Does the Automorphism Group Generate the Endomorphism Ring in Repž.S, R ?

Journal of Algebra 231, 163᎐179Ž. 2000 doi:10.1006rjabr.2000.8361, available online at http:rrwww.idealibrary.com on

Does the Automorphism Group Generate the Endomorphism Ring in RepŽ.S, R ?

Lutz Strungmann¨ 1

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Communicated by Kent R. Fuller

Received May 25, 1999

Let R be a principal ideal domain and let Ž.S, F be a finite poset. We consider the question whether the automorphism group of a representation V g RepŽ.S, R generates the endomorphism ring of V. A complete solution is given if R s k is a field except in the case of GFŽ.2 . Moreover, some partial results and helpful techniques are obtained for the general case of R being a principal ideal domain, not a field. ᮊ 2000 Academic Press

1. INTRODUCTION

In 1963 Laszlo Fuchs raised the question of when the automorphism group of an abelian group additively generates the endomorphism ringŽ see wx8. . This question can be rephrased more generally by asking when a unital ring is generated by its units. The problem of Fuchs and related generalizations have been of great interest since 1963 and the principal results on abelian groups inwx 3, 10, 12, 15 , on vectorspaces and free modules over principal ideal domains inwx 13, 18᎐20 , on general rings in w 6, 7xwx , and on Warfield modules in 16 are just some examples of research in this direction. In this paper we concentrate on free R-modules with distinguished submodules over a principal ideal domain. Therefore let R be a principal ideal domainŽ. PID and let ŽS, F .be a finite partially ordered set Ž poset . .

1 Supported by the Graduiertenkolleg Theoretische und Experimentelle Methoden der Reinen Mathematik of Essen University.

163 0021-8693r00 $35.00 Copyright ᮊ 2000 by Academic Press All rights of reproduction in any form reserved. 164 LUTZ STRUNGMANN¨

By RepŽ.S, R one usually denotes the category of representations V s g Ž.V, Vii: i S , where V is a free R-module and the V ’s are free R-submod- F : F ules of V which respect the ordering on S; i.e., VijV if i j for some i, j g S. The homomorphisms in RepŽ.S, R are the obvious ones which leave the submodules invariant; i.e., if V and W are representations in RepŽ.S, R then a homomorphism ␸ from V to W is a homomorphism ␸ g from V to W mapping Viiinto W for all i S. The empty set is allowed for S and RepŽ.л, R is just the category of all free R modules. By repŽ.S, R we denote the category of all finite rank representations, where the rank of a representation V g RepŽ.S, R is just the rank of V.Asa special case we write RepnnŽ.R and rep Ž.R instead of RepŽ.S, R and repŽ.S, R if S is an antichain of length n. For details on the categories RepŽ.S, R and rep Ž.S, R we refer towx 1, Chap. 1 . We focus on the question when the automorphism group of a representation V g RepŽ.S, R generates the endomorphism ring, i.e., when every endomorphism ␸ of V can be written as a fixed number of automorphisms of V. It is well known and was proved several times that any k-vectorspace V over a field k / GFŽ.2 has this propertyŽ seewx 13, 19, 20. ; hence the category RepŽ.л, k has this property. More generally it was proved inwx 18 that for any finite rank free R-module M of rank at least 2 we have s q End RRRŽ.M Aut Ž.M Aut Ž.M . Moreover, if M has infinite rank then every endomorphism of M can be written at least as the sum of three automorphisms of M Žseewx 18. . It is not known up to now whether the number three is minimal in this case and it is a secret conjecture that if s n g ␻ End RRŽ.M Aut Ž.M for some n then the minimal n is 2. In the present paper we show that for fields any representation V in the category s q repŽ.S, k satisfies End kkkŽ.V Aut Ž.V Aut Ž.V except in the very special case of k s GFŽ.2 . In the infinite dimensional case we show that in general everything may happen for <integers. The author thanks the referee for his very helpful and interesting comments and suggestions, in particular, for proving Example 2.4Ž.Ž v and therefore Theorem 3.4.Ž. for finite fields k / GF 2 of characteristic 2. AUTOMORPHISM GROUP 165

~~2. PRELIMINARIES~~

In this section we consider the m-sum property and the unit sum number of rings and modules. By a ring we always mean an associative, commutative ring with 1, and a module is always a two-sided module.

DEFINITION 2.1. Let R be a ring. We say that R has the m-sum property for some m g ␻ if each element of R can be expressed as the sum of m units in R.If R has the m-sum property for some m g ␻, then let USPŽ.R s Ä4m g ␻: R has the m-sum property ; otherwise let USPŽ.R s л.

DEFINITION 2.2. Let R be a ring, and let M be an R-module. Then g ␻ we say that M has the m-sum property for some m if End RŽ.M has the m-sum property. If M has the m-sum property for some m g ␻, then let USPŽ.R s Ä4m g ␻: M has the m-sum property ; otherwise let USPŽ.M s л. It is easy to see that a module or ring has the k-sum property for all k G m if it has the m-sum property Ž.m, k g ␻ . Hence the following definition makes sense.

