Reduction Techniques for Strongly Graded Rings and Finite Representation Type

194, 567]593 Ž1997. JA977031

JOURNAL OF ALGEBRA ARTICLE NO.

Reduction Techniques for Strongly Graded Rings and Finite Representation Type I. Bounded Functors and Graded Equivalence Jeremy Haefner* , † Department of Mathematics, Uni¨ ersity of Colorado, Colorado Springs, Colorado 80933 Communicated by Susan Montgomery Received October 17, 1996 DEDICATED TO PROFESSOR IRVING REINER WHOSE RESEARCH, BOOKS, AND STUDENTS HAVE INSPIRED SO MANY OF US

We present reduction techniques for studying the category of lattices over strongly graded orders. In particular, we apply these techniques in order to reduce the problem of classifying those strongly graded orders with finite representation type to the case where the coefficient ring is a maximal order in a division ring. Q 1997 Academic Press

1. INTRODUCTION AND PRELIMINARIES Let R be a Dedekind domain with quotient field K, let A be a finite-dimensional, separable K-algebra and let L denote an R-order in A. By a L-lattice, we mean a finitely generated Žleft. L-module M such that M is R-torsion free. We say that L has finite representation type provided L has only finitely many, non-isomorphic, indecomposable L-lattices. If G is a Žfinite. group and L is a G-graded R-order, we say that L is strongly G-graded provided L s [g g G L x and L x L y s L x y for every x, y g G, and we say that L has a maximal 1-component provided the identity * The author was partially supported by a grant from DGICYT ŽSpain. ŽSAB 95-0215. and by a grant from NSA ŽUSA.. † E-mail: [email protected]. 567 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.

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component L 1 is a maximal R-order in KL 1. In this paper, we are concerned with the following problem: The FRT Problem. Let G be a finite group. Classify those strongly G-graded R-orders L with maximal 1-component that have finite representation type in terms of L 1 , G, and the grading on L. This problem is motivated by the theorem of Heller and Reiner, Dade, Berman, and Gudivok, and others which states that the integral group rings ZG have finite representation type if and only if, for each prime p dividing < G <, the Sylow p-subgroups of G are cyclic of order p or p 2 w2, Theorem 33.6x. Of course, there are two major differences between the ZG result and our above problem. First is the grading and the second is the coefficient ring L 1. The following example illustrates how both the grading and the coefficient ring can affect the representation type and it reflects the disparity between the group ring and the strongly graded situations. EXAMPLE. Let R denote the 2-adic integers and let L denote the ring of 8 = 8 matrices over R. Let G be the cyclic group of order 8 and grade L in a natural way, embedding G as permutation matrices in L. This makes L into a G-skew group ring over the subring formed by the diagonal matrices in L; i.e., the 1-component is [8 R. Since R is a maximal Z-order and L is the full matrix ring over R, it follows that L has FRT. On the other hand, L 1 s [8 R and so the integral group ring L 1G is a direct sum of eight copies of the integral group ring RG. By w2, Theorem 33.6x, it follows that RG and L 1G do not have FRT. Hence, the skew group ring L 1 )G has FRT while the group ring L 1G does not. The pathology of this example arises because G, under the strongly graded situation, permutes the prime components of L 1 , whereas with the group ring ZG, G acts trivially on the Žprime. coefficient ring. Consequently, to solve the FRT problem stated above, our goal of this paper is to reduce the general FRT problem to the case where the 1-component is prime. In fact, we prove: MAIN THEOREM. Let R be a Dedekind domain with global quotient field K, let G be a finite group of order n, and let L be a strongly G-graded R-order with maximal 1-component. Then: 1. For each prime p that di¨ ides the order of the group, there exists a finite family of crossed product R p-orders Ž R p denotes the completion of R at p . of the form DŽ p, j .)GŽ p, j . where DŽ p, j . is a maximal R p-order in a di¨ ision ring and GŽ p, j . is a p-subgroup of G for 1 F j F mŽ p .. 2. L has finite representation type if and only if DŽ p, j .)GŽ p, j . has finite representation for each prime p that di¨ ides n and each 1 F j F mŽ p ..

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See Theorem 6.4 which gives the specific constructions for the DŽ p, j .)GŽ p, j .. In addition, we prove that each of these crossed products DŽ p, j .)GŽ p, j . is primary, allowing us to invoke the conditions of Drozd and Kirichenko to prove THEOREM A. For each crossed product DŽ p, j .)GŽ p, j ., let V Ž p, j . denote the intersection of the maximal orders containing DŽ p, j .)GŽ p, j ., let I Ž p, j . denote V Ž p, j .rDŽ p, j .)GŽ p, j ., and let J Ž p, j . denote the intersection of the maximal DŽ p, j .)GŽ p, j .-submodules of I Ž p, j .. Then the crossed product DŽ p, j .)GŽ p, j . has FRT if and only if each of the following conditions hold: 1. 2. 3.

V Ž p, j . is hereditary; I Ž p, j . has two generators as a DŽ p, j .)GŽ p, j .-module; and J Ž p, j . is cyclic as a DŽ p, j .)GŽ p, j .-module.

See Theorem 6.5. Finally, the proof of the Main Theorem requires knowing when a strongly graded ring is maximal. As a result, we also prove: THEOREM B. Let G be a finite group of order n, let D be a maximal order, and let L s DŽ G . be a G-strongly graded order. Then L is maximal if and only if 1rn g D. See Theorem 4.2 for the proof. A similar result for group rings appears in w2, Proposition 27.1x. To make these reductions, we use two techniques. First we introduce the notion of bounded functors; in essence, this concept means that if F: L-latt ª G-latt is a bounded functor, then if L has finite representation type, so does G. The bounded function is a weaker concept than a separable functor of which much has been written in the literature. See Section 2. Second, we extensively use the notion of graded equivalence. Graded equivalent orders yield bounded Žin fact, separable. functors and so our results about bounded functors apply; see Section 3. Finally, we set some notation that will be used throughout this paper. Let G be a group, G a ring with 1, and L a strongly G-graded ring with 1-component G; i.e., L s [g g G L g , where L g L h s L g h for all g, h g G and L 1 s G. Denote this by L s G Ž G .. For this paper, the notation G )G will denote a crossed product Ži.e., a strongly graded ring such that every component contains a unit of the ring.. When we want to specify the map s : G ª AutŽ G . and the factor set a : G = G ª UŽ G . ŽUŽ G . denotes the units of G .., then we shall use the notation G )as G; see w13x for more details. If R is a Dedekind domain and L is an R-order, then let L-latt denote the subcategory of left L-lattices.

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This paper is organized as follows. Section 2 concerns our concept of bounded and separable functors. In Section 3, we present our graded equivalence technique. Section 4 concerns the reduction to the case where R is a complete, local Dedekind domain and G is a p-group. In Section 5, we reduce to the case where the 1-component is prime. Finally, in Section 6, we reduce to the case where the 1-component is contained in a division ring. In our sequel paper w8x, we consider reduction techniques to reduce to the case where the 1-component is a commutative, maximal R-order contained in a field.

2. BOUNDED AND SEPARABLE FUNCTORS In this section we introduce the notion of a bounded functor and compare it to the notion of a separable functor. Bounded functors allow us to reduce the FRT problem from one strongly graded ring to another. DEFINITION. Let A and B be rings and let C denote a full, additive subcategory of A-mod and let D denote a full additive subcategory of B-mod; let both C and D be closed under direct sums and direct summands. If there exists an additive functor F: C ª D that preserves direct sums and that, for an indecomposable object M of D, there exists an object N of C such that M is isomorphic to a direct summand of F Ž N ., then we say that D is representation bounded by C or C is a representation bound for D, and F is a bounded functor. If both C and D are bounds for each other, then we say that C and D are representation related. If C and D are clear from context, we will say that B is representation bounded by A or A and B are representation related. There are several examples of bounded functors and categories. EXAMPLES. 1. Any equivalence functor and its inverse are bounded functors. 2. If B is a ring direct summand of A, then B-mod is bounded by A. If e is the central idempotent corresponding to B, then the functor F: A-mod ª B-mod given by X ¬ eX is a bounded functor. Moreover, if A s B [ C as rings, then the functor G: Ž B-mod. = Ž C-mod. ª A-mod given by Ž X, Y . ¬ X [ Y is a bounded functor. 3. Graded equivalent graded rings. This follows from Ž1. above and the next section. 4. Smash products and the Hopf algebra duality. These are representation related categories because of the Morita equivalence.

