The Krull–Schmidt Property for Finitely Generated Modules over Valuation Domains

Journal of Algebra 237, 533᎐561 Ž2001. doi:10.1006rjabr.2000.8592, available online at http:rrwww.idealibrary.com on

The Krull᎐Schmidt Property for Finitely Generated Modules over Valuation Domains Paolo Zanardo Dipartimento di Matematica Pura e Applicata, Uni¨ ersita ` di Pado¨ a, Via Belzoni 7, 35131 Padua, Italy E-mail: [email protected] Communicated by Kent R. Fuller Received July 22, 1999

INTRODUCTION When we are dealing with the theory of modules over a commutative ring, a major question is whether these modules decompose uniquely into direct sums of indecomposable summands, up to isomorphism. Let R be a commutative ring. We say that an R-module M satisfies the Krull᎐Schmidt property if it decomposes in a unique way, up to isomorphism, in a direct sum of indecomposable summands. An R-module is said to be a KS module if it satisfies the Krull᎐Schmidt property. We shall be interested in the Krull᎐Schmidt property for finitely generated modules over valuation domains. First, however, it is useful to recall some known results on the existence of non-KS torsion-free finite rank modules over valuation domains. The failure of the Krull᎐Schmidt property for torsion-free modules of finite rank over ⺪ p Žthe integers localized at the prime p . has been folklore for many years. Of course, the source of inspiration was given by the classical examples of torsion-free non-KS abelian groups of finite rank Žsee wFx for various references .. A first unpublished example of a non-KS ⺪ p-module, due to Butler, goes back to the 1960s. Other results and examples were found by Arnold; for instance, see Example 2.15 of wAx. Warfield wWx proved in 1978 that if R is a nonhenselian DVR Žthat is, a rank-one discrete valuation ring., and the completion of R is not algebraic over R, then torsion-free R-modules of finite rank do not satisfy the Krull᎐Schmidt property. Warfield’s argument is the following: Evans wEx proved in 1973 that, for such a DVR R, there exist a finite R-algebra E 533 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.

534

PAOLO ZANARDO

and a finitely generated non-KS E-module M; the existence of a full embedding ⌽ of a suitable category of E-modules into Mod R, first proved by Brenner and Ringel wBRx, shows that ⌽ Ž M . is a torsion-free non-KS R-module of finite rank. Recently Goldsmith and May wGMx proved the failure of the Krull᎐ Schmidt property for finite rank torsion-free modules over valuation domains much more general than DVRs. Their investigation was carried on by May and Zanardo wMZx who obtained the following result: suppose that the valuation domain R contains a prime ideal J such that RrJ is not ˆ : L x G 6, where L is the field of fractions of RrJ henselian and that w L ˆ is the completion of L in the topology of the valuation; then the and L Krull᎐Schmidt property fails for torsion-free R-modules of finite rank. It is worth noting that all the constructions of non-KS torsion-free modules given by Butler and Arnold, and subsequently by Goldsmith, May, and Zanardo rely on theorems of realization of R-algebras as endomorphism rings of finite rank torsion-free R-modules, in the spirit of the celebrated finite rank Corner theorem in wCx. Also the above mentioned embedding of categories ⌽ used by Warfield is essentially based on Corner’s ideas. It appeared to be more difficult to settle the question of whether or not finitely generated R-modules satisfy the Krull᎐Schmidt property. Indeed, in the finitely generated case there is no suggestion from abelian group theory, since finitely generated abelian groups are KS ⺪-modules. Hints of the existence of valuation domains R which admit non-KS finitely generated modules followed from results by Zanardo wZx and Salce and Zanardo wSZx, where finitely generated modules with non-local endomorphism rings were found. Nevertheless, in wSZx it is proved that, for every valuation domain R, if a finitely generated R-module decomposes into a direct sum of two-generated modules, then this decomposition is unique, up to isomorphism. This result is independent of the endomorphism rings of the 2-generated summands. As an immediate corollary we get that 4-generated R-modules are KS modules. Along similar lines of research, in wZ1x natural classes C of indecomposable finitely generated R-modules were constructed and investigated; it was proved that, if a finitely generated R-module decomposes in a direct sum of modules in C , this decomposition is unique, up to isomorphism. A positive result in the direction of the Krull᎐Schmidt property was wVx and Siddoway wSix: if R is any proved independently by Vamos ´ Ž henselian local ring not necessarily a valuation domain. and M is a finitely generated R-module, then End R Ž M . is local. It then follows from Azumaya’s theorem that the Krull᎐Schmidt property is satisfied by finitely generated R-modules.

KRULL ᎐ SCHMIDT PROPERTY

535

Vamos was the first to show the failure of the Krull᎐Schmidt property ´ for finitely generated modules over a suitable Žnonhenselian . valuation domain ŽTheorem 20 of wVx.. Vamos’ idea was to obtain a non-KS finitely ´ generated module M over a suitable valuation domain R starting with a non-KS torsion-free V-module of finite rank A, where V is a nonhenselian DVR which is a proper quotient of R. Thus V ( RrI, for a suitable prime ideal I of R; we remark that I, as well as the V-module A, must satisfy some technical conditions Žsee the preliminary section for the precise statement of Vamos’ theorem.. The rank of A as a V-module turns out to ´ coincide with the minimal number of generators of the R-module M. Getting from A to M requires clever arguments and the use of a ‘‘transfer lemma’’ wV, Lemma 1x. We remark that Vamos’ theorem was proved in order to be applied to ´ Warfield’s examples of non-KS torsion-free modules in wWx. However we have stated it ŽTheorem 1.2. in a form suitable for applications to the more general examples constructed by Goldsmith, May, and Zanardo in wGM, MZx. The purpose of the present paper is to give a direct construction of finitely generated modules Žover suitable nonhenselian valuation domains. which do not satisfy the Krull᎐Schmidt property. We start with a suitable valuation domain R containing a nonzero prime ideal I such that RrI is not henselian. The idea is to define by generators and relations an indecomposable finitely generated R-module M, with non-local endomorphism ring and to use two suitable non-units ␾ 1 , ␾ 2 g End R Ž M ., such that ␾ 1 y ␾ 2 is a unit, to show that M is a direct summand of a non-KS finitely generated R-module. In the second section we give a rather general construction of R-modules M such that End R Ž M . is not local, by adapting the techniques used extensively in wZ, Z1x Žsee the Preliminaries .. The simple key lemma of the third section shows that M is a direct summand of a non-KS module if there exist two suitable non-units ␾ 1 , ␾ 2 g End R Ž M . such that ␾ 1 y ␾ 2 is a unit and ␾ i Ž M . \ M Ž i s 1, 2.. Then we establish conditions which R must satisfy in order for the key lemma to be applicable. In this way, we show, in Theorems 3.1 and 3.2, that if R contains a prime ideal I such that RrI ( ⺪ p , with p G 5, for all m G 3 there is an indecomposable m-generated R-module M, which is a direct summand of a non-KS finitely generated module. We exhibit other significant examples of valuation domains R, which admit finitely generated non-KS modules. Sometimes their behaviour may be peculiar: in Theorem 3.7 we show that there exist valuation domains R such that every direct sum of R-modules generated by F 4 elements satisfies the Krull᎐Schmidt property, but there

536

PAOLO ZANARDO

are indecomposable 5-generated R-modules which are direct summands of non-KS modules. Our approach in the construction of finitely generated non-KS modules presents some advantages with respect to that of Vamos. An evident fact is ´ the simplicity of the structure of our non-KS modules: the ingredients are an indecomposable R-module M defined by generators and relations, together with two homomorphic images ␾ 1Ž M ., ␾ 2 Ž M . of M, where ␾ 1 , ␾ 2 are specific endomorphisms of M. Vamos’ method leads to quite ´ unmanageable finitely generated non-KS modules. For a thorough comparison between our method and that of Vamos, we refer to the Remark in ´ our fourth section. However, it is worth noticing that our results do not cover all the cases for which Vamos’ theorem is applicable. ´

1. PRELIMINARIES For general facts about valuation domains and their modules we refer to the books by Schilling wSx and Fuchs and Salce wFSx. If not otherwise specified, in the following R will denote a valuation domain. The maximal ideal of R is denoted by P, and its field of fractions by Q. For any assigned ring T, UŽT . will be the set of the units of T. We shall make frequent use of the notion of henselian valuation domain and of henselization of a valuation domain R. We refer to the paper by Ribenboim wRx for definitions and basic results on this topic. We recall this useful result on henselizations Žsee, for instance, Theorem 4.11.11, p. 179, of wNx.: let R be a valuation domain of rank one; then the completion Rˆ of R contains a copy of the henselization of R. DEFINITION. An R-module M satisfies the Krull᎐Schmidt property if it decomposes in a unique way, up to isomorphism, as a direct sum of indecomposable summands. A KS module is a module which satisfies the Krull᎐Schmidt property. wVx and We shall be interested in finitely generated R-modules. Vamos ´ Siddoway wSix independently proved the following result, which relates the Krull᎐Schmidt property with the henselianity of the ring. Note that here R is not assumed to be a valuation domain. THEOREM 1.1 wV, Six. Let R be a henselian ring and let M be a finitely generated indecomposable R-module. Then End R Ž M . is local. Consequently, the Krull᎐Schmidt property holds for finitely generated R-modules. We shall give a proof of the above result in Proposition 3.6. Theorem 1.1 has a remarkable application to our context. Recall that in a finitely generated module M over a valuation domain the torsion part