DEFINITION 2.3. Let R be a ring, and let M be an R-module. If USPŽ.M / л then USN Ž.M s minÄm g ␻ ¬ m g USPŽ.M 4 is called the unit sum number of M.IfUSPŽ.M s л then let USN Ž.M s ϱ. Let us first give some examples to get familiar with these definitions.

EXAMPLE 2.4. Following are some examples of unit sum numbers. Ž.i Let k / GF Ž2 .Žk s GF Ž2 .. be a field. Then any k-vectorspace Ž.of dimension ) 1 has unit sum number equal to 2. In particular, USNŽ.k s 2 holds for k / GF Ž.2. Ž.ii GF Ž.2 and ޚ have unit sum number equal to ϱ. Ž.Ž.iii USN ޚrnޚ s 2if n is odd and USNŽ.ޚrnޚ s ϱ if n is even. ␭ ) g ␭ Ž.iv Let R be a PID, let 0 be any cardinal, and let Ä4xi: i be wxg ␭ a set of commuting variables. Then Rxi: i has unit sum number equal to ϱ. Ž.v Let k / GFŽ.2 be a field, and let gxŽ.g kxwx_ k be any irreducible polynomial. Then kxwxr²Ž.gxe: has unit sum number equal to 2 for all integers e G 1. Ž.vi Let R be a ring such that USN ŽR .s t. Then there exists for any cardinal ␭ ) 0 a commutative R-algebra A of rank ␭ such that USNŽ.A s t and AR is free. 166 LUTZ STRUNGMANN¨

Proof. For the first three statements seew 13, Examples 1.1 and 1.4, Theorem 2.5x . Moreover, it is clear that if R is a PID and ␭ ) 0 any w g ␭ cardinal, then the only units in Rxi: i 4 are the constant polynomials; wxg ␭ s ϱ hence USNŽ Rxi: i .Ž.and iv holds. Now let k / GFŽ.2 be a field, and let gxŽ.be an irreducible polynomial in kxwxand e an integer G 1. Then k has at least three distinguished elements, a12, a , and a 3. Without loss of generality we may assume that q q gxŽ.is relatively prime to the polynomials Ž.Ž.x a12and x a . Now choose any w fxŽ.xwxg kxr²Ž.gxe:Ž.. Then gx cannot divide both fxŽ.q q q q Žx a12 .and fx Ž. Žx a .; hence we may assume that gxŽ.does not q q w xws q q xwqy q x divide fxŽ. Žx a11 .. Thus fxŽ. fxŽ. Žx a . Ž.x a1is the sum of two units in kxwxr²Ž.gxe: since every element whxŽ.x g kxwxr²Ž.gxe:Žis a nonunit if and only if it is divisible by gx.. Therefore Ž.v holds. Finally let R be a ring such that USNŽ.R s t, and let ␭ ) 0bea cardinal. If n s 1 then we choose A s R.If ␭ s n ) 1 for some n g ␻, s [ y ␭ G ␻ then let M i- n R the direct sum of n 1 copies of R.If , then s [ let M ig ␭ R. In both cases M is an R-module with componentwise addition and scalar multiplication, and we set A s ÄŽ.r, m : r g R, m g M4. An R-algebra structure on A is defined by componentwise addition, scalar ) s multiplication, and the following multiplication: Ž.Ž.r11, m r 2, m 2 q Ž.rr12, rm 1 1rm 2 2. It is easy to see that A is a commutative R-algebra of rank ␭ withŽ. 1, 0 as identity. Moreover, we will show that Ž.r, m is a unit in A if and only if r is a unit in R.IfŽ.r, m is a unit in A, then there exists Ž.s, n g A such that Ž.Ž.Žr, m ) s, n s rs, rn q sm .Ž.s 1, 0 ; hence rs s 1. Thus r is a unit in R. Conversely, if r is a unit in R and m g M, then there exists s g R such that rs s 1. Let n sys2 m; then Ž.Ž.Žr, m ) s, n s rs, rn q sm .Žs 1,yrs2 m q sm .Žs 1,ysm q sm . s Ž.1,0 and Ž.r, m is a unit in A. Since USNŽ.R s t it is now clear that USNŽ.A s t was well. Note that USNŽ.ޚ s ϱ and USN ŽGF Ž..2 s ϱ although each element of ޚ and GFŽ.2 is the sum of finitely many units. The following lemma fromwx 13 is easy but very useful.