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5. Clifford theory for group representations. This is a special example of Proposition 2.3 below. There are two critical functors, used extensively in representation theory, namely, the induction and restriction functors. We shall discuss these and another extension function later. Presently, we show why bounded functors are important. PROPOSITION. Let A, B, C , and D be as in the definition abo¨ e. Assume that e¨ ery module of D is a direct sum of indecomposable modules from D and that this decomposition is unique up to isomorphism and order. If C is a representation bound for D and C has only finitely many indecomposable modules, then D has only finitely many indecomposable modules. Proof. Let F: C ª D be a bounded functor between C and D. Given an indecomposable object from D, M, there exists an object from C , say N, such that M is isomorphic to a direct summand of F Ž N .. We can assume that N is indecomposable as follows. Suppose N s N1 [ N2 where each Ni g C . Since F preserves direct sums, we have F Ž N . s F Ž N1 . [ F Ž N2 .. Now from the hypothesis of unique decompositions, M is a direct summand of at least one F Ž Ni .. Thus, assume that N is indecomposable. Let N1 , . . . , Ns denote all the indecomposable objects in C up to isomorphism. By the unique decomposition hypothesis, write each F Ž Ni . as a direct sum of indecomposable objects from D uniquely up to isomorphism: F Ž Ni . s Mi, 1 [ ??? [ Mi, mŽ i. . The above argument shows that any indecomposable object, say M, of D is isomorphic to one of the Mi, j where 1 F i F s and 1 F j F mŽ i .. If R is a complete, local Dedekind domain and L is an R-order, then each f.g. L-module decomposes into indecomposable summands and this decomposition is unique up to isomorphism and order Žsee w2, Theorem 30.6x.. The next result, which is one of our main tools of this paper, follows immediately. COROLLARY 2.1. Let R be a complete, local Dedekind domain and let L and V be R-orders such that L is a representation bound of V Ž for the category of L-latt .. If L has FRT, so does V. For the most part, the bounded functors of this paper and other papers in the literature are various versions of the restriction and induction functors. However, these functors are often not only bounded but also separable, which is yet a stronger condition.

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DEFINITION. Following w14x or w1x, if C and D are arbitrary categories, a functor F: C ª D is said to be separable if for each pair of objects M, N g C , there is a map f : Hom D Ž FM, FN . ª Hom C Ž M, N . satisfying: 1. For all a g Hom C Ž M, N ., f Ž F Ž a .. s a . 2. If there are M9, N9 g C and a g Hom C Ž M, N ., b g Hom C Ž M9, N9., f g Hom D Ž FM, FM9., g g Hom D Ž FN, FN9. such that the diagram 6

Fa

FM

FN g

f

6

Fb

6

6

FM9

FN9

is commutative, then the diagram a

6

M

N fŽg.

fŽ f .

6

b

6

6

M9

N9

is commutative. Now let a : R ª S be a ring morphism. Then the induction and restriction functors associated to a are defined as follows: 1. Restriction. Define a Ž ] .: S-mod ª R-mod which makes an S-module M into an R-module a Ž M . via r ) m [ Ž r . a m. 2. Induction. Define S mR ]: R-mod ª S-mod in the natural way. The next result, which draws a connection between bounded and separable functors, is from a private communication with A. del Rıo ´ Mateos. LEMMA 2.2. Let C and D be arbitrary categories and suppose there are functors F: C ª D and G: D ª C so that F is left adjoint to G. Then 1. 2.

If F is separable, then G is bounded. If G is separable, then F is bounded.

Proof. Since F is the left adjoint of G, there is the unit u: 1 C ª GF and counit c: FG ª 1 D . By w16, Theorem 4.1x, F is separable if and only if u splits Ži.e., there exists a natural transformation e : GF ª 1 C such that eC ( u C s 1 C for every C g C .. Similarly, G is separable if and only if c splits Ži.e., there exists a natural transformation m : 1 D ª FG such that c D ( m D s 1 D for every D g D .. In particular, if F is separable and

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M g C , then M is isomorphic to a direct summand of GF Ž M . and so G is bounded. Similarly, if G is separable, then F is bounded. Relative to the induction and restriction functors, the lemma above affords us the following theorem: THEOREM. 1.

a

Ž ] . is separable if and only if the map C: S mR S ª S

¨ ia s m s9 ¬ ss9 splits as a map of S-S-bimodules.

2. 3. 4.

S mR ] is separable if and only if a splits as an R-bimodule map. If s mR ] is separable, then a Ž ] . is a bounded functor. If a Ž ] . is separable, then S mR ] is a bounded functor.

Proof. Parts Ž1. and Ž2. can be found in w1, Proposition II.5.1.4x or w14, Corollary 2.4 and Proposition 2.5x. Parts Ž3. and Ž4. follow from the above lemma. Suppose D is a unital ring and let L be a G-strongly graded ring with L 1 s D. Since each component L g is an invertible D-bimodule, it follows from w17x that there is a unique G-action on the center of D, ZŽ D ., satisfying for a g L g , b g Ž D .: ab s b g a. The next result follows from the above theorem and from various results in the literature; in particular, see w1, 14, 16x. PROPOSITION 2.3 Žw1, Proposition II.5.1.5x.. Let D be a unital ring and let L be a G-strongly graded ring with L 1 s D. Let DŽ ] .: L-mod ª D-mod denote the restriction functor. Then: 1. L mD ]: D-mod ª L-mod is separable and so DŽ ] .: L-mod ª D-mod is bounded. 2. DŽ ] .: L-mod ª D-mod is separable if and only if the trace map t: Ž . Z D ª ZŽ D . ¨ ia a ¬ Ý g g G a g is surjecti¨ e. Moreo¨ er, in this case, the functor L mD ]: D-mod ª L-mod is bounded. 3. If G is finite such that < G < is a unit of D, then DŽ ] .: L-mod ª D-mod is separable and so L mD ]: D-mod ª L-mod is bounded. For integral representation theory, the induction functor is often inappropriate since it may not take lattices to lattices. That is, if L ; V are R-orders in a K-algebra A, then the functor V mL ] may not take L-lattices to V-lattices since L m M may have R-torsion Žsee the example below.. In place of the induction functor, the extension functor V mL M V Ž ] . : L-latt ª V-latt via V Ž M . s , t Ž V mL M . where t Ž V mL M . denotes the torsion part of V mL M, is often useful. This functor, however, need not be left or right adjoint as the next example shows.

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EXAMPLE. Let R s Z, let p be a prime positive integer, let L s Ž x, y . g R [ R < x ' y mod p4 , and let V s R [ R. Thus, L ; V are R-orders. To show that V Ž ] . is not left or right adjoint, it suffices to show that it is not left or right exact. If we apply the functor V Ž ] . to the exact sequence of L-lattices 0 ª 0 [ pZ ª L ª Z [ 0 ª 0 we get the inexact sequence of V-lattices 0 ª V Ž 0 [ pZ. ª V ª V Ž Z [ 0 . ª 0. These R-orders also show that the induction functor does not take lattices to lattices. Indeed, let M s Z [ 0 and N s 0 [ Z so that V s M [ N. We claim that V m N contains R-torsion elements. But V m N ( Ž M m N . [ Ž N m N .. However, N ( LrŽ pZ [ 0. so M m N ( M m LrŽ pZ [ 0. ( ZrpZ. Moreover, not only is the extension functor not adjoint, it is rarely separable. This is because it is quite possible that V Ž M . ( V Ž N . while M( u N for M and N L-lattices. EXAMPLE. Let R be a local Dedekind domain with maximal ideal P and let Ls

ž

R P

R R

/

and

Vs

ž

R R

R . R

/

Let e1 and e2 denote the usual matrix idempotents. Then if M s L e1 and N s L e2 , then M and N are not isomorphic as L-lattices but their extensions to V are isomorphic. Nonetheless, the extension functor, while not adjoint or separable, is bounded. PROPOSITION. L ; V. Then:

Let L and V be R-orders in a K-algebra A such that

1. The extension functor V Ž ] . is bounded. In particular, if L has FRT, then so does V. 2. The restriction functor LŽ ] .: V-latt ª L-latt is separable. Proof. Ž1. If M g V-latt is indecomposable, then V ŽL M . s M and so Ž V ] . is bounded. Ž2. For V-lattices M and N, define a map u M , N : HomŽL M, L N . ª HomŽV M, V N . via f ¬ Ž1 K m f .< M : M ª N, where 1 K m f : K m M ª K m N. We leave it to the reader to check that u M , N satisfies the above definition for separability.