KRULL ᎐ SCHMIDT PROPERTY

537

t Ž M . splits and Mrt Ž M . is free Žand thus automatically is a KS module.. Therefore, when dealing with finitely generated modules, we may confine ourselves to the torsion modules. Thus from Theorem 1.1, it easily follows that, if R is almost henselian Žwhich means that RrI is henselian for every ideal I / 0; see wVx., then the Krull᎐Schmidt property holds for finitely generated R-modules. Therefore, if we seek non-KS finitely generated R-modules, we must start from R which are not almost henselian. By wVx we know that R is not almost henselian if and only if there exists a prime ideal I such that RrI is not henselian. wVx had the idea of obtaining examples of non-KS finitely Vamos ´ generated modules Žover a suitable valuation domain. starting from the available examples of torsion-free modules of finite rank not satisfying the Krull᎐Schmidt property. Actually he invoked the result by Warfield wW, Corollary 3.1x mentioned in the Introduction. We shall write Vamos’ result wV, Theorem 20x in a slightly more general ´ form than his. The proof is the same as in wVx. THEOREM 1.2 wVx. I such that

Let R be a ¨ aluation domain containing a prime ideal

Ži. I is the radical of aI for a suitable 0 / a g I. Žii. Setting V s RrI, there exists a torsion-free V-module A of finite rank such that Ža. A does not satisfy the Krull᎐Schmidt property. Žb. End V Ž A. is a subring of the matrix ring Mn=nŽ V .. Then there exists a finitely generated R-module N not satisfying the Krull᎐Schmidt property. Moreo¨ er rk Ž A. is equal to the minimal number of generators of N. Indeed, from a non-unique decomposition of torsion-free V-modules A s B1 [ B2 s C1 [ C2 . where Bi \ C j , Vamos obtains a non-unique decomposition of finitely ´ generated R-modules N s X 1 [ X 2 s Y1 [ Y2 , where X i \ Yj and rk Ž Bi ., rk Ž C j . coincide with the minimal number of generators of X i , Yj , respectively. A typical valuation domain R to which the above result is applicable is R s ⺪ p q t⺡ww t xx, consisting of formal power series over ⺡, in the indeterminate t, with constant term in ⺪ p . In the above notation we have I s t⺡ww t xx, V s ⺪ p , and a s t.

538

PAOLO ZANARDO

Note that the torsion-free finite rank non-KS modules constructed by Warfield wWx, as well as those constructed by Butler, Arnold wAx, Goldsmith and May wGMx, and May and Zanardo wMZx, do satisfy condition Žb. of the above assertion. For the convenience of the reader we now recall some notions and results on finitely generated modules over a valuation domain Žsee wFSx for a general exposition.. Let M be a finitely generated R-module. There exists a finite-ascending chain of submodules 0 s M0 - M1 - ⭈⭈⭈ - Mny1 - Mn s M

Ž 1.

such that Ži. Every Mi is pure in Miq1. Žii. Miq1rMi is cyclic. Moreover we may choose Ž1. in such a way that, setting AnnŽ MirMiy1 ., 1 F i F n, we have

Ai s

Ann Ž M . s A1 F A 2 F ⭈⭈⭈ F A n . A pure-composition series for M is any chain Mi of M, as in Ž1., satisfying conditions Ži. and Žii. as above. The sequence of ideals A i s AnnŽ MirMiy1 . is called the annihilator sequence of M. Any two pure-composition series of M are isomorphic. Therefore M determines its annihilator sequence Žup to the order.. We denote by l Ž M . the minimal number of generators of M; l Ž M . s dim R r P Ž MrPM . is the common length of the pure-composition series of M. When we say that M is n-generated we mean that l Ž M . is exactly n. For an assigned valuation domain R, we denote by S a fixed maximal immediate extension of R. Recall that S is not in general determined as a ring, but it is determined as an R-module: S is the pure-injective envelope of the R-module R. For every ideal I of R, SrIS contains a copy of Ž RrI .ˆ, the completion of RrI in the topology of the ideals. Let u be a unit of S not in R; the breadth ideal B Ž u. of u is defined by B Ž u . s  a g R : u f R q aS 4 . The ideal I is the breadth ideal of a unit u g S _ R if and only if u q IS g Ž RrI .ˆ; SrIS. The units u1 , . . . , u n g S _ R are said to be u-independent over the ideal I of R if I s B Ž u i . for all i F n and a0 q Ý nis1 a i u i g IS with a0 , . . . , a n g R implies that a0 , . . . , a n g P Žsee wZx or wFSx.. The notion of u-independence was used in wZx to construct indecomposable finitely generated R-modules as follows.

KRULL ᎐ SCHMIDT PROPERTY

539

Assume that u1 , . . . , u n g S _ R are u-independent over the ideal I of R. Let us choose 0 / a g I and let A s aR. Let us consider the ideal J s Ž A :: I . s  r g R : rI - A4 Žthe symbol ‘‘- ’’ denotes proper inclusion of ideals.. We set J * s J _ A; by definition ry1A ) I for all r g J *, and one can check that F r g J * ry1A s I. Since B Ž u i . s I, by the definition of the breadth ideal for every i F n there exists a family of units of R  u ir : r g J *4 satisfying the condition u ir y u i g ry1AS,

᭙ r g J *.

ŽThe u ir q I form a Cauchy net with no limit in RrI and convergent to u q IS in Ž RrI .ˆ; SrIS.. As in wZx Žsee also wLx., we define by generators and relations a finitely generated R-module M s ² x 0 , x 1 , . . . , x n :, where the generators x i are subject to the conditions n

Ann Ž x i . s A;

ŽU .

rx 0 s r Ý u ir x i , ᭙ r g J *. is1

Note that the above relations are consistent, since u ir y u is g ry1A whenever r, s g J * and s divides r. Since u1 , . . . , u n are u-independent over I, the module M turns out to be indecomposable Ž n q 1.-generated, in view of Theorem 6 of wZx. Moreover the submodule B s ² x 1 , . . . , x n : s Rx 1 [ ⭈⭈⭈ [ Rx n is pure and essential in M Žsuch a B is said to be a basic submodule of M .. The annihilator sequence of M is given by A s ⭈⭈⭈ s A - J; this fact will be crucial for our discussion in Section 3.

2. NON-LOCAL ENDOMORPHISM RINGS Our first task is to give a somewhat standard construction of indecomposable finitely generated R-modules with non-local endomorphism rings. As observed in the preceding section, R cannot be almost henselian, so that there must exist a prime ideal I of R such that RrI is not henselian. THEOREM 2.1. Let R be a ¨ aluation domain, containing a prime ideal I / 0 such that there exists a polynomial f g Rw X x, satisfying the following properties: Ži.

f is monic, of degree n q 1 G 2, and the constant term of f is a unit

of R. Žii. The reduction f of f modulo I is irreducible in Ž RrI . w X x and f has a root in the completion Ž RrI .ˆ of RrI. Žiii. The reduction of f modulo P has two distinct roots in RrP.

540

PAOLO ZANARDO

In this situation, there exists an indecomposable Ž n q 1.-generated R-module M, such that its endomorphism ring is not local. Proof. Let RrI s D, and let f s X nq 1 q bn X n q ⭈⭈⭈ qb1 X q b 0 , bi g ˆ is a root of f, R. By hypothesis, b 0 g UŽ R .. As a consequence, if ␩ g D ˆ then ␩ g UŽ D .. Moreover, if a g R is a root of f modulo P, that is, f Ž a. ' 0 mod P, then a is necessarily a unit of R. ˆ. such that f Ž␩ . s 0. Then ␩ s u q IS for a suitable unit Fix ␩ g UŽ D u of S; recall that u q IS g Ž RrI .ˆ if and only if its breadth ideal B Ž u. coincides with I. Of course, f Ž␩ . s 0 is equivalent to f Ž u. g IS. We start by verifying that u, u 2 , . . . , u n are u-independent over I. We must show that I s B Ž u k . for 2 F k F n. Assume that s f I s B Ž u.; then a y u g sS for a suitable a g R. It follows that a k y u k s Ž a y u. hŽ u. g sS, whence s f B Ž u k .. We conclude that I G B Ž u k .. Let us now suppose, by way of contradiction, that s g I _ B Ž u k .. Then b y u k g sS for a suitable b g R implies that ␩ is a root of X k y b g Dw X x, where k F n. But the irreducible polynomial of ␩ is f, whose degree is n q 1, which is impossible. We conclude that I s B Ž u k ., as desired. Moreover a0 q Ý nis1 a i u i g IS implies that a0 , . . . , a n g I - P, since 1, ␩ , . . . , ␩ n are linˆ early independent over D. We now construct an indecomposable Ž n q 1.-generated R-module as described in the preceding section. Starting with the ideal I, choose A s aR and J, as in the preliminary section. Let u r , r g J * be a family of units of R satisfying the condition u r y u g ry1AS,

᭙ r g J *.

