = the symmetric group of degree 3, H a Aj 18 s the alternating subgroup and 6 be given by:

0 1 w 0 6(12) = 6(123) = 1 0 0 w

rd A where w is a primitive 3 -root of unity. Then A q is iso morphic to the crossed product order .A(l, Z Cw3,G/H). A natural /ZCwJ -basis is given by where G/H has been identified with the representative subgroup

< ( 1 2 ) > . This is a classical crossed product order. If we call instead A q the -module generated by 6(G), then it is again a crossed product order but in our broader sense.

4.- In the typical case, the -^-completion of a crossed product

order in the classical case is a crossed product order in

our broader sense (cf. Remark ^.7)*

5.- Other examples of crossed product orders, in the classical

sense, can be found in papers of S. Williamson and M* Ha-

rada (cf. (12), (1 9), (20), (21) and (22)). 19

3. Hereditary crossed product orders*

Throughout this section ve keep the notations introduced in

2. From time to time we add special assumptions as needed*

The following theorem is well known:

Theorem 3*1* Let R be a discrete rank one valuation ring* E

a Galois extension of F (with group G) and A ^ the maxi

mal R-order in E. Then the following are equivalent:

(1) E/F is tamely ramified;

(2) ACf, A 0,g) = A is hereditary for every f;

(3) -A(l, A q,G) is hereditary.

It was proven by S. Williamson in (19) but the equivalence

between the first and the third was first given in (3)* Later

on, it was proven in (12) that proposition (2) can be weak

ened by eliminating the condition ” for every f" , provided

that the residue class field of A Q is perfect.

Now we consider again crossed product orders in our broader

sense: 20

Theorem 3«2. Lot K be a complete discrete rank one valuation rinrc and the maximal order in the commutative alprebra E.

If E^ = Ee^ is tamely ramified extension of Fe.^« then A =

A ( f . A .0 ) ia hereditary. In fact, it is a principal heredi tary order.

Proof, It well known that can be written in the form: —1 ■ ■ — is A_ o s

where A 0^ is t*10 maximal order in E^ = Ee^ and cf^( AQ^) is the maximal order in E^ = c^CE.^), If ^ is a prime ele ment generating the radical of A Q^» we call n ^ o a i ^7Ci^ n = ji^ , so that n Aq s rad( AQ).

Let us call, for brevity, fe^ the restriction of the map ping:

(o,t) > f(o,T).ei to H., Then, as it is easily seen, each fe. is a 2-cocycle: X 1

fe. : H.xH. --- >U(d. (A . )). i i x i ol

Then it makes sense to speak of the crossed product:

A x = A C f V Ao1 ,h1>

which verify the hypothesis of Th. 3 .1 . A ^ is naturally a 21 subring of A :

a and is an order in

A1 = © Ejt0 = A ( f e ltE1 ,H1 ) A. a 6 U 1

Let be the radical of A^s = rad(A-j_)* It was prov

en in (19) that is a free A^-module and, more precisely,

that = ^ i ni* Hence, A ^ a principal heredita

ry order. ) Our theorem will be proved if we can show that N a n A is

the radical of A.

It is clear that, from the invariance of rad(A^) under o G, o(tc) A q = n A Q ( Vtf €G). This implies that N is a two-

sided ideal of A. Since N = rad AQ» A , it follows from Naka-

yama's lemma that Ncrad(A).

To prove the reverse inclusion, we show that A / N is a se

misimple F-algebra, where F is the residue class field of R.

If we introduce now the notation f for the natural passage

of f to the quotient A Q/rc A , we see that

A/m s A(f * A0/n Ac,g)

and, in the same way as before, we recognize that A^/rc^A^ is

similarly a crossed product over which can be viewed as a 22 subalgebra of A/N. As we have pointed out, radC A^)=0.

Since rad(A/N) is the maximal nilpotent ideal of this al gebra, and the same is true .for the algebra A^/n^A^, we see that

rad(A/N) H (A^/it^A^) c rad( = 0 . (+)

Our goal is to prove that rad(A/N) = 0, for which we are go ing to use the fact (+).

With the obvious meaning for the notations, we can write any element in rad(A/N) in the form:

X—Xy••«|6 o € G

Hence, for each i,j = l,...,s, the element t .x*ei

A l /7liA i 60 that it is equal to 0. In fact:

v °:1 i * - i W o . = ( = 0-- 3 c € G ° 3

-1 -1 e,.a. »f(o. ,o)• e. .t ■> t

Now, all terms of this sum vanish, except those for which 23 oT^cfcr. belongs to H-. In turn, these elements o are of the i j J. form (x£H^), so that our expression reduces to:

q-1

■6 ./

Now, the fact that these elements are zero for each i and each j implies obviously that all g are equal to zero, i.e.: a = 0.

q.e.d.

Proposition 3»3« Keeping the preceding notations, if R is any local Dedekind domain, rad(A ) - ra d ( A ) A A ^.

Proof. If is the maximal ideal of R, it is clear by Na- kayama's lemma that rad( A^) and ^Ac. rad( A)» so

that we may as well work modulo . Then, using the character

ization of the Jacobson radical in the case of artinian rings,

we conclude as before that

rad(A) A A-j^ C rad( A^).

To prove the reverse inclusion, we proceed as follows. We

call the radical of A-^ N the radical of A and

we introduce the set: s 2k

J i=l i

Hi ra-1 and wo claim that for every natural number m, J = J.N^ . I t would be enough to show this for the case m-2 .

If n, n'£J, they can be written thus:

n = Z * t . i=i °i 1

. y t Z - n = a f *t • '• -j c t .0 # • / . C . S * s' TT 1 I T X X X =1 1 T £ *

so that:

TO.. to.» t X X n.n' = £ --- • t a. 8.f(x,o./)a^ . e.. t t . . . .. , o.x.Tl x 1 to.* T x,x =1,...,s x x t ,t '£ Hx

In this sum all terms vanish except those for which t g ^.£H^, v/hich implies i' = 1. Hence, it is clear that nn' can be put in the form:

nn' =

It is a straightforward computation to-see that JA is a two-sided ideal of A and, since is nilpotent, it follows that JA is nilpotent too. In other words: 25 JACK.