LEMMA 2.5. If R is a ring and USNŽŽ..RrJR s 2, where J Ž. R denotes the Jacobson radical of R, then USNŽ.R s 2. Now we will start to consider the category RepŽ.S, R , where R is a PID and S is a finite poset. AUTOMORPHISM GROUP 167

LEMMA 2.6. Let R be a ring, let S be a finite poset, and let V g RepŽ.S, R s [ g such that V ig I Wii for some finite indexset I and W RepŽ.S, R g g g g s Ž.i I . Then m USP Ž.Vifm USP Ž.Wi for all i I and USNŽ.V m s g s if USNŽ.W i m for all i I. More generally USNŽ.V maxÄ USNŽ.W i : i g I4. g g ␸ g Proof. Assume that m USPŽ.W iRfor all i I and End Ž.V . ␸ g ␸ : g s [ Then End RiiŽ.V and V V for all i S. Since V ig I W iwe write ␸ as a matrix ␸ иии ␸ 11 1k ␸ s ...... , ␸ иии ␸ 0k1 kk s ␸ g where I Ä41,...,k is an ordering of I and ij Hom RŽ.W i, W j for all g g g ␸ s ␸ j i, j I. Since m USPŽ.W iifor all i I we can write iÝ1F jF mii g ␸ j g F F g for all i I, where iiAut RŽ.W i for 1 j m and i I. Now

␸ 1 иии иии иии 11 0 0 ␸␸1 иии иии 21 22 0 0 ␸ s ␸␸␸1 иии 31 32 33 0 0 ...... ␸ иии иии иии иии ␸ 1 0k1 kk ␸ 2 ␸ иии иии иии ␸ 11 12 1k ␸␸2 иии иии ␸ 0 22 23 2 k q ␸␸2 иии ␸ 00 33 34 3k ...... 0 иии иии иии 0 ␸ 2 0kk q ␸ j ␸ j Ý diagŽ.11 ,..., kk , 3FjFk

␸ jj␸ = ␸ j where diagŽ 11,..., kk. is the k k diagonal matrix which has 11,..., ␸ j kk as entries on the diagonal and zeros otherwise. Now it is easy to see that each of the above matrices is an automorphism of V; hence ␸ g m g Aut RŽ.V and m USPŽ.V . By definition of USN it is clear that USNŽ.V s g maxÄ USNŽ.W i : i I4. In contrast to the above lemma it is not true that the m-sum property is ޚ inherited by direct summands. A counterexample is given in Rep0Ž.by 168 LUTZ STRUNGMANN¨ the integers and two copies of the integers. It is easy to see that USNŽ.ޚ s ϱ but USNŽ.ޚ [ ޚ s 2. Nevertheless, if we want to check whether an element V g RepŽ.S, k has the m-sum property Ž.k a field then the above lemma shows that it F s s suffices to consider the case when ig SiV 0 and Ý ig SiV V.

LEMMA 2.7. Let R be a ring, let S be a finite poset, and let V g RepŽ.S, R s s g s such that ViiVorV 0 for all i S. Then USNŽ.V USN Ž.V . Proof. The proof is clear since under the assumptions of the lemma we s have End RRŽ.V End Ž.V . Remark 2.8. Note that USNŽŽ.. Rep S, R s m; i.e., USN Ž.V s m for all X V g RepŽ.S, R implies that also USNŽŽ Rep S , R.. F m for all subposets X X S : S since there is a full categorical embedding of RepŽS , R. into X RepŽ.S, R for S : S Žseewx 1, Proposition 1.3.1. .

~~3. THE VECTORSPACE CASE~~

~~In this section we start considering the category RepŽ.S, k , where k is a field and S is a finite poset. Using an appropriate decomposition of a representation in RepŽ.S, k for <~~

THEOREM 3.1. Let k / GFŽ.2 be a field, let S be a finite poset of cardinality at most 3, and let V g RepŽ.S, k . Then USN Ž. V s 2. In particu- s F lar USNŽŽ.. Repn k 2 for n 3. Proof. Obviously it is enough to show the statement for <

s [ [ [ [ [ [ [ [ [ ) V W M12312321M M Q Q N N N I Ž. AUTOMORPHISM GROUP 169 and s [ [ [ [ ViijkiM N N Q I, where Ä4i, j, k is any permutation ofÄ4 1, 2, 3 . The corresponding decomposition of V shows that by Lemma 2.7 it is enough to show that the [ [ summand Q Ž.Q12123Q , Q , Q , Q has unit sum number equal to 2. By ␣ ª definition of Q13and Q there are epimorphisms i: Q3 Qi mapping an q g s element x12x Q 3to its coordinate xi for i 1, 2. Hence we may write

~~s ␣ [ ␣ ␣ ␣ Q Ž.Q31Q 32, Q 31, Q 32, Q 3.~~

␸ g Obviously any endomorphism End kŽ.Q induces an endomorphism of ␸ g ␺ ª Q3. Conversely if End kŽ.Q3 , then let : Q Q be defined by ␣ q ␣␺s ␸␣ q ␸␣ ␣ q ␣ g ␺ Ž.q11q 22q 1 1q 2 2for q 12q 22 Q. Then leaves ␺ g Q12, Q , and Q 3invariant; hence End kŽ.Q . Therefore the restriction ␾ Ž.ª Ž .␸ ¬ ␸ < Ž.( : End kkQ End Q3 , Q3 is an isomorphism and End k Q