Then we have, for all i F n, u ir y u i g ry1AS,

᭙ r g J *.

We define by generators and relations the finitely generated R-module M s ² x 0 , x 1 , . . . , x n :, where the generators x i are subject to the conditions n

Ž *.

Ann Ž x i . s A;

rx 0 s r Ý u ri x i , ᭙ r g J *. is1

Since u, u 2 , . . . , u n are u-independent over I, the module M is Ž n q 1.generated indecomposable, by Theorem 6 of wZx. Our purpose is to show that End R Ž M . is not a local ring. Of course, any ␾ g End R Ž M . can be represented Žnot uniquely., by the Ž n q 1. = Ž n q 1. matrix T s w a i j x, 0 F i, j F n, where ␾ Ž x i . s Ý njs0 a i j x j . Actually, a matrix T represents an endomorphism ␾ of M if and only if its entries a i j satisfy

KRULL ᎐ SCHMIDT PROPERTY

541

the following conditions, for each m with 1 F m F n, n

Ž m.

a0 m y

n

Ý ai m u i q u m is1

ž

a00 y

Ý ai0 u i is1

/

g IS.

The above relations may be obtained directly from the conditions Ž*., using the linearity of ␾ , or otherwise by invoking Lemma 5 of wLx Žwhere different, more general notations are adopted.. For every t g R, let us consider the matrix Tt s w a i j x, whose entries are defined as a00 s t y b1 , a01 s yb 0 , a0 j s 0, a i0 s biq1 , a ii s t , a i , iq1 s b 0 , a i j s 0

᭙j ) 1

for 1 F i - n and j / 0, i , i q 1

a n0 s 1, a n n s t , a n j s 0,

᭙ j / 0, n.

Thus the matrix Tt has the form t y b1 b2 b3 Tt s ⭈⭈⭈ bn 1

yb 0 t 0 ⭈⭈⭈ ⭈⭈⭈ 0

0 b0 t ⭈⭈⭈ ⭈⭈⭈ ⭈⭈⭈

⭈⭈⭈ ⭈⭈⭈ b0 ⭈⭈⭈ ⭈⭈⭈ 0

0 0 ⭈⭈⭈ ⭈⭈⭈ b0 t

We now check that Tt represents an endomorphism ␾ t , by verifying that the conditions Žm. are satisfied for 1 F m F n. For m s 1 we may write the relation in the form n

a01 q u Ž a00 y a11 . y

Ý Ž ai1 q aiy1, 0 . u i y an0 u nq1 g IS is2

whence, by definition of Tt , n

yb 0 q u Ž t y b1 y t . y

Ý Ž 0 q bi . u i y u nq1 s yf Ž u . g IS, is2

as desired. For m ) 1, inserting the entries of Tt in Žm. we get ny1

0 y b 0 u my 1 y tu m q u m Ž t y b1 . y

Ý biq1 u iqm y u nqm is1

s yu

my 1

f Ž u . g IS.

542

PAOLO ZANARDO

Developing the determinant of Tt along the first column, we readily see that det Tt s h Ž t . s t nq 1 y b1 t n q b 0 b 2 t ny 1 y b 02 b 3 t ny 2 q ⭈⭈⭈ " b 0n . There is an easily checked relation between the polynomials hŽ t . and f Ž X .: namely f Ž X . s "X nq 1 hŽyb 0rX .rb 0n. Observe that the property Žiii. of f is equivalent to the existence of two units a1 , a2 of R such that a1 y a2 g UŽ R . and f Ž a1 . ' 0 ' f Ž a2 . mod P. Now set t 1 s yb 0ra1 , t 2 s yb 0ra2 , and let ␾ t i be the endomorphisms associated to the matrices Tt i Ž i s 1, 2.. Now t 1 y t 2 g UŽ R ., since b 0 and a1 y a2 are units; therefore the endomorphism ␾ t 1 y ␾ t 2 s Ž t 1 y t 2 .1 is invertible. Moreover det Tt i g P Ž i s 1, 2., since "a inq 1 hŽ t i .rb 0n s f Ž a i . ' 0 mod P. Therefore neither ␾ t 1 nor ␾ t 2 is a unit of End R Ž M .. We conclude that End R Ž M . is not local, as desired. EXAMPLE 2.2. In the notation of Theorem 2.1, assume that D s RrI is as follows: D is the domain determined by a rank-one valuation on ⺢ which extends the p-adic valuation on ⺡ Ž p / 2.. Let us consider the polynomial f Ž X . s X 2 y 1 q p g Rw X x. Then f s X 2 y 1 q p is irreducible in Dw X x ; ⺢w X x. By Hensel’s lemma f has a root in the compleˆ of D. Finally f Ž X . ' Ž X y 1.Ž X q 1., so that f has two distinct tion D roots modulo P Žnote that 1 k y1, since p / 2., Therefore f satisfies conditions Ži., Žii., and Žiii. of the preceding theorem. Let us remark that if g g Rw X x is any monic polynomial of degree G 3, then g is reducible in ⺢w X x and therefore g cannot satisfy the hypotheses of Theorem 2.1. In the notation of Theorem 2.1, it is clear that the existence of a polynomial f satisfying the requirements Ži., Žii., and Žiii. is ensured if there exists a monic irreducible polynomial g g Dw X x such that the constant term of g is a unit, g has two distinct roots modulo ᑧ, the ˆ of D. maximal ideal of D s RrI, and g has a root in the completion D The existence of such g seems to be a rather common phenomenon, as partially illustrated by the next result. PROPOSITION 2.3. Let D be a nonhenselian DVR, with field of quotients K, and suppose that the residue field Drᑧ is finite of characteristic ␹ / 2. Then for all n g ⺞ there exists a monic irreducible polynomial g g Dw X x of degree G n such that the constant term of g is a unit, g has two distinct roots ˆ of D. modulo ᑧ, and g has a root in the completion D Proof. We shall denote by D H the henselization of D Žsee wRx. and by K the field of fractions of D H. Since D has rank one, we may assume ˆ that D H ; D. H

KRULL ᎐ SCHMIDT PROPERTY

543

In view of condition Ž6. in wRx, since D is not henselian, there exists an irreducible polynomial f Ž X . g Dw X x such that f Ž X . ' X m Ž X y a.

mod ᑧ ,

where m ) 0 and a is a unit of D. There exists u g D H such that f Ž u. s 0 and u ' a mod ᑧ. Let < Drᑧ < s ␳ and choose a prime number q with q ) ␳ and q ) Ž m q 1. ¨ Ž a0 ., where a0 is the constant term of f Ž X .. Observe that there exists w g D H such that w q s u. Since D H is henselian, it suffices to show that the polynomial X q y u g D H w X x has a simple root modulo ᑧ D H . In fact, X q y a g Ž Rrᑧ .w X x is separable, since q ) ␹ . Moreover, there exists z g D such that z q ' a mod ᑧ: let r,s g ⺪ be such that rq q sŽ ␳ y 1. s 1; then q

a ' Ž ar . Ž as .

␳ y1

q

' Ž ar . ' z q

mod ᑧ .

We now distinguish two cases. We first assume that X q y u has no roots in K Ž u.. It is then known that q X y u is irreducible in K Ž u.. Let us verify that f Ž X q . is irreducible over K. In fact, f Ž X q . has degree q Ž m q 1. and f Ž w q . s f Ž u. s 0. Then f Ž X q . is a multiple of hŽ X ., the irreducible polynomial of w over K. On the other hand, K Ž w . : K s K Ž w . : K Ž u.

K Ž u . : K s q Ž m q 1. ,

since X q y u is the irreducible polynomial of w over K Ž u.. It follows that hŽ X . and f Ž X q . have the same degree, so that they must coincide Žthey are both monic.. In particular, f Ž X q . is irreducible. We now consider the second case, namely that X q y u has a root w in K Ž u.. We will reach a contradiction, showing that this second case cannot occur. Let us write m

f Ž X . s Ž X y u . Ł Ž X y ␤i . , is1

where the ␤i lie in K sep , the separable closure of K. Let us pick ␥ i g K sep such that ␥ iq s ␤i . Then we have qy1

fŽ Xq. s

m

Ł Ž X y ⑀ j w . Ł Ž X y ⑀ j␥ i . , js0

Ž 2.

is1

where ⑀ g K sep is a primitive qth root of 1. Now let ¨ * be a valuation of K sep extending the valuation ¨ of K. Let D* be the valuation domain of ¨ *. Note that ⑀ g D*, whence ¨ *Ž ⑀ . s 0. Moreover ¨ *Ž ␤i . ) 0 for all i.