Now, since c jA, we get the desired inclusion:

M c N r\

q.e.d.

Theorem 3.^. With the notations already adopted, if R is a

complete discrete rank one valuation ring we have: if A is

hereditary then is hereditary. Hence, the following prop

ositions are equivalent:

(1) E^/Fe^ is tamely ramified;

(2) A ( f , A Q,G) is hereditary for all f ;

O ) A(l, A q ,G) is hereditary;

(where A ^ is themaximal R-order in E). If, furthermore.

theresidue class field of A ^ is perfect. (2) may be re

placed by:

(2)' A(f, A0,G) is hereditary.

Proof. From Th. 3*2 it follows that (l) implies (2) and,

obviously, (2) implies (3). The first part of the theorem,

together with Th. 3*1» means that (3) implies (l). So, we

only have to prove that A hereditary implies A^ heredita

ry. 26

As it is well known, an order is hereditary if and only if its radical is left projective* This means that A is hereditary if and only if N~^N = A, where N“^ =: |x 6 A/ xN c Aj,

Under this assumption we are going to prove the equivalent

statement for

By assumption, there are elements n^ 6. N, n^ € Nsuch

that:

2-* nknk = 1 * k=l s Since z e = 1, we still have: q=l q

e,n,'e t t”^e n. e, = e, <+) zj 1 k q a a qHk 1 1 k=l,...,r q q q=l,■••,s

= identity element of A^.

Now, a direct computation similar to that in the proof of Th.

3*2 shows that each t”^e n. e, 6 H A A, =* N_ (cf. Prop. 3»3)» O Q K X X X q and that each e,n,'e t . £ A,, On the other hand, it is clear 1 k q 1 that

elnkeqtO N1 ^ A1A A 3 ^ 1 q

b o that each e,n,'e t belonge to N " 1 . Therefore, (+) means

precisely that a

(j* e • d» 27

Remark 3.5. The proof indicates that the following more general statement is also true: if K is any local ^edekind domain and

A » A ( f , A o ,G), where is any order in E containing all the idempotents e^, then if A is hereditary, A ^ is here ditary too.

Remark 3.6. (On the achur index).

Let us go back for a minute to the classical case: R is a complete discrete rank one valuation ring and E is a Galois extension of F, with Galois group G. In this case, our crossed product algebra A is also a central simple F-algebra Jxf of the form D where D is a division ring which is a cen- 2 tral finite F-algebra. Under these conditions, dim^D = s is 2 2 2 the square of a natural number and dim^A = f s = n , where n = £g :iJ. The integer s is called the Schur index of the simple algebra A.

We are going to show that the Schur index can be easely com puted in the tamely ramified case, if we know f, E, G.

Let -H be the unique maximal order in D« It is well known

that if P is its unique maximal ideal (= rad(ft)) every he

reditary order in A is conjugate to one of the form: 28

— — — —> V . ______\

^ rt

n si ..... sn / p si ...... f i

p ...... SI /

Jvjf (The matrices in A = D are partitioned into blocks of sizes r^,..., r^ and the order in question is obtained as the set of all matrices whose blocks on or above the diagonal have arbitrary entries in X 2 , while its blocks below the diagonal have only arbitrary entries in P.) (cf. (11) and (17))•

The order is principal if and only if all blocks have the same size:

r^ = Tj a ... = r^ = r and, also, if and only if the radical of the order can be gen erated on either side by only one and the same element. This is precisely the case for a hereditary crossed product order in our tamely ramified case.

So, t being the number of blocks, we have the first rela 29 tion:

f = r.t.

It has been proved in (19) that t, which is also the num ber of maximal two-sided ideals of the order A * is equal to the order of the conductor group (which depends on G and f). So, we may assume that t is known.

It is also well known (cf. (7) and (16)) that the Schur index s is both equal to the ramification index and the resi due class degree of D. If F denotes as before the residue class field of R we can compute dim ^ (A/N) in two differ ent ways:

dim(A/ N ) = t.dim j (D rxr) = t.r2.s

dim(A/N) = n.dim =; (E) a n»f„ JP TC

where f^ is the residue class degree of Therefore:

2 - teT •6 — “ !• 6#f TC TC

r.r.S a f.S.f 1C

r = f TC .

This means that J a f^t and we obtain the following expression

for the Schur index: 3 0

Now let us come back to the more general case in which £ is only a commutative semisimple algebra (but we still assume that

R is a complete discrete rank one valuation ring). Here we have the algebra A and the subalgebra and to the crossed product order /\ in A we associate as before the crossed product order A-^ in A^. They are simultaneously principal hereditary orders in the tamely ramified case and, as it will follow from the proof of Prop. 4.6, they have the same number of maximal ideals: t.

If we call h = (E^Fe^ = [H:l3 and k = n/h and if f^ denotes the residue class degree of A 0]_ WQ find that:

dim = E = f .k £ TC and carrying on the same computations as for the classical case we obtain the following expression for the Schur index:

s = f^.t.k

Finally, if s^ is the Schur index of A^, we have:

n h.k f »t.k f »t.k s it n ~ ~ = ; a 1 " ~ ~~~ a 1 . s1 h h V* V* This allows us to state: 31

Pro-position 3.7. If R is a complete discrete rank one valua tion rinp; and we keep the notations of Th. 3.2. then the Schur index of the simple algebra A is equal to the Schur index of the simple algebra A^.

^t. Reduction to the classical case.

Throughout this section we keep the preceding notations, and we make only the further assumption that R is a local Dede kind domain:

R is a local Dedekind domain with maximal ideal and

quotient field F.