End kŽ.Q3 follows. But End kŽ.Q3 is a full matrix ring of a vectorspace and s s by Example 2.4Ž. i we obtain USNŽQ .USN Ž End k ŽQ3 .. 2. Therefore USNŽ.V s 2 and the proof is complete. It is clear from the proof of Theorem 3.1 that one can also obtain a similar theorem for k s GFŽ.2 but in this case one has to avoid that one of the summands in Eq. Ž.) has dimension 1. To avoid this in general one has to pose a lot of conditions on V g RepŽŽ..S, GF 2 . We at least want to state these conditions in the case <

THEOREM 3.2. Let k s GFŽ.2,let S be a finite poset, let<< S F 2, and let V g RepŽ.S, k . Then USN Ž.V s 2 if one of the following conditions is satisfied. s л ) Ž.i S and dim k ŽV . 1; Ž.ii S s Ä41 and V G ) / Ž.a codim kŽ.V1 2, dim k Ž.V111 if V Vor ) s Ž.b dim k ŽV . 1 if V V1; Ž.iii S s Ä41, 2 and ) s s Ž.a dim k ŽV . 1 if V V12Vor V q G V1 l G Ž.b codim kŽ.V12V 2 or 0, codim k Ž.V12V 2 or 0, V2 l G l G codim k Ž.V12V 2 or 0 and dim kŽ.V12V 2 or 0. 170 LUTZ STRUNGMANN¨

~~Proof. Let V g RepŽ.S, k , <~~

PROPOSITION 3.3. Let n g ގ, and let k / GFŽ.2 be a field. If R is a s g ␻ k-subalgebra of M Mat n=nŽ.Žkn .which contains the identity map, then USNŽ.R s 2. Proof. Let k / GFŽ.2 be a field. We induct on the k-dimensions of R. s s If dim kŽ.R 1, then R k id; hence R has unit sum number equal to 2 / s ) since k GFŽ.2 by Example 2.4. Therefore assume that dim kŽ.R m 1, and let A be any element of R. If the k-subalgebra ²:A of R, generated by A, is a proper direct summand of R as a k vectorspace, then A is the sum of two invertible matrices on R by the induction hypothesis. There- fore assume that ²:A s R. Then R s kxwxr²Ž.:fx , where fx Ž.is the minimal polynomial of A. Using the Chinese remainder theoremŽ seew 14, Theorem 6.2x.Ž we may assume that fx.s gxŽ.e for some irreducible polynomial gxŽ.and some integer e G 1. Now Example 2.4 Ž. v shows that USNŽ.R s 2. Consequently we obtain the following theorem.

THEOREM 3.4. Let k / GFŽ.2 be a field, and let S be a finite poset. Then the following hold: Ž.i If V g rep ŽS, k ., then USN ŽV .s 2; hence USN Ž rep ŽS, k ..s 2. Ž.ii Any finite dimensional k-algebra has unit sum number equal to 2. Proof. Both claims follow immediately from Proposition 3.3 since the endomorphism ring of any representation V g repŽ.S, k and also any finite dimensional k-algebra is isomorphic to a k-subalgebra of some full finite dimensional matrixring over k. In the infinite dimensional case for <

~~THEOREM 3.5. Let S be a finite poset, and let ␭ be any infinite cardinal. Ž.i The following are equi¨alent: Ž.a S is of infinite representation type.~~

Ž.b S contains S4 , Ž.Ž.Ž.Ž. 2,2,2, 1,3,3, 1,2,5, or N,4 as a sub- poset. Ž.c For all commutati¨e rings R and for all R-algebras which can be generated by no more than ␭ elements as an R-algebra there exists a fam- i ily of representations V␣ s ŽV␣␣, V : i g S.Ž.Ž.g Rep S, R ␣ - ␭ with s ␣ F ␤ s g ␤ ¨ Hom RŽ.V␣␤, V Aif and Hom RŽ.V␣␤, V 0 if a . Moreo er, i the submodules V␣ are R-summands of V␣ . A family of representations ha¨ing the properties of Ž.c is called a rigid family. Ž.ii If rep ŽS, k . has finite representation type, then each indecompos- F ( able V in repŽ.S, k has rank 6 and End k Ž.V k. Using partŽ. i of the above theorem we obtain an existence theorem.

THEOREM 3.6. Let R be a PID such that USNŽ.R - ϱ, let S be a finite poset of infinite representation type, and let ␭ be any infinite cardinal. Then there exist rigid families of representations V␣ , W␣ g RepŽ.Ž.S, R ␣ - ␭ of rank ␭ such that USNŽ.V␣ s USN Ž.R and USN ŽW␣ .s ϱ for all ␣ - ␭. Proof. If R is a PID then there exist by Example 2.4Ž. iv and Ž. v an R-algebra A such that USNŽ.A s USN Ž.R and an R-algebra B such that USNŽ.B s ϱ. Moreover, both algebras A and B are of rank ␭.By Theorem 3.5 both algebras can be realized as the endomorphism algebra of rigid families of representations V␣ and W␣ g RepŽ.Ž.S, R ␣ - ␭ , respectively. Hence USNŽ.V␣ s USN Ž.R and USN ŽW␣ .s ϱ for all ␣ - ␭.