544

PAOLO ZANARDO

In fact the ␤i all lie in D*, since D* is integrally closed. From f Ž X . ' X m Ž X y a. mod ᑧ, it follows that f Ž X . has a unique root, namely u, which is a unit of D*. Let hŽ X . g Dw X x be the irreducible polynomial of w over K. Since w g K Ž u., we must have deg hŽ X . s m q 1 s w K Ž u. : K x. Moreover hŽ X . divides f Ž X q . in Dw X x, since f Ž w q . s 0. Let b 0 g D be the constant term of hŽ X .. From Ž2. it follows that b 0 s ⑀ h w k Ł␥ ih i , for suitable positive integers h, k, and h i . From deg hŽ X . s m q 1 it follows that k q Ýh i s m q 1. Then we have ¨ Ž b0 . s

Ý ¨ * Ž ␥ i . h i s Ý ¨ * Ž ␤i . h irq F Ý ¨ * Ž ␤i . Ž m q 1. rq

s ¨ Ž a0 . Ž m q 1 . rq - 1.

Ž 3.

where the last strict inequality holds by our choice of q. But ¨ Ž b 0 . g ⺞, since D is a DVR, so that ¨ Ž b 0 . s 0. The relation Ž3. shows that this is possible only if h i s 0 for all i F m. We conclude that k s m q 1 and ⑀ h w mq 1 g D. Raising to the qth power we get b s u mq 1 g D, and therefore X mq 1 y b is the irreducible polynomial of u over K, whence f Ž X . s X mq 1 y b. This is an obvious contradiction, as one immediately sees by reducing modulo ᑧ. Thus, from the above arguments it follows that f Ž X q . is irreducible of degree Ž m q 1. q in K w X x, and it has a root w g K H . Let us now choose b g D such that a k b q k 0 mod ᑧ. Such a b exists since the characteristic ␹ / 2. In fact, for instance, if z q ' a mod ᑧ, then, setting b s yz, we have b q ' yz q ' ya k a mod ᑧ. Now the polynomial g Ž X . s f ŽŽ X q b . q . is irreducible in K w X x, and g Ž X . ' Ž X q b.

qm

Ž Ž X q b.

q

y a.

mod ᑧ .

ˆ > D H , namely w y b; moreover Then g Ž X . has a root which lies in D g Ž X . has two distinct roots modulo ᑧ, namely yb and z y b Ž z as above., and its constant term is congruent to b q m Ž b q y a. k 0 mod ᑧ. Therefore g Ž X . satisfies our requirements, and its degree q Ž m q 1. may be chosen arbitrarily large, as desired. 3. DIRECT CONSTRUCTION OF NON-KS FINITELY GENERATED MODULES Our strategy for obtaining finitely generated non-KS modules consists of using the non-local endomorphism ring of a suitable indecomposable finitely generated R-module. The following easy lemma, variations of which have been used in other contexts Žsee, e.g., wE, GMx., is crucial.

KRULL ᎐ SCHMIDT PROPERTY

545

KEY LEMMA. Let M be a finitely generated indecomposable R-module. Assume that there exist ␾ 1 , ␾ 2 g End Ž M . such that Yi s ␾ i Ž M . \ M and ␾ 1 y ␾ 2 is an isomorphism. Then there exists a finitely generated R-module Z which gi¨ es rise to a direct decomposition M [ Z ( Y1 [ Y2 not satisfying the Krull᎐Schmidt property. Proof. The composition of the homomorphisms M ª Y1 [ Y2 , z ¬ Ž ␾ 1 z, ␾ 2 z . and Y1 [ Y2 ª M, Ž z1 , z 2 . ¬ z1 y z 2 , is an automorphism of M. Then M [ Z ( Y1 [ Y2 , where Z is the kernel of the second map. This direct decomposition does not satisfy the Krull᎐Schmidt property. In fact M is indecomposable and M is isomorphic neither to the Yi Ž i s 1, 2. nor to any possible proper direct summand Y of the Yi , since in that case l Ž Y . - l Ž Yi . F l Ž M .. Our task is to describe a situation to which our key lemma is applicable. Under the hypotheses and with the notation of Theorem 2.1, suppose that R contains a prime ideal I / 0 such that the degree n q 1 of the corresponding polynomial f Ž X . is at least 3. We write f Ž X . s X nq 1 q bn X n q ⭈⭈⭈ qb1 X q b 0 . Let u be the fixed unit of S not in R such that B Ž u. s I and f Ž u. ' 0 modulo IS. Let M s ² x 0 , x 1 , . . . , x n : be the indecomposable Ž n q 1.-generated module, defined as in the proof of Theorem 2.1. Recall that the annihilator of M is a principal ideal A s aR, that the annihilator sequence of M is given by A s ⭈⭈⭈ s A - J, and that ² x 1 , . . . , x n : s Rx 1 [ ⭈⭈⭈ [ Rx n . We know that, for all t g R, the matrix t y b1 b2 b3 Tt s ⭈⭈⭈ bn 1

yb 0 t 0 ⭈⭈⭈ ⭈⭈⭈ 0

0 b0 t ⭈⭈⭈ ⭈⭈⭈ ⭈⭈⭈

⭈⭈⭈ ⭈⭈⭈ b0 ⭈⭈⭈ ⭈⭈⭈ 0

0 0 ⭈⭈⭈ ⭈⭈⭈ b0 t

represents an endomorphism ␾ t of M. We set y 0 s Ž t y b1 . x 0 y b 0 x 1 yi s biq1 x 0 q tx i q b 0 x iq1 , yn s x 0 q tx n . Then ␾ t Ž M . is generated by y 0 , y 1 , . . . , yn .

1FiFny1

546

PAOLO ZANARDO

From now on we will assume t to be a unit of R. Step 1. AnnŽ yi . s A, for 2 F i F n. Moreover C s ² y 2 , . . . , yny1 : s Ry 2 [ ⭈⭈⭈ [ Ryn . 1 i Ž n If 0 s ryn s r Ž x 0 q tx n ., then r g J and we have 0 s Ý ny is1 ru r x i q r u r q t . x n , which implies r g A. Assume now that 2 F i F n y 1. Then 0 s ryi s r Ž biq1 x 0 q tx i q b 0 x iq1 . implies that s s rbiq1 g J, whence rbiq1 x 0 s sÝ nis1 u is x i . We conclude that rbiq1 x 1 s 0 and r Ž biq1 q t . x i s 0. Since t is a unit of R, this is possible only if r g A, as desired. Suppose now that ␭2 y 2 q ⭈⭈⭈ q␭ n yn s 0, for suitable ␭ i g R. By the definition of the yi we obtain the following relation

ny1

0s

žÝ

␭ i biq1 q ␭ n x 0 q t ␭2 x 2 q Ž ␭2 b 0 q t ␭3 . x 3 q ⭈⭈⭈

/

is2

q Ž ␭ ny 1 b 0 q t ␭ n . x n .

Ž 4.

1 Ž . Set s s Ý ny is2 ␭ i biq1 q ␭ n . From 4 and the relations defining M we get

su s x 1 s 0, Ž su 2s q t ␭ 2 . x 2 s 0, Ž su sj q ␭ jy1 b 0 q t ␭ j . x j s 0

2 - j F n.

From the above relation one gets s g A, so that t g UŽ R . implies that ␭2 , . . . , ␭ n g A. We conclude that C s Ry 2 [ ⭈⭈⭈ [ Ryn , as desired. Step 2. C is a pure submodule of ␾ t Ž M .. Assume, by way of contradiction, that C is not pure in ␾ t Ž M .. Then there exist s g R, q g P, and z g ␾ t Ž M . such that sqz s sÝ nis2 a i yi / 0, for suitable elements a2 , . . . , a n g R not all lying in P. Since A is a principal ideal, we may choose a multiple ␳ of s such that ␳ f A and ␳ q g A. But then 0 s ␳ qz s ␳ Ý nis2 a i yi / 0, since ␳ a j y j / 0 when a j is a unit. We have thus reached the desired contradiction. We seek a sufficient condition to ensure that AnnŽ yi q C . ) A, for i s 0, 1. Let us introduce a notation: for 1 F i F n we set f i s bi X i q biy1 X iy1 q ⭈⭈⭈ qb1 X q b 0 . Step 3. In the above notation, let us suppose that there exist a unit t g R and a suitable s g J _ A such that n

gŽ t. s tn q

Ý t nyj Ž yb0 . jy1 Ž f Ž u s . y f j Ž u s . . ru sj s 0. js1

Then AnnŽ yi q C . ) A, for i s 0, 1.

Ž 5.

KRULL ᎐ SCHMIDT PROPERTY

547

Let ␭1 , . . . , ␭ n g R be such that Ý nis1 ␭ i yi s 0. By the definition of the yi , this relation holds for a ␭1 f A if there exists a suitable s g J _ A such that the following relations are satisfied Žcf. Ž4..,

␭1 t s su s , ␭1 b 0 q ␭2 t s su 2s , . . . ␭ ny1 b 0 q ␭ n t s su ns , ny1

Ý ␭i biq1 q ␭ n s ys. is1

In fact, in this case, necessarily ␭1 s su srt f A. For i G 2 we may calculate the ␭ i recursively from the first through the nth by application of the above equalities. One finds that iy1

␭i s s

j Ý tyjy1 u iyj s Ž yb 0 .