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3. GRADED EQUIVALENCE AND A SKEW GROUP REDUCTION As mentioned in the Introduction, the second technique we employ is graded equivalence and we discuss this here. In light of bounded representation introduced in the previous section, graded equivalent functors are bounded functors. Our main result of this section is Theorem 3.1. For graded rings, an important application of graded equivalence is the Cohen]Montgomery duality theory, which enables us to reduce the study of modules over graded rings to the study of modules over skew group rings ŽProposition 3.4.. DEFINITION. Let G be a finite group. We say that two G-strongly graded rings L and G are graded equi¨ alent provided there is a left L 1-progenerator P1 such that G ( EndŽL L mL 1 P1 . as graded rings. See w6x for more details and applications. If H is a subgroup of G, then we denote the H-truncation of L by L H . The essence of graded equivalence is that the rings L H and GH are Morita equivalent for each subgroup H of G; that is, the categories of modules are equivalent. In particular, this means that L H has finite representation type if and only if GH has finite representation type. Our interest lies mostly in the case when H s G, except in the p-group reduction. The next result is one of our fundamental tools for reductions. THEOREM 3.1. Let G and D be any rings with identity and let L s G Ž G . be a strongly G-graded ring o¨ er G. Suppose G and D are Morita equi¨ alent ¨ ia the Morita context w G, D, G PD , D Q G ; m : P m Q ª G, t : Q m P ª D x. Set V s Q mG L mG P. Then V is a G-strongly graded ring o¨ er D, V s DŽ G ., and L and V are graded equi¨ alent. In particular, L and V are representation related. Proof. We define the multiplication in V as Ž q m l m p .Ž q9 m l9 m p9. s q m lm Ž p m q9. l9 m p9 for all q, q9 g Q, l, l9 g L, and p, p9 g P. Using this multiplication, we can grade V by G so that the g th component is V g s Q m L g m P. Clearly, we have V s Q m L m P s Q m Ž[g g G L g . m P s [g g G Q m L g m P and it is easy to check that V is strongly graded. ˜ s Q mG L. Q˜ is an V]L-bimodule under the multiplication Next set Q Ž q m l m p .Ž q9 m l9. s q m lm Ž p m q9. l9 for all q, q9 g Q, l, l9 g L, and p g P; we leave it to the reader to check that the action is associative. ˜ is a left progenerator for V. Pick a left G-epimorphism We claim that Q f : G Ž n. ª P, which exists since P is finitely generated as a left G-module. ˜ gives 1 m f : Q˜Ž n. ª Q˜ mG P. But Q˜ m P s V Tensoring on the left by Q ˜ is a generator for V. Now choose a left split so we have shown that Q

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epimorphism g: P Ž m. ª G, which exists since P is a generator and since G ˜ to obtain a split epimorphism is projective. Again, tensor on the left by Q Ž m. ˜ ˜ 1 m g: V ª Q; this shows that Q is left projective. ˜ so that the g th component is Q˜g s Q m L g . This is a Next we grade Q ˜h s Q˜g h . grading that agrees with the grading of V; in particular, V g Q ˜ Thus, we have that Q is a graded, left progenerator for V and so by w6, ˜. are graded equivalent. We finish the proof Theorem 2.6x, V and EndŽV Q ˜ by showing that EndŽV Q . and L are isomorphic as graded rings. ˜. via l ¬ rl , where rl is right multiplication by Define r : L ª EndŽV Q l. It is straightforward to see that r is well defined. To see that r is a ˜h . Then Ž q m graded ring homomorphism, let l g L g and let q m l9 g Q ˜ ˜ ˜ . l9 l s q m l9l g Q h g and so rl : Q h ª Q h g , as desired. To see injectivity, ˜ ª P m Q˜ is a left homomorphism let l g L and note that 1 m rl: P mD Q L ª L under the identification given by m. Moreover, this homomorphism is right multiplication by l, which we can denote by rlL . Thus, rl s 0 m ˜., rlL s 0 m l s 0 and so r is injective. In a similar way, if f g EndŽV Q L ˜ ˜ then 1 m f : P m Q ª P m Q gives rise to rl : L ª L and it follows that f s rl. This shows that r is surjective and so the proof is complete. As an application of the above result, we prove next that if H and H9 are two conjugate subgroups of G, then the rings L H and L H 9 are Morita equivalent. This will be important in our first reduction in the next section. COROLLARY 3.2. Let G be a finite group and let L s G Ž G . be a G-strongly graded ring. Suppose H and H9 are subgroups of G that are conjugate; i.e., H9 s xHxy1 for some x g G. Then L is graded equi¨ alent to L x m L m L xy1 in such a way that L H and L H 9 are Morita equi¨ alent. In particular, L H and L H 9 are representation related. Proof. We have a Morita context w G, G, L x , L xy1 , m : L x m L xy 1 ª G, t : L xy1 m L x ª G x for G where L x is an invertible G-bimodule. By the theorem above, V s L x m L m L xy1 is graded equivalent to L. Moreover, by w6, Theorem 3.1x, the truncations L H and V H are Morita equivalent. But a simple computation shows that V H s L x m L H m L xy1 ( L x H xy1 s L H 9. Hence, L H and L H 9 are Morita equivalent. An important application of graded equivalence is that every graded ring by a finite group is graded equivalent to a skew group ring. There is a tremendous amount in the literature concerning this correspondence; see w4, 6x for details and a list of other articles. We shall need this equivalence in several places later in the paper so we turn to the Cohen]Montgomery duality and the notion of a smash product. DEFINITION. Let L be a G-graded ring, where G is a finite group of order n. Let MG Ž L . denote the G = G matrices that are indexed by G

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with entries from L and set L a G to be the subring  X g MnŽ L . < X g, h g L g hy1 4 of MnŽ L .. We call L a G the smash product of L by G. It is worth remarking that there are generalizations of the smash products using Hopf algebras. However, since we need it only for graded rings, we shall use this simple matricial version. Remark 3.3. We record some general facts here about the smash product. Again, we refer the reader to w4, 6x for details. 1. The group G acts faithfully via conjugation by permutation matrices on L a G. 2 The skew group ring Ž L a G .)G is isomorphic to MnŽ L .. 3. L and Ž L a G .)G are graded equivalent. ŽHere graded equivalence is defined for non-strongly graded rings; see w6x.. 4. If L is strongly graded, then the L 1-progenerator, as mentioned in the definition of graded equivalence, can be taken to be L itself. Now we return to the graded order situation. PROPOSITION 3.4. Assume L is a strongly G-graded R-order with maximal 1-component inside the K s QŽ R .-algebra A. 1. 2.

A is strongly graded ¨ ia A s [g g G K ? L g . A a G s K Ž L a G . ( MG Ž KL 1 ..

L a G is a maximal R-order in the separable K-algebra A a G. 4. L and Ž L a G .)G are representation related. In particular, L has finite representation type if and only if Ž L a G .)G has finite representation type. 3.

Proof. Ž1. If L s [g g G L g , then A s KL s [g g G KL g so A is G-graded with the g-component given by A g s KL g . Ž2. Since K lives in the center of A, it is easy to see that A a G s K Ž L a G .. Ž3. By w2, Theorem 26.21x, it suffices to assume R is a complete, local Dedekind domain and L a G is prime. Since L 1 and L a G are Morita equivalent, L a G has precisely one indecomposable lattice, which is projective. Thus, L a G is hereditary and by the Structure Theorem for hereditary orders w2, Theorem 26.28x, L a G is maximal. Ž4. This follows immediately since graded equivalent categories are representation related categories.