Note that the unit sum number cannot depend on the dimension of the representation as the above theorem shows. Nevertheless, for finite representation type we at least get an upper bound for the unit sum number.

THEOREM 3.7. Let k / GFŽ.2 be a field, and let S be a finite poset of finite representation type. Then USNŽŽ.. Rep S, k F 3. Proof. Bywx 17 each representation V g RepŽ.S, k decomposes into a direct sum of finite rank indecomposables, each of which has endomorphism ring isomorphic to k by Theorem 3.5Ž. ii . If V has finite dimension then USNŽ.V s 2 by Theorem 3.4. If V has infinite dimension then Corollary 4.12 shows that one may assume that V is of countable dimension. Now the statement follows from the more general Theorem 4.14. 172 LUTZ STRUNGMANN¨

~~Obviously there are simple posets for which any representation over a field k / GFŽ.2 has unit sum number equal to 2; e.g., S is a chain. Thus the following question remains open.~~

~~QUESTION 3.8. Let k / GFŽ.2 be a field. For which finite posets S do we ha¨e USNŽŽ.. Rep S, k s 2?~~

~~4. THE CASE OF REPŽ.S, R , WHERE R IS A PID~~

In this section we return to the case of an arbitrary principal ideal domain R. Therefore for the rest of this section R denotes a PID if nothing else is stated. By Theorem 3.6 it is clear that we cannot expect to obtain satisfying results for <

THEOREM 4.1. Let R be a PID, not a field, let ␭ be any infinite cardinal, and let A be any reduced and R torsion-free R-algebra. Then there exists a g s representation V Rep2Ž.R such that End R Ž.V A. Using the same arguments as in the proof of Theorem 3.6 we obtain another existence theorem.

THEOREM 4.2. Let R be a PID, not a field, and let ␭ be any infinite cardinal. If USNŽ.R s m g ␻ then there exist representations V, W g s s ϱ Rep2Ž.R such that USNŽ.V m and USNŽ.W . Let us now consider first the category repŽ.S, R . Using a result by Ehrlich we get a first lemma about finite rings.

LEMMA 4.3. Let R be any finite ring with 1 such that 2 is a unit of R Ž.Ž.e.g., the integers modulo some n g ގ not di¨isible by 2.Then USN V s 2 for all V g repŽ.Ž.S, Rng ގ and all finite posets S. In particular R itself has unit sum number equal to 2. [ Proof. Since R is finite we know that E End RŽ.V is finite and hence artinian. Thus ErJEŽ.is a semisimple artinian ring and 2 is a unit in ErJEŽ.. Bywx 6, Theorem 7 we obtain that USNŽŽ..ErJE s 2 and hence by Lemma 2.5 USNŽ.E s USN Ž.V s 2 also follows. If R is a finite ring with 1 and 2 is not a unit in R then the above result is no longer true as the following example shows. AUTOMORPHISM GROUP 173

EXAMPLE 4.4. Let R s ޚr2ޚ = ޚr2ޚ with coordinate multiplication and addition. Then the R-subalgebra A of Mat 4=4Ž.R generated by 00 0 0 00,10Ž. 0 ␸ s 00Ž. 1,00 000 0Ž. 1,1 has unit sum number equal to infinity. Proof. By easy calculations it follows that the only units of A are of the form r )id for some r g R; hence ␸ cannot be written as the sum of two units in A. An easier example is the ring of integers modulo 6, but Example 4.4 is some kind of motivation for our next theorem, which provides a sufficient condition for an endomorphism ␸ of some representation V g repŽ.S, R to be a sum of two units.

THEOREM 4.5. Let R be any commutati¨e ring with 1, and let V g ¨ ␸ g ␹ repŽ.S, R for some finite poset S. Moreo er let End RŽ.V , and let ␸ be the characteristic polynomial of a representing matrix of ␸. If there exists U U r g R j Ä40 such that ␹␸Ž.r g R and either r / 0 or 2 is a unit in R, then ␸ is a sum of two automorphisms of V. ␺ g Proof. First we show that if an endomorphism End RŽ.V is invert- s g ␻ [ ible in M Mat n=nŽ.ŽRn ., then it is also invertible in A ␺ g ␺y1 End RRŽ.V . Therefore let End Ž.V such that exists in M. Then ␹ ␺ n i the characteristic polynomial ␺ of is of the form Ýis0 sxi , where g s s ␺ g U si R for i 0,...,n and s00detŽ., hence s R . Moreover we have ␹␺s wx␺y1 s n y r ␺ iy1 g ␺ Ž.0 Ž see 2, Chap. 7. . Thus Ýis1Ž.si s0 A. U U Now let ␸ g A and r g R j Ä40 such that ␹␸Ž.r g R .If r / 0, then ␸ y r )id is invertible in M and hence ␸ s Ž.␸ y r )id q r )id is a sum of two automorphisms of V by the above arguments. If r s 0 and 2 is a unit U in R then ␹␸Ž.0 g R implies that ␸ is already an automorphism; hence ␺ s 11␸ q ␸ 22is again a sum of two automorphisms of V.