1 F i F n.

js0

Substituting ␭1 , . . . , ␭ n in the last equality, one checks easily that the result is consistent precisely when t satisfies relation Ž5.. We have thus seen that AnnŽ y 1 q C . ) A. It remains to prove also that AnnŽ y 0 q C . ) A: in fact, n

n

n

n

sy 0 s su s y 1 q s Ý u is yi s t ␭1 y 1 q s Ý u is yi s yt Ý ␭ i yi q s Ý u is yi g C is2

is2

is2

is2

and s f A. It is worth noticing that the coefficient b1 does not appear in the formula for g Ž t . in Ž5.. Step 4. Assume that relation Ž5. holds for suitable units t and u s . Then M is not isomorphic to ␾ t Ž M .. In view of Step 3, we have A i s AnnŽ yi q C . ) A, for i s 0, 1. Moreover C is pure in ␾ t Ž M ., by Step 2. It follows that AnnŽ ␾ t Ž M .rC . s A 0 l A1 ) A. Consequently, the annihilator sequence of ␾ t Ž M . has the form A s ⭈⭈⭈ s A - A 0 l A1 F B and it is certainly distinct from A s ⭈⭈⭈ s A - J, the annihilator sequence of M. Therefore M \ ␾ t Ž M .. Step 5. Assume that there exist two units t 1 , t 2 of R such that t 1 y t 2 is a unit and, for a suitable s g J _ A, g Ž t 1 . s 0 s g Ž t 2 ., where g Ž t . is as in Ž5.. Set Yi s ␾ t Ž M .. Then there exists a finitely generated R-module Z i which gives rise to a direct decomposition M [ Z ( Y1 [ Y2 not satisfying the Krull᎐Schmidt property. Note that ␾ t 1 y ␾ t 2 s Ž t 1 y t 2 .1 is a unit of EndŽ M .. By Step 4 we see that all the hypotheses of the key lemma are satisfied. The desired conclusion follows.

548

PAOLO ZANARDO

We shall now discuss the existence of valuation domains R to which the above arguments are applicable. In the next results we will make use of the following fact, which is a consequence of the Hilbert irreducibility theorem Žsee, for instance, Chapter 9 of Lang’s book wLax.: for all prime numbers p and for every polynomial X m q a my 1 X my1 q ⭈⭈⭈ qa1 X q a0 g ⺪ p w X x, there exist infinitely many values of ␭ g ⺪ such that the polynomial X m q a my1 X my 1 q ⭈⭈⭈ qŽ ␭ p q a1 . X q a0 is irreducible in ⺪ p w X x. Clearly the next two theorems may be joined into a single result, as is obvious from their statements. Nevertheless, we have thought it convenient to present them separately. Indeed, apart from the fact that the simpler case when M is 3-generated is of particular interest and merits special treatment Žsee the next section., the proof of Theorem 3.2 is valid only for n q 1 G 4. THEOREM 3.1. Let R be a ¨ aluation domain containing a prime ideal I / 0 such that RrI ( ⺪ p , where p G 5. There exists an irreducible 3-generated R-module M which is a direct summand of a non-KS finitely generated module. Proof. We may assume that ⺪ p ; R; note that P s pR. Let us consider the polynomial in Rw X x, f Ž X . s Ž X y 1 . Ž X q 1 . Ž X y 2 . q ␭ pX s X 3 y 2 X 2 q Ž ␭ p y 1 . X q 2, where ␭ g ⺪ is suitably chosen to render f Ž X . irreducible in ⺪ p w X x. It follows that f is irreducible modulo I, since RrI ( ⺪ p , so that f is also irreducible in Rw X x. Let us now fix a henselization R H of R. Recall that any maximal immediate extension S of R contains a copy of R H ; more generally, for every ideal B of R, SrBS contains a copy of Ž RrB . H . Let us then assume that Ž RrB . H ; SrBS, for every ideal B. Since X ' 1 is a simple root of f Ž X . ' Ž X y 1.Ž X q 1.Ž X y 2. mod pR, by Hensel’s lemma there exists an element u g S _ R such that f Ž u. s 0 and u ' 1 mod pS. Note that u q IS g Ž RrI . H ; SrIS. Let us verify that the breadth ideal of u coincides with I. In fact RrI ( ⺪ p is a DVR, whence Ž RrI . H ; Ž RrI .ˆ; SrIS. Therefore u q IS g Ž RrI .ˆ, which is equivalent to B Ž u. s I Žcf. the Preliminaries .. We have to fix a unit u s in the set U s  u r : r g J *4 . Note that u r ' 1 mod P, for all r g U ; therefore we may assume that u s s 1 g U , for a suitable s g J *.

KRULL ᎐ SCHMIDT PROPERTY

549

For n s 2 formula Ž5. becomes g Ž t . s t 2 q t Ž u 2s q b 2 u s . y b 0 u s s t 2 y t y 2 s Ž t q 1 . Ž t y 2 . . Thus g Ž t . has two roots in R, namely t 1 s 2 and t 2 s y1. Note that t 1 y t 2 is a unit of R since p / 3. We are now in the position to construct a 3-generated indecomposable module M as above. All the preceding arguments work in our situation, so that, from Step 5, we finally conclude that M is a direct summand of a non-KS module. THEOREM 3.2. Let R be a ¨ aluation domain containing a prime ideal I / 0 such that RrI ( ⺪ p , where p G 5. For all n G 3 there exists an irreducible Ž n q 1.-generated R-module M which is a direct summand of a non-KS finitely generated module. Proof. We may assume that ⺪ p ; R. We first verify that, for all n G 3, there exists a polynomial f Ž X . s X nq 1 q bn X n q ⭈⭈⭈ qb1 X q b 0 g R w X x , satisfying the following requirements: Ža. f Ž X . is irreducible in Rw X x and X ' 1 is a simple root of f modulo P s pR. Žb. Let g Ž t . be the polynomial in the formula Ž5., where we set u s s 1; then g Žy1. s 0 and g Ž t . has another root a g ⺪ ; R such that a, a q 1 k 0 mod p Žthe choice of a will depend on p .. We will write f Ž X . in the form f Ž X . s Ž X y 1 . Ž X n q c 2 X 2 q c1 X q 1 . q ␭ pX , where the c1 , c 2 g ⺪ p are suitably chosen, and, once the c i are fixed, ␭ is an integer such that f Ž X . is irreducible in ⺪ p w X x and, hence, also irreducible in Rw X x. In our notation we get b 0 s y1, b 2 s c1 y c 2 , b 3 s c2 , b4 s ⭈⭈⭈ s bny 1 s 0, and bn s y1 Žor bn s y1 q c 2 just in the case when n s 3.. If in Ž5. we set u s s 1 we get g Ž t . s t n q t ny 1 Ž 1 q bn q bny1 q ⭈⭈⭈ qb 2 . y b 0 t ny 2 Ž 1 q bn q ⭈⭈⭈ qb 3 . q ⭈⭈⭈ q Ž yb 0 .

ny 2

t Ž 1 q bn . q Ž yb 0 .

ny1

550

PAOLO ZANARDO

whence we readily obtain g Ž t . s t n q c1 t ny 1 q c 2 t ny 2 q 1. We have to take the c i in such a way that g Žy1. s 0 s g Ž a., for a suitable a g ⺪. The integer a must be chosen in such a way that a, a q 1 f p⺪ Žwhence a, a q 1 are units of R . and that X ' 1 is a simple root of f modulo P s pR. This last condition causes some further work, but it is crucial to allow us to apply Hensel’s lemma to the polynomial f Ž X .. It is clearly equivalent to ask that 2 q c1 q c 2 k 0 mod pR. For the moment, let us just assume that a is such that a, a q 1 f p⺪, this is possible since p G 5. Thus the conditions g Ž a. s 0 s g Žy1. reduce to the linear system in the indeterminates c1 , c 2 , a n q 1 q c1 a ny 1 q c 2 a ny 2 s 0 n

ny1 ny2 q c 2 Ž y1 . s 0. Ž y1. q 1 q c1 Ž y1.

The above system has a solution in ⺪ p ; R: in fact, its determinant is Žya. ny 2 Žya y 1. and hence a unit of ⺪ p by our assumption on a. However, there is no guarantee yet that 2 q c1 q c 2 k 0 mod pR. Here a suitable choice of a plays its role. We must distinguish cases according to the parity of n. Assume first that n is odd, and solve the linear system. One finds that c1 s c 2 s y

an q 1 a ny2 Ž a q 1 .

.

After easy calculations, we see that 2 q c1 q c 2 k 0 mod pR is equivalent to a n y a ny 1 y a ny 2 q 1 k 0

mod p.

Ž 6.