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4. THE COMPLETE, LOCAL AND THE p-GROUP REDUCTIONS To pass the FRT problem to the situation where R is a complete, local Dedekind domain, we assume that quotient field K of R is a global field; i.e., K is either an algebraic number field Ža finite field extension of Q. or a function field Ža finite extension of k Ž X . where k is a finite field.. With these assumptions, the Jordan Zassenhaus Theorem holds: If A is any f.d., semisimple K-algebra, K a global field, and L is any R-order in A, then for each f.g. A-module V, there are only finitely many isomorphism classes of left L-modules M such that KM ( V. Let L denote an arbitrary R-order. For a prime P of the Dedekind domain R, let L P denote the completion of L at the prime P and let S Ž L . denote the set of prime ideals P of R for which L P is not a maximal order. By w2, Statement 31.1x, S Ž L . is a finite set. We now state a result that enables us to reduce to the complete, local case: THEOREM 4.1 ŽJones’ Theorem w2, Theorem 33.2x.. only if L P has FRT for each P g S Ž L . of R.

L has FRT if and

We wish to make some remarks about the proof that appears in w2, Theorem 33.2x relative to our discussion about bounded functors. Remark. 1. In the argument given for the proof of the fact that if L has finite type, then L P has finite type, the authors show that given an indecomposable L P-lattice X, X is a direct summand of YP s L P mL Y, where Y is some indecomposable L-lattice. This means that the induction functor L P mL ]: Ž L-latt. ª Ž L P-latt. is bounded. 2. For the converse argument, the authors consider a subcategory Z of the product of categories =P g S ŽL . Ž L P-latt. which is defined by Z s  Ž L P m M . Pg S Ž L . for some M g Ž L-latt . 4 . They define a functor H: Z ª Ž L-latt. via Ž L P m M .P g S ŽL . ¬ M and they show that this functor preserves and reflects indecomposable lattices and that Z has only finitely many indecomposable objects up to isomorphism. That is, they show that H is a bounded functor. For the proof of our main theorem, we would like to identify the set S Ž L . where L is a G-strongly graded order. As in the group ring case, this turns out to be the set of primes that divide the order of G. This is where we prove and use Theorem B from the Introduction. THEOREM 4.2. Let G be a finite group of order n, let D be a maximal order, and let L s DŽ G . be a G-strongly graded order. Then L is maximal if and only if 1rn g D.

REDUCTION TECHNIQUES, I

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Proof. By Remark 3.3, we know that L is graded equivalent to a skew group ring. If we can prove the result for skew group rings, then it holds for strongly graded rings. ŽNote that if 1rn g L a G, then 1rn appears in D, and conversely.. Thus, we assume that L is a skew group ring, L s D )G. Assume L is a maximal order and let e s ny1 Ý x g G x. As in w2, p. 583x, e is a central idempotent of KL s Ž KD .)G and it follows that L q D e is also an R-order in Ž KD .)G. By maximality of L, e g L. Since elements of L s D )G can be written uniquely as D-linear combinations of the group elements, 1rn g D. Now assume that 1rn g D. By w2, Theorem 26.21x, it suffices to prove that L P is maximal where L P denotes the completion of L at a prime P from R. Replacing L with L P we may assume that R is a complete, local Dedekind domain. By the assumption and Proposition 2.3, we know that the restriction functor DŽ ] .: Ž L-latt. ª Ž D-latt. is separable. By w1, II.5.1.3x, we know that given a L-homomorphism f : M ª N such that D f : D M ªD N splits, then f splits. This implies that if D N is projective as a D-lattice, then N is projective as a L-lattice. However, since D is maximal, every D-lattice is projective and so every L-lattice is projective. Consequently, L is hereditary. In addition, if M and N are L-lattices such that they are isomorphic as D-lattices, then they are isomorphic as L-lattices. ŽThis can be seen directly: if f : M ª N is a D-isomorphism, then f can be extended to a L s D )G-isomorphism Ž1rn.Ý x g G xfxy1 .. Let e be a central, primitive idempotent of L Ži.e., eL e is a prime ring.. By w2, Theorem 26.20x, it suffices to prove that eL e is maximal. Let e1 and e2 be two primitive idempotents of L such that e i s ee i for i s 1, 2. Since L is hereditary over a complete, local Dedekind domain, it suffices to show that L e1 ( L e2 as left L-lattices by the structure of hereditary orders w2, Theorem 26.28x. However, since eL e is prime, K Ž L e1 . ( K Ž L e2 .. Since D is maximal, we have by w2, Exercise 11, p. 581x, DŽ L e1 . (DŽ L e2 .. By our observation about the separable restriction functor above, L e1 ( L e2 and so the theorem is proved. Using the above result, we can embed S Ž L . inside the Žfinite. set of primes that divide the order of the group G. COROLLARY 4.3. Let G be a finite group of order n and let L s DŽ G . be a G-strongly graded order with maximal 1-component D. Then S Ž L . ;  p prime < p < n4 . In particular, L has FRT if and only if L p has FRT for each prime p that di¨ ides n. Proof. If p does not divide n, then 1rn g R p ; D p . By Theorem 4.2, L p is maximal and so S Ž L . ;  p prime < p < n4 . The final statement follows from Theorem 4.1.

580

JEREMY HAEFNER

In any event, we now can assume, in order to solve the FRT problem, that K is global Žor at least satisfies the JZ theorem., R is a complete local Dedekind domain, and G s L 1 is a maximal R-order. As in the proof of the classification of which integral group rings ZG have FRT, we now reduce the strongly graded order FRT problem to p-groups. THEOREM 4.4. Let p be a prime integer such that p is a non-unit of R. Let H and H9 be p-Sylow subgroups of G. Then: 1. L H and L H 9 are representation related so that L H has FRT if and only if L H 9 has FRT. 2. L and L H are representation related. In particular, L has FRT if and only if L H has FRT for any p-Sylow subgroup H of G. Proof. Ž1. This first statement follows from the fact that H and H9 are conjugate and from an application of Corollary 3.2. Ž2. This result has been proved in w6, Corollary 7.6x. We present a different proof here using both our tools of bounded Žseparable. functors and graded equivalence. the idea is to show that the induction and restriction functors between L-mod and L H -mod are separable functors. The key fact is that, by using Proposition 3.4 and the notion of graded equivalence, L is representation related to a skew group ring over G, say V )G. Now using arguments similar to those for Proposition 2.3, it is possible to show that Lm ]: L H -mod ª L-mod is separable. This means that the restriction functor is bounded which proves that if L has FRT, then so does L H . For the converse, note that since p is a non-unit of R, then the index of the order of H in G is relatively prime to p and so the index is a unit of R. Consequently, by statement Ž3. of Proposition 2.3, the restriction functor is separable and so the induction functor is bounded. Hence, if L H has FRT, then L has FRT. Finally, by Ž1., if any L H has FRT, then so does L H 9. We can now assume that G is a p-group, G is a maximal order, and that R is a complete local Dedekind domain.

5. THE REDUCTION TO CROSSED PRODUCT ORDERS In this section, we show how to reduce from the strongly graded situation to the case of a crossed product order over a prime, maximal 1-component. Then using our assumption that R is complete and G is a p-group, we are able to show that such an order is primary. This will allow

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us to use a well-known result about finite type of Drozd and Kiricenko to obtain a generalization of Theorem A mentioned in the Introduction. We begin by asking how close L is to a crossed product order. Here a crossed product order is a graded order L such that each component L g contains a unit of the order L. For example, we have the following result: PROPOSITION 5.1.

Let L be a strongly graded order o¨ er R.