£ is a commutative semisimple F-algebra of finite dimen

sion.

e.,...,e are the primitive central idempotents of E. X 6

6 is a (finite of order n) group of automorphisms of E

whose fixed subalgebra is equal to F.

H. is the G-stabilizer of E. a Ee.. 1 I X

cr^ is an element of G with the property that o^(e^)»e^. 32

A 0 is an R-order in E which is G-invariant.

XI is the unique maximal order in E.

A 0i is the order in E^.

f is a 2-cocycle of the G-module E with values in U(AQ)

which is furthermore normalized.

A is the crossed product: A(f»E,G)»

is the crossed product:

A is the order A(f, A q»G) in A.

t is a natural basis for any of these crossed products.

First of all, we wish to describe here the natural way of climbing (in the tamely ramified case) from A to the hered itary order A(f,^,G).

We remark first that S. Williamson, in (19), proved the general version of Th. 3*1 without the assumption of complete ness. Therefore (cf. Remark 3*5) A(f,J2,G) is hereditary if and only if Atfe^ft j,^) is hereditary and if and only if

Ej/Fe^ is tamely ramified (even if R is not complete). The only thing that is missed in the non-complete case is the ^prin cipality” of the hereditary order.

Given A , each A . = A e. = a . (A .,) is an order in ' o oi o i 1 ol E. and, furthermore, "T.A . « is aa order containing X * *T-/ Ol ^ ox o 32 which contains all the idempotents e^. It is a crossed product order too and is contained in A(f,£l,G).

Remark. In the sequel the phrase: ” A 0 1® equal to the sum of its simple components” will mean that A 0 contains all the idempotents e^.

Now we recall some terminology and introduce some additional notations:

If M is any R-lattice in any F-algebra A, the left and right orders of M are defined thus:

[m/m] = (xeA / xHCm J ;

[m \ m] = (x a / m x c m } .

If A is an R-order in A and OL is a A -ideal in A*

the idealizer of 01 in A (denoted by IA ( ^ 0 briefly,

1(00) is defined as the maximal order of A whioh contains

01 as an ideal, namely:

iA (tfU = l a / m ] a £ a \ o i ] .

It is known (cf. (23)) that, in the local case, an order

A in A with radical N is hereditary if and only if Ia (N)s

A * or» equivalently, if and only if [N/n] a A or if and

only if [n\n] =A. This leads to the following construction (the *’ idealizer construction” ) of an invariant hereditary or der containing a given order A • Considering the chain defined inductively by:

r „ = A >

Mi »raS(ri)i ^1*1 = W the first order *or which hereditary and contains A .

Proposition b.l. With the notations introduced at the beginning of ...this , ...section, — , j .let -1 i.Ol 0 be an —ideal — " of A o which... is. ■— G- ■ invariant and let 01 be the crossed product A(f.flf0«G).

Then OH is an ideal of A and we have:

[ o/ m 0 ],G) ;

[ > \ r c ] = A ( f , | l ^ 0 \cJi0 ],G) i

IA (0l) = ACf, Ie ( ^ 0 ),G).

Proof. Let x 6 A qi cf 6 G, a = 2 ^ 0 e ^ • Then we liave:

xt^a = xt a t = xa^f(0|x)t 6 0 1 ; O — 0 T X ^ T OX X

axt„ = J'* a t xt = ^ » O L—-» XT O ' X XO X

so that 01 is an ideal in A . 35

Now let ye [0lQ/ 0lQ]» o6Gy a as before. Then:

ytaa » y a ' a 2 1 ya^ T = Z N tT

(where a'=tffa, a' £ 0lQ, a " 6 CKQ).

This shows that A (f, £0lQ/ $ 0]tQ) c [ ° l /<%}• Conversely, let

X = 2 x0*0 (X0 ^ E) be any element in £cfc/0l], and aQ any element in ClOl. Then from xa , follows “ o o^6

Z *0*0**® 6 which implies that each X

The proof for the right order is entirely similar. We only have to keep in mind that, being G-invariant, both

\ & Q/ 6lQ] and [0lo\ 0lo] are G-invariant.

cj* o • d*

Now, if r = A . I\ . ..., is the chain of the oo o lo ko ” idealizer construction” applied to A Q , the last term is a hereditary order in E, so that = £1 • Hence, if =

A(f, r. ,G) we get an increasing chain of orders starting with 3*0 A and ending up at /\(f iQ ^G). In the tamely ramified case,

this is therefore an ” idealizer construction” to obtain a he reditary order containing the given crosse^l product order A • 36

Proposition 4.2. If A Q is equal to the sum of its simple com ponents, then A _ '/>' \ t^ Ajt and, a fortiori: A ° A A ^ A. o ,t £G

Proof. Let us call, for brevity, A the set on the right hand side. We first observe that, since f(o,t)€ A,

f(cJ,x)Ax C A x and

A 1.f(o,T)c A^.

Given a £ A , a direct computation shows that for each i,j:

to_lQia6JtO;, « V

Hence:

ct* . — • t . .a.e..t . >* C /\ •, XQ . -1 X J <*.{ rt-l 1 1 --1 d. 3 -1 flj f(oi%a^ ) i 1 ^ which implies that

a b ^ t„ .a. ..t , 6 A* . 4 4 cr. x3 -1 i , j i

This shows that A £ A and the reverse inclusion is obvious.

£[• ©* dc

Proposition 4.3. Given any order A ^ in containing A ^,

we define: 37

A' = A A { A . if A o is equal to the sum of its simple components. A* ia an order in A, containing A , such that

A' a = A^ .

Proof. It is clear that A* is a finitely generated B-module containing A • It is also clear that A £ C A* HA^. To prove that A' is closed under multiplication, it suf fices to show:

A A'A A'A ^ AA'A .