COROLLARY 4.6. Let p ) 2 be a prime, and let R be the ring of integers localized at p or the p-adic integers. Then USNŽ.V s 2 for all V g rep ŽS, R . of rank - p, where S is any finite poset. Proof. First note that any element r g R is a unit if and only if it is not divisible by p. Moreover, 2 is a unit in R.If V g repŽ.S, R is of rank - p ␸ g ␹ ␸ and End RŽ.V , then the characteristic polynomial ␸ of is of degree - p. Thus ␹␸ has at most p y 1 roots modulo p. Therefore there exists UU r g R j Ä40 such that ␹␸Ž.r g R . Now apply the above Theorem 4.5. 174 LUTZ STRUNGMANN¨

~~Unfortunately the sufficient condition of Theorem 4.5 is not necessary for an endomorphism to be a sum of two automorphisms as the next example shows.~~

~~EXAMPLE 4.7. Let F s Ä40, 1, x, y be the field of four elements, i.e., x 2 s y, y 2 s x, xy s yx s 1, x q 1 s y, y q 1 s x, and x q y s 1. If~~

~~0000 0100 ␸ s , 00x 0 0000y~~

then for all f g F we have ␹␸Ž.f s 0. Let A be the F-algebra generated by ␸; then A is the endomorphisms ring of some representation V g ␸ rep5Ž.F , and is a sum of two automorphisms of V. Proof. Easy calculations show that ␸ s Ž y␸ q ␸ 2 q id.Žq x␸ q ␸ 2 q id. is a sum of two units in A. Bywx 1, Example 7 it follows that A is the g endomorphism ring of some V rep5Ž.F . Inwx 18 it was proved that for any PID R and any free R-module M of finite rank G 2 we have USNŽ.M s 2. Moreover, it was shown that USNŽ.M F 3ifrk Ž.M is countable. Using an argument by Castagnawx 3 this result could be extended to the case of arbitrary rankŽ seewx 18. .

THEOREM 4.8. Let M be a free R-module, let R be a PID. Then the following hold: Ž.i If rk Ž.M s 1, then USN Ž.M s USN Ž.R . Ž.ii If 2 F rk Ž.M - ␻, then USN Ž.M s 2. Ž.iii If rk Ž.M G ␻, then USN Ž.M F 3. s It is our aim to give a natural extension of this theorem for S S1 using the well-known stacked bases theorem for modules over PIDs by Cohen and Gluck fromwx 4 . Recall that two free R-modules A : B have stacked bases if and only if their quotient BrA is a direct sum of cyclic R-modules bywx 4 . s g s THEOREM 4.9. Let USNŽ.R 2 and V rep1Ž.R . Then USN Ž.V 2. r Proof. Since V is of finite rank the quotient V V1 is finitely generated and hence a direct sum of cyclic R-modules by Gauß’s theorem. Thus V1 and V have stacked bases by the theorem of Cohen and Gluckwx 4 . Therefore V decomposes into a finite direct sum of representations of the s [ ¨ s [ ¨ g form Ž.R, rR , say V ig Iii R and V1 g I riiR for some r i R g ¨ ¨ Ž.i I . Now each summand Ž.iiiR, r R obviously has unit sum number equal to 2 and thus USNŽ.V s 2. AUTOMORPHISM GROUP 175

For the case of infinite rank representations we show that it suffices to consider representations of countable rank by extending a result of Castagna on free modulesŽ seewx 3. . s [ ¨ DEFINITION 4.10. Let V be a free R-module; i.e., V jg J j R for some ¨ g g ¨ g ¨ ¨ s ¨ ¨ j VjŽ.J and V. Then has a unique representation Ý jg Jjjr , ¨ g g wx¨ V [ g ¨ / where rjjR for all j J. The set Ä j J: r 04 is called the support of ¨ in V.