1. If L 1 is a prime, maximal order o¨ er R and R is a complete local Dedekind domain, then L is a crossed product. 2. If L 1 s MnŽ D . where D is a local ring, then L is a crossed product. Before we prove this result, we give an example that shows that the prime hypothesis of Ž1. is necessary. EXAMPLE. An example of a strongly graded order with maximal 1-component o¨ er a global field that is not a crossed product. Let R be any complete local Dedekind domain with global quotient field K. Let L s M3 Ž R . be the ring of 3 = 3 matrices with entries from R. Grade L by the cyclic group of order 2 as follows: Set R L1 s R 0

ž

R R 0

0 0 R

/

and

Lg s

ž

0 0 R

0 0 R

R R . 0

/

Since L g does not contain any units of L, L is not a crossed product. To prove the proposition, we use the notion of the smash product from Section 3. Proof. Ž1. It suffices to show that each component L g contains a unit of L. Our argument uses the smash product L a G; for each x g G, let eŽ x, x . denote the usual matrix idempotent of L a G. Let e be a primitive idempotent of L 1. It follows that eeŽ x, x . s eŽ x, x . e is a primitive idempotent of L a G because w eeŽ x, x .xL a Gw eŽ x, x . e x s eL 1 e is primitive. Since L 1 and L a G are Morita equivalent due to the strong grading hypothesis of L, and since L 1 has a unique Žup to isomorphism., indecomposable projective module w15, Theorem 18.7x, the order L a G also has a unique Žup to isomorphism., indecomposable projective module. Consequently, ee Ž x, x . Ž L a G . ( ee Ž y, y . Ž L a G . for all x, y g G and for all primitive idempotents e g L 1. We claim that eŽ x, x .Ž L a G . ( eŽ y, y . L a G for all x, y g G. We may write eŽ x, x . s Ýe i eŽ x, x . and eŽ y, y . s Ýe i eŽ y, y . for a finite collection of primitive idempotents  e1 , . . . , e n4 . Then e Ž x, x . Ž L a G . s Ž Ý e i e Ž x, x . Ž L a G . . (

Ý ei e Ž y, y . Ž L a G . s e Ž y, y . Ž L a G . .

582

JEREMY HAEFNER

It follows that there are elements u g eŽ x, x .Ž L a G . eŽ y, y . and ¨ g eŽ y, y .Ž L a G . eŽ x, x . such that u¨ s eŽ x, x . and ¨ u s eŽ y, y .. Yet u g L x yy1 eŽ x, y . and ¨ g L y xy 1 eŽ y, x . and hence u s a x yy 1 eŽ x, y . and ¨ s a y xy1 eŽ y, x . for some a x yy 1 g L x yy 1 and a y xy 1 g L y xy 1 . Since u¨ s eŽ x, x . and ¨ u s eŽ y, y ., we have that a x yy1 a y xy 1 s 1 s a y xy 1 a x yy 1 so that a x yy 1 and a y xy1 are units of L inside L x yy1 and L y xy1 , respectively. As a result, L s L p is a crossed product. Ž2. Let X g PicŽ L 1 ., the Picard group of L 1. It follows easily that X ( MnŽ Y . where Y g PicŽ D .. By the local hypothesis, Y ( D as left modules and so X ( MnŽ D . s L 1 as left modules. This shows that X s L 1 x where x is some invertible element in Xy1 . In particular, if X s L g , then L g s L 1 g and g g UŽ L .. Next we proceed to show that even in the situation when the 1-comonent is not prime, we can still reduce to crossed products. LEMMA 5.2. Let G be a direct sum of t prime rings, G s G1 [ ??? [ Gt with central idempotents  e1 , . . . , e t 4 and let P g PicŽ G .. Then P induces an automorphism a P of the center of G, ZŽ G ., such that a P permutes the set  e1 , . . . , e t 4 under the rule e i x s xe ia P for all x g P. Proof. Define the automorphism of ZŽ G . in the usual way w3, Lemma 55.7x: For c g ZŽ G ., let l c : P ª P denote left multiplication by c. Since l c g EndŽG P . and since G ( EndŽG P . via g ¬ rg Žwhere rg denotes right multiplication., there exists g g G such that l c s rg . Further, g g ZŽ G . because l c centralizes EndŽG P .. Consequently, the map c ¬ g defines an automorphism of ZŽ G ., which we denote by a P . For notational simplicity, set c P [ c a P . For 1 F i F t, let e i be the central idempotent of G corresponding to Gi ; we have e iP g ZŽ G .. First we show that these are the only primitive, central idempotents of G. If e is a primitive, central idempotent, then so is ee i and e s Ýee i . By primitivity, there is some i, 1 F i F t, such that ee j s 0 for all j / i, and e s ee i . But e i s ee i q Ž1 y e . e i and by the primitivity of e i and the fact that ee i / 0, we have Ž1 y e . e i s 0. Hence, e s ee i s e i . Now since automorphisms take primitive, central idempotents to primitive central idempotents, the lemma is proved. Although clear from the result above, the action of P on the idempotents e i is given by e i x s xe iP for all x g P. Now we fix some notation. Notation 5.3. Let G be a direct sum of prime rings, G s G1 [ ??? [ Gt , with central idempotents  e1 , . . . , e t 4 , and let L s G Ž G . be a strongly G-graded ring with 1-component G. For g g G, the component L g is an element of PicŽ G . and so L g determines an automorphism of the center of G, ZŽ G ., which we denote by a g where c a g s c g .

REDUCTION TECHNIQUES, I

583

By Lemma 5.2, G acts on the set V s  e1 , . . . , e t 4 . Assume that there are m orbits to this action given by V 1 , . . . , V m and choose one representative e i from each orbit i s 1, . . . , m. Since G acts transitively on any orbit, we have V i s  k < e k s e ig for some g g G4 . For 1 F j F m, let Gj denote the stabilizer of e j ; i.e., Gj s  g g G < e jg s e j 4 . Finally, for 1 F j F m, let m j s < G : Gj <, let Ej s Ý k g V j e j , and let D j s L Ej . THEOREM 5.4. Let L and G be as in Notation 5.3. Then L is Morita equi¨ alent to a direct sum of crossed products. In particular: 1.

m Each D j is a strongly G-graded ring, and L s [js1 D j.

2. D j is Morita equi¨ alent to e j L G j e j where L G j denotes the truncation of L at the subgroup Gj . Ge j .

3.

e j L G j e j is a crossed product with grading group Gj and 1-component

Proof. Ž1. We claim that the elements Ej are central idempotents of L. As noted in Notation 5.3, G permutes the e j and in fact G transitively acts on each orbit V j , 1 F j F m. Since the Ej s Ý k g V j e k , it follows that Ejg s Ej for each g g G. Hence, each Ej centralizes L g for each g g G and so Ej belongs to the center of L. Since the Ej are also orthogonal and sum to 1L , we decompose L as m m L s [js1 L Ej s [js1 D j , as desired. To see that D j is strongly G-graded, note that D j s L Ej s [g g G L g Ej and L g Ej L h Ej s L g h Ej for all g, h g G and so the g th component of D j is L g Ej . Ž2. We shall prove the result for j s 1; that is, we prove that D s D 1 is Morita equivalent to e 1 L G 1 e 1. Set Q s e 1 L G 1 e 1 , P s e 1 D, and Q s D e 1. We want to show that w D, Q, P, Q, m : Q m P ª Q, t : P m Q ª D x, where m Ž d 1 e 1 m e 1 d 2 . s d 1 e 1 d 2 and t Ž e 1 d 1 m d 2 e 1 . s e 1 d 1 d 2 e 1 , is a Morita context with both m and t are surjective. To do this, it suffices to show that D s D e 1 D and Q s e 1 D e 1. To see that D ; D e 1 D, we first show E1 g D e 1 D. Note that for any g g G, e 1g g Ge 1g s L gy 1 L g e 1g s L gy 1 e 1 L g ; L e 1 L s L E1 e 1 L E1 s D e 1 D. This shows that each e 1g g D e 1 D and so E1 g D e 1 D Žsince G acts transitively on V 1 .. Now for any g g G, the g th component of D is D g s D g E1 ; D e 1 D and so D s [g g G D g ; D e 1 D. The opposite inclusion is straightforward. Finally, to see that Q s e 1 D e 1 , we first show that e 1 D e 1 is strongly G1-graded. The components of D are D g s L g E1 and so the components of e 1 D e 1 are e 1 D g e 1. Now using the action of L g on the idempotents e k , we have that e 1 D g e 1 s e 1 L g E1 e 1 s e 1 L g e 1 s e 1 e 1gL g for each g g G.