Let a, a', a"fcA» P» P*£ A-£* As follows from the proof of Prop. **-.2, there are then elements p. . £ such that: 13 At X

a' = 2* t Pi;jt • ifj °i 10 Oj

We observe now that if P & A ', t p is of the form: 1 oi

'i x~ 3xd x vi

so that for any P^ 6 A£ ,

Plt0 p = (Kronecker delta). 1

Using these remarks, we obtain:

e *» aPa'p'a" o Z P.*** = ZaPPijt .jP#a* i,i i d i 5 3 8

On the other hand, there is an index, k, and an element T of

such that o”^ a 0]C‘C» 80 that:

%-X ■ *(<£„> V * t ' and, if we call Pxj*"f 6 A £ and tT0# a P'cA£» 1c we get:

e = 2 , “P P ^ o pTa" = aPPixP^a"^ AA{ A . j k

The proof will be complete if we show that A * A .

V/e observe that if a 6 A » once we write a in the fora:

a * i7 ti i V -i i - i V «3 i • (s« € v it is obvious that ©xael “ ^11^ ^ l ^ ^ l * Then» given any ele ment of A ^ A ' = A A ^ A a A.: 7 a B a' (a .a'tAi P„& A O 1 X x^qqq <1 <1 <1 1 we have:

Z “qV « ° " l - Z V q V el ■ Z A £ . ^ Q >

€[• e • d«

Theorem k.k. Let A ^ be equal to the sum of its simple compo nents. Let (0 be the set of all orders in A which contain

A and ^ the set of all orders in which contain A ^ « 39

Then there is a 1-1, order preserving, correspondence between ll) and given by the following functions;

h ^ — » 0

— > IPX

r(A{) « A A£A

# ( A ' > = A'r\Av

Proof. Apply Props. *f.2 and

Corollary 4-. 5. A* A is a maximal order if and only if A £ is a maximal order.

Remark. One should observe that the function $1 has another al

gebraic expression, namely:

= J2C A') = A'n Ax = ex A'e^

In fact, since e^ is the identity element of A^, A £ **

el ^ l el *- ei A #e^» the other hand, e ^ A * ^ ^ A*" ^ d direct

computations show that A^ = e^Ae^.

This fact allows for a simpler proof that if Aj[ then

A A£ A is closed under multiplication. For we have: 40

AA£AAA£A = AA£AA£A = A A ^ A e ^ i »

= AA{A1AiA = A A'A .

As for the fact that ( A A £ A ) ^ 3 A£» we have:

(A A{ A ) X = ei.A A^Ae^^ = e1A e 1Aie1A e 1 = A 1A £ A 1 »A£.

Proposition 4,6. If A is equal to the snm of its simple com ponents, both functions (R. , /L of Th. 4.4 carry hereditary

orders into hereditary orders. So, A * is hereditary if and on

ly if is hereditary.

Proof. Let N' a rad A * » a rad A£ * We claim that N£ a

N ' A A ^ = e^N'e^. In fact, if I is any ideal ofA*» then 1^

a ir\Ax a I n A£ a is an ideal of A£:

A ^ 1! = e1 A'e1Ie1 C oêî XiAx3 eiIei A e1 C eÎe^.

Conversely, if 1^ is an ideal in A^ » * = A* I ^ A ' .

Then I is an ideal in A'•

Mow we show that these two correspondences between ideals are

inverses of each other. In one direction:

e1 ( A ' I 1 A')e1 a ei A'e1Ixe1 A'«x = A ^ A ^ = V to

And, in the other: /V(e^Ie^)A' is clearly contained in I and the reverse inclusion follows by using the argument in the proof of 4.2.

The fact that these correspondences are order preserving and

1-1 implies then that N', intersection of the maximal ideals of A' corresponds to N', the intersection of the maximal ideals of A£. Hence it is true indeed that N£ = e-jN'e^.

If A' is hereditary, N' is left projective, i.e. there are elements y^, x^ such that:

*i6 N ', y±N' A ' and 2 yixi * 1#

Then, for all q's:

8114 V i V « " i c A i tV i V « eAi) q q q

and, finally:

i,qZ Vi*,Vl H ' ? i el^ixiel * V

This means that is a left projective A i “m°dule and

therefore that A ^ is hereditary.

The same way of proof can be used for the converse. If A £

is hereditary, there are yf, xT such that:

x- 6 H' , y?N' C A £ and ^ y![x? a e^.

Then x ^ t ^ ^ N ' and t0 yfN' d/\'. In fact, any element a'eK' b z can be written (cf. proof of Prop. b.Z) in the form:

2 n^ t i (with nf,6N') iTj ffi 1* o~ 13 1 3 so that:

* .2. TkVo * Jrk»i)t,-lc ? Alt A '' V ! Oj % J 0 5 J 0j

Finally, we have:

2 *« - 2 t ^ t ; 1 = 2 T . = i. q,l q q q q q q

This says that H' is a left projective //-module and hence that /\' is hereditary.

q. e. d.

Remark A. 7. (On the reduction to the complete case.)

The preceding theorems allow us to show up to what extent

the study of crossed product orders in the local complete case

i6 enough to study global crossed product orders.

Let us consider a classical crossed product: A = A(f,E,G)

(i.e. E is a Galois extension of F with group G), where we

assume that R is the ring of integers in some algebraic num

ber field F . Let us assume furthermore that A Q * the

maximal R-order in E. We know that the main question of whether A is hereditary or maximal can be answered if we know what happens with all the completions of A for the valuations corresponding to the prime ideals of R. Let us fix a prime ideal ^ of R. Then, if R^ denotes the ^.-completion of R, the ^-completion of A is given by:

A ^ = = 0 where the crossed product on the right is defined by the cocy cle 1 ® f , and the action of G on R^ ®j.p. is given (cf. section 5) by the action on the ”12-component” of each ten sor.