LEMMA 4.11. Let V g RepŽ.S, R for some finite poset S such that ␬ s G ␻ ␴ g ␴ ¨ rkŽ.V . Then there exists for any End RŽ.Va-in ariant filtra- tionÄ4 W␣ : ␣ g ␬ of V with the following properties: i Ž.i W␣ s ŽW␣␣, W : i g S.Ž.g Rep S, R for all ␣ g ␬;

~~Ž.ii V s D␣ g ␬␣W ; s D i g Ž.iii Vi ␣ g ␬␣W for all i S;~~

Ž.iv W␣ s D␤ g ␣␤Wif␣ F ␬ is a limit ordinal; s [ g Ž.v W␣ W␣y1 C ␣␣, where C RepŽ.S, R is of at most countable rank G 1 if ␣ is a successor ordinal; ␴ < g ␣ g ␬ Ž.vi W␣ End RŽ.W␣ for all . g ␴ g Proof. Let V RepŽ.S, R and End RŽ.V . We choose bases for V g s [ ¨ s [ ¨ i g and Vii Ž.S , e.g., V ␣ g ␬␣␣ R and Vi g ␬ ␣ R for Ž.i S . Note i that also ¨␣ s 0 is allowed to simplify indices and that the support of an g element w Vi only refers to the nonzero basis elements. Now we define s our filtration inductively. Let W00be the trivial representation, i.e., W Ž.0, 0, . . . , 0 . Then obviously Ž.Ž. i ᎐ vi hold. Now assume that W␤ is con- structed for ␤ - ␣ - ␬.If ␣ is a limit ordinal then we put W␣ s D␤ g ␣␤W andŽ. i ᎐ Žvi . are fulfilled. Therefore let ␣ s ␤ q 1 be a successor ordinal. If W␤ s V then we are done. Hence let W␤ / V. Then there exists an / g _ / g _ i element 0 u V W␤ . Furthermore we choose 0 uiiV W␤ for g / i s ␴ i S if Vi W␤ ; otherwise let ui 0. We now take the -closure of W␤ , ¨ ¨ i g 0 s wxV j wxV j u, ui, ␤␤R, and RiŽ.S for W␣. Therefore let X u W␤ ␤ 0 s wxVi j w i xVi j ␤ Ä4and Xiiu W␤ Ä4. Inductively we increase the supports j j s by the following procedure. If X and Xi are defined then let B i [␥ g ji¨␥ R and let B s [␥ g ¨␥ R. Now put X i X j

~~nn jq1 s j j wx␴ VVVj w␴ xj wx X X B DDBiiB is1 is1 and~~

~~jq1 s j j wx␴ Vi XiiiX B . 176 LUTZ STRUNGMANN¨~~

s D j s D j s [ ¨ i s Finally let X jg ␻ X , Xijg ␻ Xi , W␣␥␣␥ g X R, and W i 1 n [␥ g ¨␥ R. Obviously W␣␣␣␣s ŽW , W ,...,W .Ž.g Rep S, R is ␴ invariant X i ii Ž.᎐ Ž . s [ ¨ s [ i ¨ and i vi hold with C␣ ␥ g X _wW␤ x ␤␣R, C ␥ g X _wW␤ x ␤R and 1 n i C ␣ s Ž.C␣␣, C ,...,C ␣.

COROLLARY 4.12. Let USNŽ.R s m and V g RepŽ.S, R for some finite poset S. If USNŽ.W s m for all subrepresentations W : V of at most countable rank, then USNŽ.V s m. Proof. The proof is very similar to the proof ofwx 13, Corollary 2.4 but for the convenience of the reader we state it briefly. Let V g RepŽ.S, R be of arbitrary infinite rank ␬ such that USNŽ.W s m for all subrepresenta- : ␸ g tions W V of at most countable rank and let End RŽ.V . By Lemma 4.11 we may write V as a ␸-invariant filtration V s D␣ g ␬␣W satisfying propertiesŽ. i ᎐ Živ . from Lemma 4.11. We define inductively a chain of i automorphisms ␺␤ Ž.i s 1,...,m of W␤ for ␤ - ␬ such that

~~m ␸ ° s ␺ iii␺ ° s ␺ W ␤ Ý ␤␤and W ␣ ␣ is1 for all ␣ - ␤ and i s 1,...,m. Ž.)~~

i Now suppose that ␺␣ are defined for all ␣ - ␤ and i s 1,...,m.If ␤ is a limit ordinal, then we know by Lemma 4.11 that W␤ s D␣ - ␤␣W and we ii i may put ␺␤ s D␣ - ␤␣␺ for all i s 1,...,m. Obviously all ␺␤satisfy Ž.) by induction hypothesis. Next let ␤ s ␣ q 1 be a successor ordinal. Then by Lemma 4.11 W␤ s W␣␣[ C for some subrepresentation C ␣; V of at ␲ ␲ most countable rank. Let 12and be the projections of W␤␣onto W Ž.␺ ° ␸ and C ␣ , respectively. Then C ␣ is an endomorphism of C ␣ ; hence ␾ s there exist automorphisms i Ž.i 1,...,m of C ␣ such that

~~m ␺ ° ␸ s ␾ Ž.Ci␣ Ý . is1 g ¨ g ¨ s ␸␲ For each element c C␣ we choose c W␣ such that c Ž.c 1 and i define ␺␤ Ž.i s 1,...,m on W␤␣␣s W [ C by~~