584

JEREMY HAEFNER

Since the e k are orthogonal, e 1 D g e 1 / 0 if and only if g g G1. It follows that e 1 D e 1 s [g g G e 1 D g e 1 s [g g G 1 e 1 D g e 1 and so e 1 D e 1 is strongly G1-graded. Now e 1 D e 1 s [g g G 1 e 1 D g e 1 s [g g G 1 e 1 L g E1 e 1 s [g g G 1 e 1 L g e 1 s Q. Ž3. Set Qj s e j L G e j so that it is clear that Qj is a Gj-strongly graded j order. Since the 1-component is e j L 1 e j s Ge j , which is prime and maximal, we know that Qj is a crossed product order by Proposition 5.1. Crossed product versions of the above result Žwhich holds for strongly graded rings. have appeared in w18, Proposition 2.3x and w9x. In w9x, the authors consider the interesting idea of when a sub-crossed product ‘‘induces’’ the full crossed product and they prove an existence and uniqueness result. For this, they need to make the assumption that G acts transitively on the central idempotents. This might not occur in our scenario so we need Theorem 5.4. above. Applying our bounded representation ideas, we have the following immediate corollary. COROLLARY 5.5. j s 1, . . . , m.

L has FRT if and only if each e j L G j e j has FRT for

So we now can assume that L s G )as G, where G is a prime, maximal order over a complete local Dedekind domain R, G is a p-group where p is a non-unit of R, s : G ª AutŽ G . is a map from G into the automorphisms of G, and a : G = G ª UŽ G . is a factor set into the units of G, relative to the action s . With these assumptions, we can show that the order L is primary: THEOREM 5.6. Let G be a prime, maximal R-order, and let G be a p-group for some prime integer p Ž p g R .. If L s G )G is a crossed product of G o¨ er G, then L is primary. Proof. First let s : G ª AutŽ G . be a map and let a : G = G ª UŽ G . be the 2-cocycle associated to s . Since p g R and since G fixes the elements of R, the maximal 2-sided G-ideal pG is s Ž G .-invariant and so s induces s : G ª AutŽ GrpG .. Moreover, a induces a : G = G ª UŽ GrpG . and this is compatible as a cocycle with s . Set G s GrpG. Define L s Ž G )G .rpŽ G )G . and note that L ( Ž G .)G. By w2, Proposition 5.22x, we know that L rad L

(

L rad L

and so L is primary if and only if L. Since L is Artinian, the prime radical of L equals the Jacobson radical and so L is primary if and only if there is a unique prime ideal of L.

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585

Now since G is simple w2, Theorem 26.23x, let C denote the center of G; C is a field. Let Ginn s  g g G < s Ž g . is inner4 . From w13, Theorem 15.8x, we conclude that L has a unique prime ideal if and only if the twisted group ring C t w Ginn x has a unique G-prime ideal. ŽA G-prime ideal is a 2-sided ideal I which is invariant under the action of G and such that whenever AB ; I for any two ideals A and B that are also invariant under G, then either A ; I or B ; I.. However, from w12, Lemma 3.3x, we see that C t w Ginn x is local and so it is has a unique prime ideal. ˜ g where Q˜ is a If Q is a G-prime ideal of C t w Ginn x, then Q s Fg g G Q tw tw x x prime ideal of C Ginn . Since C Ginn has exactly one prime ideal, ˜ g s Q˜ and so Q s Q. ˜ This shows that C t w Ginn x has exactly one G-prime Q ideal. We conclude that L is primary and hence so is L. The above result allows us to apply the work of Drozd and Kiricenko which determines when a primary order has FRT. This is a preliminary version of Theorem A mentioned in the Introduction. THEOREM 5.7 ŽDrozd and Kiricenko w5x.. Let L s G )G be a crossed product where G is a prime, maximal order o¨ er a complete, local Dedekind domain R, and G is a finite group that acts faithfully as automorphisms on G. ˜ be the intersection of the maximal orders containing L, let I s LrL, ˜ Let L and let J be the intersection of the maximal L-submodules of I. Then L has FRT if and only if the following conditions hold: 1. 2. 3.

˜ is hereditary. L I has two generators as a L-module. J is cyclic as a L-module. 6. THE DIVISION ALGEBRA REDUCTION

We now have that R is a complete, local Dedekind domain, G is a prime, maximal R-order, and L s G )G is a crossed product. In this case, G s MnŽ D . where D is a maximal order in a division algebra. Our goal is to reduce from L s G )G to V s D )G, another crossed product but such that the 1-component is a maximal order in a division ring. Using Theorem 3.1, we have the following corollary: COROLLARY 6.1. If L s G )G is a crossed product such that G is a prime, maximal R-order where R is a complete, local Dedekind domain, and if G is Morita equi¨ alent to D, then L is graded equi¨ alent to a crossed product of the form D )G. In particular, if G s MnŽ D . where D is a maximal order o¨ er a complete, local Dedekind domain R and if L s G )G is a crossed product, then L is graded equi¨ alent to a crossed product of the form D )G.

586

JEREMY HAEFNER

Proof. The graded equivalence follows from Theorem 3.1. Since D is also prime, then DŽ G . is a crossed product by Proposition 5.1. So from the above result, we know that we can reduce to the division algebra case; that is, we can study the FRT problem for strongly graded rings by studying crossed products whose coefficient rings are orders inside division algebras. However, crossed products are characterized by a map s from the group G into the group of automorphisms and by a factor set a relative to the map s . In passing from MnŽ D . to D, we would like to know what happens to s and a . Since Proposition 3.1 does not keep track of s and a , we explore this situation for the remainder of this section. We begin with an example to illustrate the importance of s and a . EXAMPLE. The following skew group ring appears in w11, Example 3.10x; we shall use it to illustrate the FRT problem. Let a s 'y 5 , R s Zw a x, I s ²2, 1 q a :, K s QŽ R ., G s M2 Ž R ., and A s M2 Ž QŽ R ... Define Q to be the 2 = 2 matrix Qs

ž

2 1qa

y1 q a y2

/

which belongs to G and is a unit of A but not of G. Let g denote conjugation of G by Q. Set G s ² g :. Note that Q 2 s Žy2. ? I. Thus, since y2 ? I commutes with all elements of A, conjugation by Q 2 leaves G fixed elementwise and so g has order 2. Form L s G )G, a skew group ring by G where the action is defined by g g s gg g s gQy1g Q. Since G fixes R ? I, L is an R-order. Now replace R by the localization of R at the maximal ideal I and replace both L and G by L I and GI , respectively. Let P s p R s Rp denote the maximal ideal of R, where p s 1 q a . Since 2 completely ramifies in R with ramification index 2, we can write 1 y a s p u and 2 s p 2 ¨ where u and ¨ are units of R. As a result, Qs

ž

¨p 2

p

up ¨p s 1 y¨ p 2

/ ž

u ?p. y¨ p

/

Set Ls

žp ¨

1

u g G. y¨ p

/

Notice that detŽ L. s y¨ 2p 2 y u, which is a unit of R. Consequently, Ly1 exists. Moreover, Q s L ? p so conjugation by Q equals conjugation by L. Hence, for L s GI )G, G acts as inner automorphisms on G; i.e., g g s gg g s gLy1g L.