It is clear that if F denotes the quotient ring of Ry v - o and = F^, A (the .^-completion of A), then is an

R^-order in A^ . We find easily (cf. section 5) that A^ =

A ( 1 ® f» E^»G)* where now E^ « E is the direct sum:

E = E^ © ... © E^ a of the Pi-completions of E, E^, where P^ are the different prime ideals of £1 lying above ^ (i.e. the completions of E corresponding to all the extensions to E of the valuations associated to ^ in F).

Correspondingly, A (1 ~~f»S^,G), where the completion of H., is the unique maximal order in the semisimple, commutative algebra E^ and, moreover, 0 where hk is the P^-completion of -Q- (or the unique maximal order of~~

Therefore, Ay is a crossed product order in our broader sense. Here the stabilizer under G of the ” first” compo- rionf- o-P P -i/s tho HonnmTinBi t.inn irrmin 7, of* P . T +: f fil 1 nufi then from Th. b.k and Prop. A-.6 that the study of / L n regard to hereditary or maximal orders which contain it) can be effectively reduced to the study of A^t where

~~is a classical crossed product order in the local complete case.~~

~~5.- Reduction to the totally ramified case.~~

~~Throughout this section we keep the notations introduced at the beginning of section h but we assume furthermore that R is complete and that E is a Galois extension of F with group~~

~~Let us begin with a study of the algebra E = : E E and the algebra A =: E ® ^ A and their orders in relation with the orders of E and A.~~

~~It is well known that, as an E-algebra, E 5 E ® ... © E~~

(as many copies as n = (E:F) = £G:l]). We will index the dif- ±5 ferent copies of E with the elements of 6 and denote each element of that direct sura as a family: (x„)w - „ (x £ E =E). 0 0 c u o O The following is one way of writing the above isomorphism ex plicitly:

~~Let a be a primitive element of £ over F, with irredu cible polynomial P(X) = "fT (X - o(a)). Then we have: or e G~~

~~- „ _ „ ~ _ f[x] - eCx] - C& e[x! „ E = E®VE = E «i' Tp T B T - THxTT = ®a X x - c U T ) V~~

~~Hence, if we represent an element of £ in the form 53 xi ® a , i its image on the right is given by the family:~~

~~( 2 xi .cr(ai))~~

~~Since the map:~~

~~(x,y) --- » (x.o(y))o is balanced, we see that our isomorphism can be set up (inde pendently of a) as:~~

? * i ® yi — * (Zv^i^o (xi»yi6E>*

~~Now, if S^, S2 are any two orders in E, it is obvious~~

~~that after a natural identification S^ &LS2 is an order in E.~~

~~We want to investigate the conditions under which this order~~

~~contains all the irreducible idempotents of E. As an answer,~~

~~we have: *f6~~

~~Proposition 5.1. Let S ' be an order in E. The idempotents e of E are Riven by the formulas:~~

~~< < . 0 •— 1 i—1 i+1 zx~~

~~°i °i °i °i •••• y. y « : . x 1 y 4 x ••~~

~~Cf Cf Cf n a r, n~~

~~fli i = l rr~~

~~—Where — — onx ,.*.,o n are " 1 all■ ■ —the elements of G, —y 0^—. is — —an — — R- basis of S and D is the disoriminant of that basis.~~

~~Proof. Let us put ^ xT ® y^. The conditions for being th X € G the o -idempotent amount to:~~

~~(Kronecker delta) T x crcp T £ G~~

~~The determinant of this system is det(y^) = Vd £ 0 and the u- nique solution is obtained by Cramer's rule as written above.~~

~~q.e.d. 47~~

~~Corollary 5.2. If E/F is unramified, each order of E con taining is equal to-the sum of its simple components.~~

~~Proof. (This is also a corollary of lemma 5*10) E/F unram- i f i e d means th a t the d is c rim in a n t D o f D- is a unit of Q - ,~~

~~We can generalize the construction above in the following way. Let H be a normal subgroup of G and let L be the~~

~~fixed subfield of E corresponding to H, which is a Galois~~

~~extension of F with Galois group G/H. We can consider the~~

~~F-algebras:~~

~~EK = L ®p E ; AH = L A ;~~

~~so that if 1 is the null subgroup of G we have E = ;~~

~~A = A . If a is a primitive element of E over F with~~

~~irreducible polynomial P(X), we can introduce, for each co~~

~~set a = cH <= G/H, the polynomial P_(X) =: "TT_ (X - cp(ot)) a~~

once that each of the P_ 's is irreducible over L, that a L[X}/(P_00) is naturally isomorphic to E under the map X a > o(a) and that P =~[|X_ is the decomposition of P o over L. Moreover, it is clear that we have:

~~_H T ~ „ -T ^ FIX] - L fX] - LCxi - p \ (p(x)) - (p(x)) = =- ^ hE (+) 48~~

If we pick a representative a for each coset a and wo use the notation (x )— for the elements of the direct sum on the a a right, we obtain one explicit expression of the isomorphism above by setting:

~~(+) X xi ® y i *--- * (Z xi »oCyi))_ (x^L. y ^ E ) i l a~~

~~Proceeding in the same way as we did in the proof of Prop.~~

5.1, we may now compute the idempotents of E . In fact, the algebra L L is naturally an L-subalgebra of L <2^, E and its idempotents are also the irreducible idempotents of the lat ter. A direct application of Cor. 5*2 yields:

Proposition 5.5« If S is an order in E, invariant under G, such that S a L = -£"2- „ is the maximal order of L and if L n — ■''' ' 1 ' ' 1 t is an unramified extension of F, then <2^ S contains all the idempotents of E and is therefore equal to the sum of its simple components. In this case, all these components are isomorphic to S.