~~q ␺ 1 s ␺ 1 q ␾ q ¨ q ␺ i s ␺ i q ␾ Ž.x c ␤ x ␣ c 1 c and Ž.x c ␤␣x c i for i s 2,...,m.~~

i i Clearly all ␺␤ are homomorphisms of W␤␣and extensions of ␺ for s ¨ s s i 1,...,m since c 0 for c 0. Moreover, it is easy to check and also wxi completely similar to the proof of 13, Corollary 2.4 that all ␺␤ are ␺ ° automorphisms of W␤ and that their sum equals W ␤ . Finally we get AUTOMORPHISM GROUP 177

␸ s m ␺ i ␺ i s D ␺ i s Ýis1 taking ␣ - ␬␣for i 1,...,m which are automorphisms of V. ␴ ° Remark 4.13. Note that only one needs that C ␣ is a sum of two automorphisms for all ␣ in the proof of the above corollary to show that ␴ itself is a sum of two automorphisms.

THEOREM 4.14. Let R be a PID, and let S be a finite poset. Let g s [ / g V RepŽ.S, R such that V ig ␻ W ii, where 0 W Rep Ž.S, R has unit sum number less or equal to 3 for all i g ␻. Then USNŽ.V F 3.

Proof. Since all W i are non-trivial we know that V must have infinite rank. Now the proof is completely similar to the one given inw 13, Theorem 2.9x for free modules of countable rank. For the convenience of the reader ␸ g ␸ we state the arguments briefly. Let End RŽ.V and recall that is called an ␣ ␸ : [ g ␻ Ž.i -endomorphism if W ikk ) i W for all i ; ␤ ␸ : [iy1 g ␻ Ž.ii -endomorphism if W ikks1 W for all i ; ␸ : g ␻ Ž.iii d-endomorphism if W iiW for all i . Obviously ␸ can be written as the sum of an ␣-endomorphism ␩,a d-endomorphism ␦, and a ␤-endomorphism ␾. Since we have assumed

that each W i has unit sum number less or equal to 3 we may write the ␶ ␦ ␦ d-endomorphism as a sum of three d-automorphisms , 12, and . Now it is easy to see that ␶␾y1 is locally nilpotentŽ i.e., for each x g V there exists k g ގ such that xŽ␶␾y1 .k s 0. and thus Id q ␶␾y1 is an automorphism. Hence also ␶ q ␾ is an automorphism of V. Next we consider the ␣-endomorphism. Since, for any i - ␻, there is a - ␻ ␩ : [m minimal m such that W ijjsiq1 W there exists a strictly ascend- ing sequence s - - - иии 0 r012r r

~~y ␩ : [rsq 2 1 F - of integers having the property that W ijsjsiq1 W whenever r i ␩ ␩ rsq11. We define the mappings and 2by~~

~~W ␩ for r F i - r ␩ s i 2 t 2 tq1 W i 1 F - ½ 0 for r2 tq12i r tq2~~

~~and~~

~~0 for r F i - r ␩ s 2 t 2 tq1 W i 2 ␩ F - ½ W i for r2 tq12i r tq2 , 178 LUTZ STRUNGMANN¨~~

s ␩ ␩ where t 0, 1, 2, . . . . It follows immediately that 12and also are again ␣-endomorphisms and that their sum is ␩. Moreover, an easy calculation - - иии ␩ shows, using the definition and the sequence r01r above, that 1 ␩ ␣ ␩ and 2 are locally nilpotent. Thus we can write the -endomorphism as a sum of two locally nilpotent ␣-endomorphisms. Now it is easy to check wx␦ q ␩ ␦ q ␩ Žsee 13, Theorem 2.9. that 11and 12are automorphisms of V. Hence ␸ s ␶ q ␾ q ␦ q ␩ q ␦ q ␩ Ž.Ž.Ž.11 22 is a sum of three automorphisms of V and USNŽŽ.. Rep S, R F 3 follows.

An immediate corollary is the following theorem. s g THEOREM 4.15. Let USNŽ.R 2 and let V Rep1 Ž.R be of infinite r F rank such that V V1 is a direct sum of cyclic R-modules. Then USNŽ.V 3. r Proof. Since V V1 is a direct sum of cyclic R-modules it follows by the wx stacked bases theorem 4 that V and V1 have stacked bases. Hence V decomposes into an infinite direct sum of representations of the form Ž.R, rR for some r g R. It is easy to see that one may assume that V is of countable rank by applying Lemma 4.11 and Corollary 4.12. Note that one 1 can choose the C ␣ in the proof of Lemma 4.11 such that C␣␣rC is a direct r sum of cyclic R-modules since V V1 is a direct sum of cyclic R-modules. Therefore let us assume that V is of countable rank and that V and V1 have stacked bases. Now the statement follows by the above theorem since each representation Ž.R, rR has an endomorphism ring isomorphic to R and hence has unit sum number equal to 2.

~~REFERENCES~~

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