Since a py 1 ' 1 mod p, Ž6. is equivalent to a ␤ y a␥ y a ␦ q 1 k 0

mod p.

where ␤ , ␥ , and ␦ are the positive remainders of the division by p y 1 of n, n y 1, and n y 2, respectively. It is important to note that ␤ , ␥ , and ␦ are pairwise distinct, since, by hypothesis, p y 1 G 4. Let us now consider the polynomial h Ž X . s X ␤ y X ␥ y X ␦ q 1.

KRULL ᎐ SCHMIDT PROPERTY

551

hŽ X . has degree - p y 1 and we readily see that hŽ1. s hŽy1. s 0, since ␤ and ␦ are odd and ␥ is even. In particular, hŽ X . has fewer than p y 1 roots modulo p. It follows that there exists a g ⺪ _ p⺪ such that hŽ a. k 0 mod p; necessarily a k "1, so that a q 1 f p⺪. For such a choice of a the relation Ž6. holds, and we are done. In the case when n is even, the discussion follows the same track as for when n is odd. Here we get c1 s y

a n y 2 a n y2 q 1 a ny2 Ž a q 1 .

,

c 2 s c1 y 2

and the relation for a, corresponding to Ž6., gets the form a n y 2 a ny 2 q 1 k 0

mod p.

Again we may find a such that 2 q c1 q c 2 k 0 mod pR. Now fix a henselization R H of R and a maximal immediate extension S of R, containing R H . Since by construction f Ž X . has 1 as a simple root modulo P, there exists an element u g S _ R such that f Ž u. s 0 and u ' 1 mod PS. Note that the breadth ideal of u coincides with I Žsee the proof of Theorem 3.1.. As in the preceding proof, we have to fix a unit u s and we may choose u s s 1. By construction, the polynomial g Ž t . has two nonzero roots t 1 , t 2 g R, with t 1 y t 2 k 0 mod P, namely t 1 s a, t 2 s y1. In view of Step 5, we can construct a Ž n q 1.-generated indecomposable module M which is a direct summand of a non-KS finitely generated module. Remark. It is not possible to apply the method described above for constructing non-KS R-modules, when the residue field RrP is either F2 or F3 , the fields with 2 or 3 elements, respectively. In fact, it is crucial in the construction that g Ž t . have two roots t 1 , t 2 which are distinct and nonzero modulo P. This is clearly impossible if RrP ( F2 . The problem is subtler when RrP ( F3 . For instance, let us try to construct a 3-generated module as in the proof of Theorem 3.1. The argument does not work for the following reason: assume that f Ž X . ' Ž X y 1.Ž X 2 q a1 X q a0 . modulo P Žwhere RrP ( F3 . and that X ' 1 is a simple root of f mod P, so that we are able to find u g R H such that f Ž u. s 0 and u ' 1 mod PS. Then u s y u g sy1AS - PS implies that u s ' 1 mod P. From the formula Ž5. for g Ž t . we obtain g Ž t . ' X 2 q a1 X q a0

mod P.

Since X ' 1 is a simple root of f mod P, it cannot be a root of g Ž t . mod P. Since 1, y1 are all the units in F3 , we conclude that g Ž t . cannot have two roots t 1 , t 2 which are nonzero and distinct modulo P.

552

PAOLO ZANARDO

A longer but not difficult examination shows also that the technique used in Theorem 3.2 to construct Ž n q 1.-generated modules does not work for p s 3. Nevertheless, it is quite possible that the characteristic of RrP be either 2 or 3; see Example 3.4. Let us give other examples of valuation domains R to which our method of constructing non-KS modules is applicable. EXAMPLE 3.3. Let R be a valuation domain which contains a nonzero prime ideal I such that D s RrI ( ⺓w y xŽ y . , where y is an indeterminate over ⺓. For all n G 2, let us consider the polynomial

␾ Ž X . s Ž X y 1. Ž X y 2. Ž X q 1.

ny 1

s X nq 1 q bn X n q ⭈⭈⭈ qb 2 X 2 q cX q b 0 , where the bi and c are in ⺪. Let us set b1 s c q y and let f Ž X . s X nq 1 q bn X n q ⭈⭈⭈ qb1 X q b 0 g R w X x . Note that f Ž X . is irreducible modulo I, since f Ž X ., regarded as an element of ⺓w X, y x, has degree 1 with respect to y. Let us fix a henselization R H of R and a maximal immediate extension S of R, containing R H. Since f Ž X . ' ␾ Ž X . mod P, we see that X ' 1 is a simple root of f modulo P. Then there exists an element u g S _ R such that f Ž u. s 0 and u ' 1 mod PS. As in the proof of Theorem 3.1, we see that the breadth ideal of u coincides with I. As above, we may choose u s s 1, and the relation corresponding to Ž3. becomes g Ž t . s t n q t ny 1 Ž 1 q bn q ⭈⭈⭈ b 2 . y ⭈⭈⭈ q t Ž yb 0 .

ny 2

Ž 1 q bn . q Ž yb0 .

ny1

.

Since g Ž t . g ⺓w t x, it splits completely in ⺓ ; R. However, to apply Step 5, we need g Ž t . to have two nonzero roots t 1 , t 2 g R, with t 1 y t 2 k 0 mod P. Let us first observe that b 0 s 2, so that g Ž t . has no zero roots. It is then enough to verify that there are no ␣ g ⺓ such that g Ž t . s Ž t q ␣ . n : in fact, if this were true, we would obtain ␣ n s Žyb 0 . ny 1 s Žy2. ny 1 and 1 q bn q ⭈⭈⭈ qb 2 s n ␣ , whence ␣ g ⺡, which is impossible. Thus, as above, we can construct an Ž n q 1.-generated indecomposable module M which is a direct summand of a non-KS module.

KRULL ᎐ SCHMIDT PROPERTY

553

EXAMPLE 3.4. Let K be any field of cardinality G 4 and let R be a valuation domain which contains a nonzero prime ideal I such that D s RrI ( K w y xŽ y . , where y is an indeterminate over K. Then there is a 3-generated indecomposable R-module which gives rise to a non-KS direct sum. As we have repeatedly seen above, it is enough to choose f Ž X . g Rw X x and the corresponding g Ž t . suitably. Let us pick ␣ , ␤ g R such that ␣ , ␤ and 1 are nonzero and pairwise distinct modulo P; this is possible since K s RrP has at least 4 elements. Let us set f Ž X . s Ž X y 1 . Ž X y ␣ . Ž X y ␤ . q yX . Then f Ž X . is irreducible Žsee Example 3.3., and X ' 1 is a simple root modulo P. For u s s 1 we get gŽ t. s Ž t y ␣ . Ž t y ␤ . . The desired conclusion follows. Note that there is no restriction on the characteristic of the residue field K. EXAMPLE 3.5. Let L ; ⺓ be the union of all the radical extensions of ⺡. Let us extend the p-adic valuation on ⺡ to a rank-one valuation of L, and let D ; L be the corresponding valuation domain. Let R be a valuation domain such that RrI ( D for a suitable prime ideal I of R. Note that the residue field RrP is infinite. In fact, for all prime integer q / p, the polynomial X q y 1 has q distinct roots in L, and these roots all lie in D ( RrI. Then X q y 1 splits completely also in RrP, and it is separable over RrP; hence its roots are distinct. We conclude that RrP is infinite. There exist irreducible monic polynomials in ⺪w X x of arbitrarily large degree with simple roots mod p and non-solvable Galois group Žto find examples which avoid the use of deep results, one may adapt the arguments in the book of Jacobson wJ, pp. 107᎐108x.. Therefore w LH : L x s ⬁. Note that every polynomial in Rw X x of degree F 4 splits completely modulo I. Therefore we cannot follow the above-described method to construct examples of non-KS direct decompositions M [ Z ( Y1 [ Y2 , with l Ž M . F 4. Actually, we shall prove a sharper result in Theorem 3.7. Nevertheless, we can ask that l Ž M . s 5. It suffices to consider the polynomial 5

4

f Ž X . s Ž X q 1 . y q Ž X q 1 . q qp. where q ) p is a prime number such that p does not divide q y 1. Then the constant term b 0 of f is a unit of R. Eisenstein’s criterion shows that f Ž X . is irreducible in ⺪w X x and an inspection of its graph reveals that it has exactly three real roots. By a classical result, the Galois group of f is