REDUCTION TECHNIQUES, I

587

Next we diagonalize L by trivial units; i.e., we change basis by sending 1 ¬ 1 and g ¬ gLy1 s g 9. It is straightforward to check that g 9 generates a group isomorphic to G and since Ly1 belongs to G, we have left L unchanged; that is, L s G ) ² g 9: . However, g g 9 s g gL y 1 s Ž gLy1 .g LLy1 s g 9g and so g 9 commutes with elements from G. This makes L into a twisted group order. Moreover, 2 Ž g 9 . s Ž gLy1 .Ž gLy1 . g

s g 2 Ž Ly1 . Ly1 s g 2 Ly1 Ly1 LLy1 2

s g 2 Ž Ly1 . . But Ž Ly1 . 2 s I ? v where v s Ž u q ¨ 2p 2 .y1 , which is an element of R. It follows that L s G t w G x where t Ž g 9 2 . s v . Now since the factor set t: G = G ª UŽ R . maps to the units of R, it follows that L I s GIt w G x ( M2 Ž R t w G x.. Consequently, L has FRT if and only if R t w G x has FRT. But R is local, and G is cyclic of order 2 so it is easy to see that R t w G x is a Bass order and has FRT by w2, Theorem 37.18x. The above example is an example where the group G is acting as X-inner automorphisms on MnŽZw'y 5 x.. An automorphism of G is X-inner provided it is conjugation by a unit u of the algebra K G. However, as the above example indicates, if R is local, then every X-inner automorphism is inner. To see this, let u be a unit of K G and let p generate the unique maximal ideal of R. We can write u as u s p m u9 where m is an integer and u9 is a unit of G. Since R centralizes G Ž R is contained in the center of G ., conjugation by u equals conjugation by u9 and so any X-inner automorphism is inner. It is important that our reduction made in Corollary 6.1 preserves inner automorphisms as the above example indicates. Along these lines, we shall use the following notation: If s : G ª AutŽ G . s is a map, then set Ginn s  g g G < s Ž g . is inner4 . Our next result shows that automorphisms of MnŽ D . are either inner or come from D. PROPOSITION 6.2. Let D be a maximal R-order inside a di¨ ision algebra Ž R a complete local Dedekind domain. and let G s MnŽ D .. Gi¨ en u g AutŽ D ., define the induced automorphism x Ž u . of G that maps a matrix X s Ž d i j . to x Ž u . X s Ž u Ž d i j ..; this gi¨ es rise to a group monomorphism x : AutŽ D . ª AutŽ G .. Then 1. E¨ ery automorphism f of G is the product of an inner automorphism i u of G and an induced automorphism x Ž u .; i.e., AutŽ G . s InnŽ G . ? x ŽAutŽ D ...

588 2. on D.

JEREMY HAEFNER

Moreo¨ er, if f is X-inner on G, then u can be chosen to be X-inner

Proof. Ž1. Let f g AutŽ G .. If f is inner, then we are done. Otherwise, f induces a non-trivial element Gf of PicŽ G . Žw3, Theorem 55.11x. This element Gf equals G as a left module and as a right module, it has the multiplication: g )g 9 s g wf Žg 9.x. Claim. For any prime, maximal R-order V, PicŽ V . ( OutŽ V .. Let X g PicŽ V .. By w3, Theorem 55.12x, it suffices to show that X and V are isomorphic as left V-modules. However, V has, up to isomorphism, one projective, indecomposable left module P and so we may write Ž n. for some n. Similarly, we may write V X ( P Ž m.. Since EndŽV X . VV ( P ( V ( EndŽV V ., we must have m s n. Thus, V X (V V; this completes the claim. Now since D and G are Morita equivalent, PicŽ D . ( PicŽ G . via the map X ¬ MnŽ X .. By the claim, we see that OutŽ G . ( PicŽ G . ( PicŽ D . ( OutŽ D . with the isomorphism induced from x . In particular, the map OutŽ D . ª PicŽ G . via u ¬ MnŽ Du . is surjective. It is readily seen that MnŽ Du . s MnŽ D .x Ž u . s Gx Ž u . . As a result, given Gf g PicŽ G ., there exists u g AutŽ D . such that Gf ( Gx Ž u . as G-bimodules. Now, using w3, Theorem 55.11x, we see that f s i u ? x Ž u ., where i u denotes conjugation by the unit u. Ž2. Let D be the division ring such that QŽ R . ? D s D and let A s MnŽ D .. We show that if f g AutŽ G . is X-inner, then f s t ku where u g InnŽ G . and t is conjugation by p , the prime element of D. Since f is X-inner, f is conjugation by U1 g GLnŽ D .. Since D s K m D where K s QŽ R ., we can clear denominators with central elements and assume U1 g G Žbut U1 need not be a unit in G .. Let P s p D s Dp be the unique maximal ideal of D generated by the prime element p . Factor out the largest common prime factor from all entries in U1 to obtain U1 s p k U where U has relatively prime entries Žthat is, at least one entry of U is a unit of D .. Now Uy1 g GLnŽ D .. Let m be the smallest non-negative integer such that p m Uy1 g G. Set V s p m Uy1 so Uy1 s Vpym s pym V. Further, at least one entry of V is a unit in D; otherwise, we can write V s p V9, a contradiction on m. Let J Ž ] . denote the Jacobson radical and recall that J Ž G . s p ? G. If G s GrJ Ž G ., then G ( MmŽ DrJ Ž D ... Set D s DrJ Ž D .. Note that V GU s p m Uy1 GU s p m G since conjugation by U normalizes G. If m G 1, then, using Ž ] . to denote modulo J Ž G ., we have that V GUs 0, where V, U g G. But G is simple and so is prime. Hence V s 0 or U s 0.

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But both U and V have an entry that is a unit in D. This contradiction implies that m s 0 and so Uy1 s V g G. As a result, U g GLnŽ D .. We now have U1 s p k U s Up k where U is a unit of G. Hence, let u g AutŽ G . be conjugation by U and let t g AutŽ D . be conjugation by p . Now t induces an automorphism t g AutŽ G . by applying t to each entry of a matrix. It now follows that X f s U1y1 XU1 s Ž Up k .

y1

XUp k

s pyk Uy1 XUp k s t ku Ž X . s Ž Xu .

tk

and so f s t ku . We use the above result to keep track of the map s : G ª D and the factor set a : G = G ª UŽ D . when making the reduction from MnŽ D .)G to D )G. THEOREM 6.3. Let D and G be as in Proposition 6.2 and let L be a crossed product L s G )as G of G with a group G by the action map s : G ª AutŽ G . and s-factor set a : G = G ª UŽ G ., where UŽ G . denote the units of G. Then there exists a G-action map s 9: G ª AutŽ D . and a s 9-factor set a 9: G = G ª UŽ D . in such a way that L is graded equi¨ alent to the crossed product D )as 99 G. Moreo¨ er, Gisn n s Gisn 9n . Proof. By Proposition 6.2, for each g g G, there exists a unit uŽ g . g Ž U G . and an automorphism s 9Ž g . g AutŽ D . of D such that s Ž g . s i uŽ g . ? x Ž s 9Ž g ... Fix a G-action map s 9: G ª AutŽ D . in this way. In particular, for g g G, s Ž g .g s uŽ g . x Ž s 9Ž g ..g uŽ g .y1 . The first step is to change basis so that L s G )as G where the image of s is contained in x ŽAutŽ D ... Pick a basis  g < g g G4 for L over G so that L g s G ? g and gg ss Ž g .g g for all g g G and g g G. Now set g 9 s uŽ g .y1 g for each g g G. Then g 9g s u Ž g .

y1

s uŽ g .

y1

gg s u Ž g .

y1 s Ž g .

gg

u Ž g . Ž x Ž s 9Ž g ..g . u Ž g .

y1

g sx Ž s 9Ž g ..g g 9.

With this new G-action map x Ž s 9.: G ª AutŽ G ., we also have a new factor set a : G = G ª UŽ G . given by

a Ž g , h. s uŽ g .

y1 s Ž g .

Ž u Ž h . y1 . a Ž g , h . u Ž gh . y1 .