~~Proof. Y/e know that <~ ^ contains all the idempotents of L which are the idempotents of E^. According~~

~~to (+), the a ^-component is equal to I7jj.o(S) = S.~~

~~q.e.d. 4-9~~

~~Keeping these notations we turn now our attention to the al gebra AH a L A = L &>„ ( @ EtO. We claim that it is a F F ff£G 0 crossed product of the form:~~

~~AH 5 A(f,I?,ttf).~~

~~In fact, the map~~

~~x ® Z yo ^ •'— * Z~~

~~(x ® y)° = x ® y° (x~~

~~Now we can see that our mapping is also multiplicative:~~

~~(x y»ta ). (x' (g) y't0 ,) b (x.x' ® ytay ,t(J») = xx'(5p yy't0ttf,~~

~~J > (xx' ® yy'°).tcftc(, = (~~

~~= ((x ® y).(x' (g) y' ) °).ta .t0<. =~~

~~a (( x & y)ta).((x" ®~~

~~We have proved:~~

~~JJ Proposition 5»^-« The L-algebra A is a crossed product a H H A ( f , E ,G). The stabilizer of the first component of E , 50 which is equal to the L-algebra E, is H.~~

~~Proposition 5»5« Let A = . A a ,G) as before. Let =~~

~~A A . I = A.Q AL,_ Then:~~

~~A ==~~

~~H f\ is an order in A . As before, we call e/ the family of all orders in A which contain A, and we introduce now the fam ily of all orders which have the following two properties:~~

~~C D rH^ A H i~~

~~H (ii) each element in I admits an expres~~

~~sion of the form T,w. <& x. (w. € ut X X X A Oil x. €A) such that each 1 <& x. is also X * X an element of T •~~

~~Then there is a 1-1, order preserving, correspondence between~~

~~$ and $ H given by the functions:~~

~~f : iS> --- > ;p =: rH =: aoH<% r s~~

~~3crH) ■== r Ha a . 51 H H Proof. Given Pei#* it is clear that P contains /\ . On the other hand, by the definition* the condition (ii) is sat-~~

~~H a H isfied, so that p €- $ . It is even more trivial to see that 3( P H ) belongs to $. Now we show that (P and 3 are inverses.~~

~~Let r € \9 and let us show that ( A qjj <5^ V ) A A » P .~~

Obviously, T C ( A 0jj ® D H A . Conversely, let y be any element of (AoH® D a A; let w^ be an R-basis of ^ o H and a. 6 R be such that T* a.w, = 1. Our assumptions on y X ' 4 X X mean that there are elements x.^6 P, x £ A such that:

~~Y = Z wi® xi = 1® x = 2 1 aiwi $ x = 2 / w. ® a. x .~~

~~It follows that a.x = x. € T • which implies: x =/ , w.x. 6 T • X X X X~~

~~Now let P H £ (i)H. We want to show that P** = A).~~

~~The order on the right is contained in the order on the left. m H Conversely, if y € P , it admits a representation of the form~~

~~/ 1 w^ ® x^ such that each 1 (g> x^ £ P . This means 1 x^ H m H is an element of P A A, whence y €A0g ( P AA). .~~

q.e*d*

~~Remark 5.6. In the previous situation, if we had that A Qjj®A0~~

~~is equal to the sum of its simple components, we could also~~

apply Th. and we would have that the lattice of orders A H containing A is isomorphicaly represented by the lattice 52 H H a of orders of (A containing (A In that case 0 would be isomorphic to the image of (0 in that lattice of orders containing

~~H H Now let us look more closely to A and (A )^. According H to the definition following Prop. ^.2, (A >1 is precisely~~

~~A(f,E,H) in which E is considered as an L-algebra. In other words:~~

~~(AH ), = A (f,E,H) =: A~ = 0 E.t A. A t U t~~

IT (A is the subalgebra of A obtained by restricting the sum £9 Et^ to the o's belonging to H and by considering it as an L-algebra. This algebra, which we now denote by AH , is a central simple L-algebra.

~~As we said before, if A qH is equal to the sum of its H simple components and if we denote with 0 ^ the image of 0 under the mapping JZ introducen in Th. 4,*f, we have the com mutative diagram: * 53~~

~~But, in principle, there should be a more” natural” way for going from A to A^. Namely, given T € 0 , its ” natural”~~

~~correspondent should be T O A^. We carry out this idea in the~~

~~following:~~

~~Proposition 5.7« Keeping the preceding notations, we call /\^~~

~~the— order ... A A A „ n ——= ^rr —A o t X~ and n the family of all or- T & H ders of A^ , which verify the following two conditions:~~

~~(i) P H 3 A h ;~~

~~(ii) r w is invariant under G, in the sense that si~~

~~tcr Htol c r a <*»««•>•~~

~~Then we introduce the following two functions:~~

j: $ --- *

~~J(T)=: P jj e P ^ ;~~

~~p: 0 H --- ► \9~~

~~p ( r H) = P h ^»~~

~~and we have:~~

~~J ° p a id~~

~~p(J(D)<=r (VreO).~~

~~More precisely, p(J(P)) is the minimum of the elements of (9~~

~~whose image under J is J(P). In particular, p is monic and J is epic.~~

~~It follows also that for all £ ^IT ^11^ ~~~

~~Proof* If r € (9, J ( D is an order in containing A H »~~

~~Also, since t X - t ”1 = Au , each «J(P) is invariant under G. ’ O H O n Secondly, given e ’ P HA is a finitely generated R-module containing A* Next we observe:~~