554

PAOLO ZANARDO

S5 Žfor instance, see wJx.. Then f has no roots in L and so it is irreducible in Lw X x. We have f Ž X . ' Ž X q 1 y q .Ž X q 1. 4 mod P; since q y 1 k y1 mod P, by Hensel’s lemma f has a root u g R H with u ' q y 1 mod P. As usual, we must not forget to check that B Ž u. s I. This fact can be proved as in Theorem 3.1, since the valuation on D ( RrI has rank one. In the notation of Step 3, let us take u s s q y 1. The corresponding polynomial g Ž t . as in Ž5. has degree 4; hence it splits completely in R. Since b 0 Ž q y 1. is a unit in R, the roots of g Ž t . are all units of R. Moreover, arguing as in Example 3.3, we see that g Ž t . has at least two distinct roots t 1 , t 2 g R. Thus we may construct an indecomposable 5-generated R-module M which is a summand of a non-Krull᎐Schmidt direct decomposition. Let R be a local ring Žnot necessarily a valuation domain. with maximal ideal P. Following Azumaya wAzx, we say that Hensel’s lemma holds for a polynomial f Ž X . g Rw X x if, whenever f ' g 0 h 0 mod P, where g 0 , h 0 are monic polynomials coprime modulo P, there exist f, g g Rw X x such that f s gh, g ' g 0 , and h ' h 0 mod P. The following result is a consequence of the proofs of wAz, Theorems 20 and 22x and wSi, Lemma 3 and Theorem 4x Žsee also wV, Lemmas 12 and 13x.. For the convenience of the reader, we include a more direct and somewhat shorter proof. We remark that Theorem 1.1 is a particular case of the next proposition. PROPOSITION 3.6. Let R be a local ring such that Hensel’s lemma holds for polynomials in Rw X x of degree F n. Then e¨ ery indecomposable R-module M generated by F n elements has a local endomorphism ring. Proof. Let us assume that E s End R Ž M . is not a local ring. We shall prove that M is decomposable, which yields the desired conclusion. There exists ␾ g E such that ␾ and 1 q ␾ are both non-units of E. Let f Ž X . s X m q a my 1 X my1 q ⭈⭈⭈ qa0 , a i g R be a minimal polynomial of ␾ , that is, a monic polynomial of minimal degree which annihilates ␾ . Since M is generated by F n elements, we know that any element of E is integral of degree F n over R; in particular we have m F n. Since ␾ is not a unit and f Ž ␾ . s 0, necessarily a0 g P, the maximal ideal of R. However, not all the a i lie in P, since otherwise f Ž X y 1., which is a minimal polynomial of 1 q ␾ , has a unit as a constant term, but then 1 q ␾ is invertible in E, against our assumption. So let 0 - k F m y 1 be minimal such that a k f P. We obtain the relation, f Ž X . ' X k Ž X my k q ⭈⭈⭈ qa k .

mod P.

KRULL ᎐ SCHMIDT PROPERTY

555

By hypothesis, the Hensel lemma holds for f, and therefore we may write f s f 1 f 2 , where f 1 ' X k and f 2 ' X my k q ⭈⭈⭈ qa k modulo P. Moreover, there exist polynomials g 1 , g 2 g Rw X x such that f1 g1 q f2 g2 ' 1

mod P ,

the degrees of g 1 , g 2 are strictly less than the degrees of f 2 , f 1 , respectively, and their leading coefficients are units of R. Set h i s f i g i Ž i s 1, 2.. We may write the preceding relation in the form h1 q h 2 s 1 q ␳ , where ␳ is a suitable polynomial in P w X x. Let us evaluate the above relation for X s ␾ . Note that, for i s 1, 2, h i Ž ␾ . / 0, since the degrees of the h i s f i g i are less than the degree of f, a minimal polynomial of ␾ , and their leading coefficients are units of R. Moreover h1Ž ␾ . h 2 Ž ␾ . s g 1Ž ␾ . g 2 Ž ␾ . f Ž ␾ . s 0. Since ␳ g P w X x, it is well known that 1 q ␳ Ž ␾ . is invertible in Rw ␾ x ; E. Let us set e i s h i Ž ␾ .Ž1 q ␳ Ž ␾ ..y1 g E. Then we have e1 q e2 s 1 and e1 e2 s 0. In conclusion, M s e1 M [ e2 M is the required nontrivial direct decomposition of M. The next easy lemma is well known; its proof is straightforward. We shall need it in Theorem 3.7. LEMMA. Let R be a ¨ aluation ring, not necessarily a domain, and let f g Rw X x. Let us suppose that R contains a prime ideal I such that Ža. Žb.

Hensel’s lemma holds for the reduction f of f in Ž RrI .w X x. The localization R I is henselian.

Then Hensel’s lemma holds for f. Our Example 3.5 is the core of the following result. THEOREM 3.7. There exist ¨ aluation domains R such that Ži. Any indecomposable R-module generated by at most 4 elements has a local endomorphism ring. Žii. The finitely generated R-modules do not satisfy the Krull᎐Schmidt property; more precisely, there exist 5-generated indecomposable R-modules which are summands of non-KS direct decompositions. Proof. Let L, D be as in the preceding Example 3.5. Let R be any valuation domain containing a prime ideal I / 0 such that RrI ( D and the localization R I is an almost henselian valuation domain Žsee the preliminaries.. For instance, it is enough to ask that the valuation on R I has rank one; see wVx Žto fix the ideas, one may consider R s D q YLww Y xx, where Y is an indeterminate over L; here I s YLww Y xx.. The assertion Žii.

556

PAOLO ZANARDO

is now immediate from the arguments in Example 3.5. We must verify assertion Ži.. Let M be an indecomposable R-module with l Ž M . F 4. We may assume that M has nonzero annihilator A, so that it is canonically an RrA-module. In view of Proposition 3.6, it suffices to prove that Hensel’s lemma holds for every monic polynomial f Ž X . g Ž RrA.w X x of degree F 4. If A ) I, then RrA is isomorphic to a proper quotient of D. Since the valuation on D has rank one, D is almost henselian. Thus RrA is henselian and Hensel’s lemma holds for any polynomial. Assume now that A F I, and set ᑣ s IrA. Since degŽ f . F 4 and D ( Ž RrA.rᑣ, it is clear that f splits completely modulo ᑣ. As a consequence, Hensel’s lemma holds for the reduction of f modulo ᑣ. Moreover, the localization Ž RrA. ᑣ is henselian, since it is isomorphic to a proper quotient of the almost henselian valuation domain R I . From the preceding lemma it follows that the Hensel lemma holds for f, as desired.

4. THE CASE OF 3-GENERATED MODULES It is natural to seek information on the smallest possible positive integer m such that there exists a non-KS m-generated R-module for a suitable choice of the valuation domain R. We recall that an easy consequence of Theorem 12 of wSZx is that every R-module generated by F 4 elements satisfies the Krull᎐Schmidt property. On the other hand, in view of Theorem 3.1, we know that there exist valuation domains R which admit non-KS modules N with l Ž N . F 6. In this final section we want to show that in fact l Ž N . s 6; more precisely, in the notation of Section 3, we have N s M [ Z ( Y1 [ Y2 , where M, Z, Y1 , and Y2 are all 3-generated indecomposable. In other words, the method developed in the preceding section can issue non-KS modules which are 6-generated but not 5-generated non-KS modules. We use the same notations as in Section 3 for the case n s 2. Thus f Ž X . s X 3 q b 2 X 2 q b1 X q b 0 , f Ž u. ' 0 modulo I, M s ² x 0 , x 1 , x 2 : is the indecomposable 3-generated R-module defined by Ann x i s A s aR, and

Ž *.

rx 0 s ru r x 1 q ru 2r x 2

᭙ r g J *.

KRULL ᎐ SCHMIDT PROPERTY

557

For every unit t g R, we consider the endomorphism ␾ t of M represented by the matrix t y b1 Tt s b2 1

yb 0 t 0

0 b0 . t

Our aim is to prove that the submodule ␾ t Ž M . of M is 3-generated indecomposable, for every t g UŽ R .. In particular, the modules Y1 , Y2 in Step 5 are 3-generated indecomposable, as well as M, by construction. Consequently, Z also turns out to be 3-generated indecomposable. Otherwise, Z must have a cyclic direct summand, say C. Then C has a local endomorphism ring, and therefore either Y1 or Y2 has a direct summand isomorphic to C, which is impossible. Let us set ␾ t Ž M . s Y. Then Y s ² y 0 , y 1 , y 2 :, where y 0 s Ž t y b1 . x 0 y b 0 x 1 , y 1 s b 2 x 0 q tx 1 q b 0 x 2 , y 2 s x 0 q tx 2 . LEMMA 4.1. The determinant of Tt does not lie in I. Proof. We have det Tt s h Ž t . s t 3 y b1 t 2 q b 0 b 2 t y b 02 . Then

Ž t 3 y b1 t 2 q b0 b2 t y b02 . b0rt 3 s f Ž yb0rt . k 0

mod I,

since yb 0rt g R and f Ž X . is irreducible modulo I. LEMMA 4.2. Y is 3-generated. Proof. It suffices to show that y 0 , y 1 , and y 2 are independent modulo P. Let Ýri yi s 0. Since J s AnnŽ x 0 q Rx 1 [ Rx 2 ., from the definitions of the yi we get the linear system

Ž r 0 , r1 , r 2 . Tt ' Ž 0, 0, 0 .

mod J.

Ž 7.

Recall now that J F I, since I is a prime ideal and r g J implies A ) rI, which is possible only if r g I. Multiplying both members of Ž7. by the adjoint of Tt , we get h Ž t . Ž r 0 , r 1 , r 2 . ' Ž 0, 0, 0 .

mod I,

whence r 0 , r 1 , r 2 g I, since hŽ t . f I in view of Lemma 4.1.