590

JEREMY HAEFNER

So with this change of basis, we have L s G )ax Ž s 9. G. Changing notation, we may assume that L s G )as G so that s Ž G . s x Ž s 9Ž G ... This completes the first step. Identify D with eGe where e s e11 s e1. Let e i s e i i g G be the other matrix units for i s 1, . . . , n. Our immediate goal is to construct D )as 99 G. A key observation is that since x Ž s 9Ž g .. is an induced automorphism from an automorphism s 9Ž g . of AutŽ D ., then x Ž s Ž g .. e i s e i for all i s 1, . . . , n. In particular, each idempotent e i commutes with each g, g g G, because ge ss Ž g . eg sx Ž s 9Ž g .. eg s eg. Claim. In D, s 9Ž g .d s eg d egy1 for each g g G and d g D. Here we are identifying D with eGe. Embed d as d e s d e11 . Then s 9Ž g . d ss Ž g . Ž d e . s g Ž d e . gy1 s Ž eg .Ž d e . gy1 e. Now define a 9Ž g, h. s e a Ž g, h.. Since a Ž g, h. s ghŽ gh.y1 , it follows that a 9Ž g, h. s e a Ž g, h. s a Ž g, h. e s e a Ž g, h. e. So Im a 9 ; D and we have a map a 9: G = G ª D. Moreover, a 9Ž g, h. g UŽ D . because e a Ž g, h. e a Ž g, h.y1 s e a Ž g, h. a Ž g, h.y1 e s e and so a 9: G = G ª UŽ D .. We leave it to the reader to show that a 9 is a s 9-factor set for D. This is a nasty computation but requires showing that 1. 2. 3.

s 9Ž x . s 9Ž y .

d . s a 9Ž x, y .Žs 9Ž x y .d . a 9Ž x, y .y1 ; a 9Ž x, y . a 9Ž xy, z . ss 9Ž x .a 9Ž y, z . a 9Ž x, yz .; and a 9Ž1, x . s a 9Ž x, 1. s 1. Ž

Consequently, we can form L9 s D )as 99 G with basis  ex < x g G4 , action s 9: G ª AutŽ D . via s 9Ž g .d s eg d egy1, and s 9-factor set a 9: G = G ª UŽ D . such that a 9Ž g, h. s e a Ž x, y .. Observe that

Ž ex . Ž ey . Ž exy .

y1

s e a Ž x, y . s a 9 Ž x, y . .

Now we show that L and L9 are graded equivalent. Set P s DŽ n. which we make into a D y G bimodule in the usual way and set P˜ s P mG L. We have P˜ s [x g G P m x. Claim. P˜ is a L9 y L bimodule. It suffices to show P˜ is a L9-module. But, for x g G, first define an automorphism of P, which we shall also denote by s Ž x .. That is, for p s Ž d 1 , . . . , dn ., set s Ž g . p s Žs 9Ž g .d 1 , . . . , s 9Ž g .dn .. ŽThis is another induced automorphism from s 9Ž g ... Now define the L9-action on P˜ as ex ? Ž p m g . ss Ž x . p m x ? g ss Ž x . p m a Ž x, y . xg. We leave to the reader the task of verifying that associativity holds. Claim. P˜ is a G-graded L9 y L bimodule via the grading given by P˜g s P m g for each g g G and the action DŽ ex . ? P˜g ? L y s P˜x g y . Again we leave to the reader the check of associativity.

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Claim. L9 P˜ (L9 L9Ž n.. Observe that L ; EndŽL9 P˜. via l ¬ rl , where rl denotes right multiplication by l. Moreover,

ž P˜ / e s Ž P m g . e s P m ge s P m e g g

i

i

i

i

s Pe i m g ( ePe m g ( D m g .

˜ ( [g g G D m g ( L9 and P˜ ª Pe ˜ is a split epimorSo it follows that Pe n Ž n. ˜ ˜ phism such that P ( [is1 Pe i ( L9 , which completes the claim. Thus, P˜ is a left, graded L9-module that generates L9-mod and such that L9 P˜ is projective. Hence, P˜ is a graded, locally-projective generator so that by w6, Theorem 2.6x, L9 and EndŽL9 P˜. are graded equivalent. Claim. L ( EndŽL9 P˜. via r : l ¬ rl , where rl denotes right multiplication by l. ˜ Then e1i l s 0 To see that r is injective, suppose ˜ pl s 0 for all ˜ p g P. for all i s 1, . . . , n and so l s 0. For surjectivity, we identify EndŽL9 P˜. with MnŽ L9. Žsince P˜ ( L9Ž n. .. Suppose b g MnŽ L9. so that b s Ž bi j . where bi j s Ý g g G bi j g eg. Define T : MnŽ L9. ª L via b ¬ Ý g g G Ž bi j g . g, where we view Ž bi j g . g G. We show that T is the inverse of r . Let l g L, and write l s Ý g g G l g g where l g s Ž l g i j . g G. Then rl s ŽŽ r l . i j . where ˜ 1 i ª Pe ˜ 1 j for all i, j via Ž rl . i j s r e il e j . But e i l e j s e i Ý g g G l g ge j Ž rl . i j : Pe s Ý g g G e i l g e j g by the commutativity of e i and g. Hence r

l ª rl s

T

Ý ggG

e i l e j eg ª

Ý

ei l g e j g s l

ggG

which shows that r T is the identity map. In a similar way, Tr is the identity map and so we have L ( EndŽL9 P˜.. This shows that L and L9 are graded equivalent. The second statement follows immediately. COROLLARY. Let L, V, G, and D be as abo¨ e. If s Ž G . consists of inner automorphisms of G, then s 9Ž G . consists of inner automorphisms of D. Finally, we state and prove our Main Theorem from the Introduction. Recall that the notation L p means the completion of L at the prime p. THEOREM 6.4. Let R be a Dedekind domain with global quotient field K and let G be a finite group of order n. Let L s G Ž G . denote a G-strongly graded R-order with maximal 1-component G. For each prime p that di¨ ides n, let H Ž p . be a p-Sylow subgroup of G and let e1 , . . . , e mŽ p. denote a complete, irredundant list of representati¨ es of the orbits of the action of H Ž p .

592

JEREMY HAEFNER

on the set of central, primiti¨ e idempotents of Gp . Let GŽ p, j . be the stabilizer of the action of H Ž p . on e j for 1 F j F mŽ p .. Let e j be a primiti¨ e idempotent of e j Gp e j . Then: 1. The R p-order DŽ p, j . s e j Gp e j is a maximal order in a di¨ ision ring and there exists a crossed product of the form DŽ p, j .)GŽ p, j . for e¨ ery p that di¨ ides n and for e¨ ery 1 F j F mŽ p .. 2. L has finite representation type if and only if each DŽ p, j .)GŽ p, j . has finite representation type for e¨ ery p that di¨ ides n and for e¨ ery 1 F j F mŽ p .. Proof. Ž1. The existence of the e j , the GŽ p, j ., and the e j Ž1 F j F mŽ p .. comes from Lemma 5.2, Notation 5.3, and Theorem 5.4. The fact that DŽ p, j . s e j Gp e j is a maximal order in a division ring comes from the fact that e j Gp e j is a maximal order in a simple algebra Žsince G is maximal. and from the structure of maximal orders in simple algebras over complete, local Dedekind domains Žsee w2, Theorem 26.23x.. The existence of the crossed product DŽ p, j .)GŽ p, j . follows directly from Theorem 6.3. Ž2. By Corollary 4.3, we know that L has FRT if and only if L p has FRT for each prime p that divides n. By Theorem 4.4, we know that L p has FRT if and only if the truncation Ž L p .H Ž p. has FRT for any p-Sylow subgroup H Ž p .. Corollary 5.5 shows that Ž L p .H Ž p. has FRT if and only if e j wŽ L p . GŽ p, j. x e j has FRT for 1 F j F mŽ p . where Ž L p . GŽ p, j. is the truncation of L p at the subgroup GŽ p, j .. Finally, e j wŽ L p . GŽ p, j. x e j has FRT if and only if e j wŽ L p . GŽ p, j. x e j has FRT by Corollary 6.1 or Theorem 6.3. Moreover, by Theorem 6.3, e j wŽ L p . GŽ p, j. x e j ( DŽ p, j .)GŽ p, j .. Finally, we have Theorem A from the Introduction. THEOREM 6.5. Assume the notation of Theorem 6.4. For each crossed product DŽ p, j .)GŽ p, j ., let V Ž p, j . denote the intersection of the maximal orders containing DŽ p, j .)GŽ p, j ., let I Ž p, j . denote V Ž p, j .rDŽ p, j . )GŽ p, j ., and let J Ž p, j . denote the intersection of the maximal D Ž p, j .)GŽ p, j .-submodules of I Ž p, j .. Then each crossed product DŽ p, j .)GŽ p, j . has FRT if and only if each of the following conditions hold: 1.

V Ž p, j . is hereditary;

2.

I Ž p, j . has two generators as a DŽ p, j .)GŽ p, j .-module; and

3.

J Ž p, j . is cyclic as a DŽ p, j .)GŽ p, j .-module.

Proof. This follows from Theorem 5.7.

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593

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