~~1 H ^ = ^^U^QtaC ^l£~~

~~so that~~

~~p.Zr,vo~~

~~Then:~~

~~pcrfl>-pcrH) = = Z r li»t0 r Ht~~

~~c Z r Ht0tT c p( r H ). o, x~~

~~This completes the proof that p and J are well defined. The~~

~~remainder of the proof is straightforward.~~

~~c^« e. d •~~

~~Proposition 5.8. If A is such that A ^ <3^ Afl is equal to~~

~~the sum of its simple components, the following diagrams are~~

~~commutative: 55~~

~~H H 0 P O'~~

~~AH 0 H~~

~~0 p~~

~~Proof. Given p£ $ , (PkY) = A qH and ^ A q h ^ T ^ is~~

~~the set of all elements of the form which satisfy a.,0 the conditions:~~

~~v±e A oH’ 0 --- 0 H~~

~~This says that ^ ( ^ ( D ) = J ( D , which is the commutativity~~

~~of the first diagram.~~

~~Now, given any order V* in it contains~~

~~so that:~~

~~J(Z( D ) - < A oH ^rA. r ' . A o H <^r A ) A a 56~~

~~= < M o h A o h )(2r( A r'A))nA = (AoH®r(A P'A ) ) a A.~~

~~The order on the right hand side contains p( T*) = A P* A •~~

~~Conversely, if x £ ( A qR ®r ( A r*A))AA, x can be written thus:~~

x = 2 w-?0yT = 1 ^ z j ■ where (w.). is an R-basis for A 0ti and the y-'s are 0 0 ® J suitable elements of A P*A , while z belongs to A. If 2 is the expression for 1 in terms of the w.'s, we j ^ 0 have then that y. = a.z, whence 25 = A w *y-i* that is z £ J J j 3 0 A P ' A . This proves the commutativity of the second diagram.

~~But, if the first two diagrams are commutative, since P~~

~~^ are raonic, the third diagram is commutative too.~~

~~^•6 id<~~

~~Corollary 5.9. If. keening the notations above. L is an un ramified extension of F and if A qjj = then the lat tice 0 of all orders in A containing A is icomorphic~~

~~(i.e. is in 1-1 . order preserving, correspondence with) to the lattice of the & ^ orders of A^ which contain A ^ and are~~

~~G-invariant. This correspondence preserves hereditary orders in both directions. 57~~

~~Proof. The first statement is the commutativity of the third diagram of Prop. 5.8, which can be applied because of Prop.~~

~~5-3.~~

~~As for the proof that J and p preserve hereditary or ders, we recall that, from Prop. 5*8, J = and from~~

Prop. ^-.6 follows that as well as its inverse X-, car ries hereditary orders into hereditary orders. So, it will be enough to show that both P and its inverse carry hereditary orders into hereditary orders. This fact is a particular case of the following:

Lemma 5.10. Let R be a complete local Dedekind domain with quotient field F and A an R-order in a simple algebra A. which is central over F. Let ep be the maximal ideal of R and L an unramified extension of F with maximal order

~~Then the order Gcfo A is hereditary if and only if A~~

~~is. (Note: A more general statement has been proved by Aus-~~

~~lander and Goldman (cf. (2), Prop. 8 .6 )).~~

~~Proof. Let N be the radical of A and NL that of A1*. We~~

~~claim that NL = <3^ N. We know that N is ^-nilpotent,~~

~~in the sense that for some integer m we have NracT ^ A . Then~~

~~a ls 0 ^ -nilpotent, which implies that ^ N~~

~~C N^. The reverse inclusion follows from the fact that A** /~~

~~semisimPle» For we have: 5 8~~

~~--AL._ a A. Q. (56. N ~ N ’ U L ®R K (as F=R/| -algebra) ? L~~

~~Now, L/F being unramified, a separable field extension of R/^ . ^ince A/ N is semisimple, the tensor prod uct is semisimple too.~~

~~A is hereditary if and only if N~^N = A , where N-^ =~~

~~[A/ nJ = {x€A/xNcA\. If this is true, we have:~~

(ClL % N_1).NL = = A l which means that A"^ is hereditary. Conversely, if A** hereditary, any element in (N^)-^ can be written in the form: z’'7, w. I&y., where (w.). is an R-basis of Q ~ r with w, = 1. i x x x L l -1 It follows readily that each y^ belongs to N from which we get that (N1*)”1 = 12. ^ • In turn, it follows from here that N-1N = A or that A is hereditary.

~~q.e.d.~~

Corollary 5.11. Let . For each unramified subnormal extension L the lattice 0 of orders in A is isomorphic to the lattice of -Q^-orders in . This isomorphism pre serves hereditary orders in both directions. 5 9

Corollary 5.12. If I is the inertia group of E/F and Fj. =L the inertia field and if /\ contains = .Q.(inertia ------o — — ------1 1 L ------ring of Q .). then the lattice (!) of orders in A is iso morphic to the lattice of .Sl^-orders in Aj which contain and are G-invariant. This isomorphism preserves hereditary or ders in both directions.

Corollary 5.13. If E/F is an inertial extension, ACf.Cl.G) is a maximal order*

~~6. Non-commutative crossed products.~~

~~We are going to introduce here a generalization of the idea of a crossed product which seems to play an important role in the study of the structure of crossed product orders.~~

~~Let R be a local, complete, Dedekind domain with maximal ideal jej> and with quotient field F. Let B be a finite dimen sional F-algebra and G a finite group. We consider a fixed group extension:~~

~~1 --- » InAut(B) --- > G > G --- ► 1 60 where G is a subgroup of Aut(B). Let~~

~~a: GxG --- > InAut(B) :~~

V * T 3 aO fT*VOt *

~~We assume that the cocycle a is normalized.~~

~~Let us suppose that there exists a normalized 2-cocycle f:~~

~~f: GxG U(B)~~

~~such that each inner automorphism a^ ^ is precisely conjuga~~

~~tion by f (cf *0»~~

~~Let g Q be any set in 1-1 correspondence with the ele~~

~~ments of G and let A be the free B-module with basis {t0}»~~

~~Then the rules:~~

~~t .b » (p.(b).t cf o c~~

~~t0 .tT = f(cf,x).tOT ( V o,T £ G; \/b~~

~~define a multiplication in A which makes A into a finite di~~

~~mensional F-algebra. This algebra will be called a non-commuta~~

~~tive crossed product and will be denoted by: A(fia,BtG).~~

~~Note. When we say that a is a 11 2-cocycleM we mean that 61~~

~~InAut(B) is considered as a G-set by means of the f Va\a£ G and the a verify the associative relations: C,X~~

. "I acr,x*aox,e ~ ax,e'ao,xe a ’ ^a&x ,e^cr acr,xe*

~~Similarly, saying that f is a " 2-cocycle” amounts to say~~

~~f(o,x).f(ox,e) =~~

~~The assumption on these two cocycles means that for every x € B~~

~~a is the inner automorphism given by: o, x~~

~~aQ T(x) = f(a,x).x.f(o,x)”1.~~

~~It is very easy to see that this condition is an if and only if~~

~~condition in order that the multiplication defined above in A~~

~~be associative. For we have:~~

~~(bltob2tx)(b3tc) = ^bi«~~

~~= b^.tp^Cbg) .f(o,x). (b^) .f(crx,e). t^Xg .~~

~~On the other hand:~~

~~Cb1t(J)(b2tT .b3te) = (b1ta )(b2 .(px(b3 ).f(x,s).txc)~~

~~= b-^Cbg.cp^b^.fCx^J.fCa.xeJ.t^g =~~

~~= V > a (b2 ).atf>T.cpaT(b3 ).9tf~~

~~Therefore, we will have associativity if and only if:~~

~~f(o,T).OT(b^).<|>^(f(T,e)).f(o1'cs) or, equivalently:~~

~~f(o1x).~~

~~.f(ar,e) = a.Q T *‘P~~

~~ao,x#<<>ax(b3 ) = fCo,T).(j>OT(b5 ).f(o,T)_1.~~

~~If A is an jR-order in B which is G-invariant (that is:~~

~~A is invariant under each of the tp^'s) and if, furthermore, the image of f is in U(A)» then A = A ( f , a,A,G) is an order in A which we will call a non-commutative crossed prod uct order.~~

~~Proposition 6.1. Keeping the notations above, if B is a simple algebra then A is a simple algebra.~~

Proof. We begin with the following remark: since B is simple, every automorphism which leaves its center element-wise fixed is an inner automorphism. This implies that s z, f°r all z in the center of B, implies 0 = 1, It is clear that A has an identity element, namely t^ = 1.

~~Let X be a two-sided non-zero ideal of A and let 0 ^ x£X, x = Z V o be an element of X of minimal *’ length” (i.e. a such that the set of all o such that b0 / 0 is minimal).~~

~~Let us number or^ the set of all a for which btf /0.~~

~~We can assume without loss of generality that = 1.~~

Let M be the set of all elements of X which can be writ- k ten in the form: 2, a_ • Then M is a two-sided B-mod- 6 1 *1 °i ule. We now call tL the set of all elements of B which appear as the i coefficient of an element of M. is a non zero two-sided ideal of B. Hence, each is equal to B.

~~This means that there is an element m^ in M whose first coefficient is 1. It follows from the minimality of -fc^,...~~

~~..., ck } that if a^ is the first coefficient of an element~~

~~then m = = mlal (wkere, as ra runs through M, a^ runs through the whole of B). If we now equate the i coef ficients of an<* ra;Lai we S®*:~~

~~at >~~

~~It follows that X = A, and the proposition is proved.~~

Convention. In the sequel, we will find highly convenient to consider an even more general idea of crossed product: namely, by removing the requirement that the automorphisms associated to G are pairwise inequivalent modulo inner automorphisms of

~~B . In all what follows, unless advise on the contrary, we will consider only this more general class of crossed products.~~

We present now the broadest situation we can think of as for the validity of Maschke's theorem. This will be a step further from the generalisations already given by S. Williamson (19) and M. Harada, (12).

~~« vucw* cm/# \ J. f c*. * .h* t WLW characteristic p of F (F is here any field) does not divide n = [G:i1 , then. if B is semisimple, A is semisimple too.~~

~~Proof. It is enough to prove that if M is a left ideal of A then M is a direct summand of A, as a left A-module. In turn this is true if there exists a projection 7i: A --- > M which is A-linear.~~

Since A is a left B-module, M is a left B-raodule which is finitely generated. Since B is semisimple, M is a direct summand of A as B-modules. In other words, there exists a projection it': A —— which is B-linear.

~~By hypothesis, is a unit in the center of A.~~

~~This allows us to define another map:~~

~~Since it' is the identity on M, it is the identity on M:~~

On the other hand, since Im(it') = M, Im(it) = M. Hence, we on ly have to show that it is A-linear, for which it suffices to show that it preserves multiplication by elements of the form b.t^ (b € B, t gG).

~~This is a tedious but straightforward computation, as shown:~~

~~From the relations defining the multiplication on A we get:~~

~~t“X = t ,.f (o.cT1)-1 o -1 a so that: and, since it' is B-linear and~~

— , (a-1 . (b))).t •n'.t'^.t •: 11 o€G a a"1 a.o-1 0 0 *

~~= b . ~ — / , t u't^t X = b.it(t x). n oC-G ° 0 T T~~

~~Now we observe again that from the relations defining the multiplication on A it also follows that:~~

~~t ^ t = t f (o,a“1)"1t = O T 0-1 X~~

~~so that:~~

~~i c ( b t x) = b . ~ - 2 t u'cp 1( f ( 0 , 0 “ 1))" 1f ( o “ 1, T ) . t t .x = T n oeG a 0 " 1 o ” t~~

~~= b.— 2jtm .(f(ata"1r 1)f(a“1tx),n'.t ..x a n oeG 0 0 - 1 0-1t~~

~~= b . - i - 2 W 1( f ( < J » o " 1)" 1)«Pa (f(o“ :L.'t))t ll't -l.X . O e G 0~~

~~Here we use again the fact that~~