558

PAOLO ZANARDO

PROPOSITION 4.3. Y is an indecomposable module. Proof. By way of contradiction, we assume that Y s Rz 0 [ ² z1 , z 2 :, where z i s Ý j a i j y j , 0 F i, j F 2, for a suitable matrix L s Ž a i j . invertible over R. We distinguish two cases. Case 1. Ann z 0 G J. Then, for all r g J *, we have r Ý2js0 a0 j y j s 0, which is equivalent to the relation r Ž a00 Ž t y b1 . q a01 b 2 q a02 . x 0 q r Ž ya00 b 0 q ta01 . x 1 q r Ž b 0 a01 q ta02 . x 2 s 0.

Ž 8.

From Ž8. and the defining relations Ž*. of M, we get u r Ž a00 Ž t y b1 . q a01 b 2 q a02 . y a00 b 0 q ta01 g ry1A, u 2r Ž a00 Ž t y b1 . q a01 b 2 q a02 . q b 0 a01 q ta02 g ry1A. From the above relations, since u y u r g ry1AS, for all r g J *, we obtain the following relations u Ž a00 Ž t y b1 . q a01 b 2 q a02 . y a00 b 0 q ta01 g

F ry1AS s IS,

rgJ *

u 2 Ž a00 Ž t y b1 . q a01 b 2 q a02 . q b 0 a01 q ta02 g IS. Since u, u 2 are u-independent modulo I, we easily get the linear system

Ž a00 , a01 , a02 . Tt ' Ž 0, 0, 0 .

mod I ;

as in Lemma 4.2 we conclude that a00 , a01 , a02 g I, whence det L g P, a contradiction. Case 2.

J ) Ann z 0 G A s aR.

Let us choose s g J such that sR ) Ann z 0 . Note first that ars f I: in fact, sy1A s sy1 aR ) I, since s g J *. In particular, since I is a prime ideal, we have Ž sra. I s Ž ars. I s I. Now, for every r g J *, let

Ž b 0r , b1r , b 2r . s Ž y1, u r , u 2r . Ly1 . By direct substitution we see that 2

r Ý bir z i s yry 0 q ru r y 1 q ru 2r y 2 s 0. is0

KRULL ᎐ SCHMIDT PROPERTY

559

Since Rz 0 is a direct summand of Y, we get b 0r g ry1Ann z 0 F ry1 sR, whence yc0 q c1 u r q c 2 u 2r g ry1 Ž sra. A, where the c i are the entries of the first column of Ly1 . Observe now that, since sra g Q _ R, we have u y u r g ry1AS - ry1 Ž sra. AS. Thus, since r g J * was arbitrary, we obtain yc0 q c1 u q c 2 u 2 g sra

F ry1AS s Ž sra. IS s IS.

rgJ *

But u, u 2 are u-independent over I, whence necessarily c 0 , c1 , c 2 g P. We conclude that det L g P, which is impossible. Thus in both cases we have obtained a contradiction. Therefore Y must be indecomposable, as desired. We remark that the proof of Case 2 in the above proposition is an adaptation of that of Theorem 6 in wZx. Remark. It appears desirable to give a detailed comparison between our method of constructing non-KS finitely generated R-modules and that wV, Theorem 1.2x. It is clear that our direct construction presents of Vamos ´ some advantages. We have seen that there is a sort of standard definition of M and of the endomorphisms ␾ 1 , ␾ 2 which makes the key lemma work. Thus the objects involved in our construction are rather easy to handle. The finitely generated non-KS R-modules constructed by Vamos are less ´ manageable: they are obtained in a rather clever way by torsion-free non-KS V-modules of finite rank Ž V s RrI .. Moreover, it is worth noting that the non-KS torsion-free V-modules employed in the construction are far from being easy. As already noted in the Introduction, their constructions by Butler and Arnold and subsequently by Goldsmith, May, and Zanardo all rely on Corner-type theorems of realization of R-algebras as endomorphism rings of finite rank torsion-free R-modules. Also the embedding of categories ⌽ used by Warfield is essentially based on Corner’s ideas. A second advantage arises if we are interested in the minimal possible number of generators of a non-KS module. We are able to construct directly a 6-generated non-KS module. Vamos’ method gives, at best, a ´ 10-generated non-KS module, since 10 is the minimal known rank of a non-KS torsion-free module. We can say more: Arnold has proved that a torsion-free ⺪ p-module of rank F 7 has the Krull᎐Schmidt property. Thus, when RrI s V s ⺪ p , Vamos’ technique is intrinsically unable to get ´ a 6-generated R-module.

560

PAOLO ZANARDO

Finally, Vamos’ argument also needs the prime ideal I to satisfy the ´ technical condition Ži. of Theorem 1.2, which is equivalent to requiring that I not be the union of a proper ascending chain of prime ideals. This requirement is crucial to make the proof work. We do not have any sort of limitation on I. The other technical hypothesis in Vamos’ theorem, namely ´ condition Žb. on End V Ž A., is also essential for the argument, but it does not seem to be a substantial limitation. Indeed, to our knowledge, all known examples of non-KS torsion-free V-modules A Že.g., those in wA, W, GM, and MZx. satisfy the property End V Ž A. ; Mn=nŽ V .. On the other hand, it is important to notice that Vamos’ theorem is not ´ covered by our results. For instance, the hypothesis RrI ( ⺪ 3 is acceptable in Vamos’ theorem. Moreover, it is applicable whenever V s RrI is ´ ˆ : L x G 6, due to the results in wGMx Žhere L is the field of such that w L ˆ is its completion.. We do not know if our method is fractions of V and L applicable to a valuation domain R such that V s RrI satisfies such a general condition. We end the paper with two questions. Ž1. Do 5-generated modules, over any valuation domain, satisfy the Krull᎐Schmidt property? We have seen in this section that a strict application of our method is unable to give an answer to Ž1.. Ž2. Are there valuation domains which are not almost henselian and whose finitely generated modules satisfy the Krull᎐Schmidt property? Possible candidates could be the valuation domains of type Žy. constructed by Vamos in wVx: they are not almost henselian valuation domains ´ ˜ of D has rank 2 as a D such that any maximal immediate extension D D-module. Vamos has proved that torsion-free finite rank D-modules do ´ satisfy the Krull᎐Schmidt property.

REFERENCES wAx wAzx wBRx wCx wEx

D. Arnold, ‘‘Finite Rank Torsion Free Abelian Groups and Rings,’’ Springer-Verlag, BerlinrHeidelbergrNew York, 1982. G. Azumaya, On maximally central algebras, Nagoya Math. J. 2 Ž1950., 119᎐150. S. Brenner and C. Ringel, Pathological modules over tame rings, J. London Math. Soc. Ž 2 . 14 Ž1976., 207᎐215. A. L. S. Corner, Every countable reduced torsion-free ring is an endomorphism ring of a countable reduced torsion-free Abelian group, Proc. London Math. Soc. Ž 3 . 13 Ž1963., 687᎐710. E. G. Evans, Krull᎐Schmidt and cancellation over local rings, Pacific J. Math. 46 Ž1973., 115᎐121.

KRULL ᎐ SCHMIDT PROPERTY

wFx

561

L. Fuchs, ‘‘Infinite Abelian Groups,’’ Vol. II, Academic Press, New YorkrLondon, 1973. wFSx L. Fuchs and L. Salce, ‘‘Modules over Valuation Domains,’’ Dekker, New YorkrBasel, 1985. wGMx B. Goldsmith and W. May, The Krull᎐Schmidt problem for modules over valuation domains, J. Pure Appl. Algebra 140 Ž1999., 57᎐63. wJx N. Jacobson, ‘‘Lectures in Abstract Algebra,’’ Vol. III, Van Nostrand, Princeton, NJr New Yorkr Toronto, 1964. wLax S. Lang, ‘‘Fundamentals of Diophantine Geometry,’’ Springer-Verlag, New Yorkr Berlinr Heidelberg, 1983. wLx M. Lunsford, A note on indecomposable modules over valuation domains, Rend. Sem. Mat. Uni¨ . Pado¨ a 93 Ž1995., 27᎐41. wMZx W. May and P. Zanardo, Nonhenselian valuation domains and the Krull᎐Schmidt property for torsion-free modules, Forum Math. 10 Ž1998., 249᎐257. wNx M. Nagata, ‘‘Theory of Commutative Fields,’’ AMS Translations, Am. Math. Soc., Providence, 1993. wRx P. Ribenboim, Equivalent forms of Hensel’s lemma, Exposition Math. 3 Ž1985., 3᎐24. wSZx L. Salce and P. Zanardo, On two-generated modules over valuation domains, Arch. Math. Ž Basel . 46 Ž1986., 408᎐418. wSx O. F. G. Schilling, ‘‘The Theory of Valuations,’’ AMS Surveys IV, Am. Math. Soc., New York, 1950. wSix M. F. Siddoway, On endomorphism rings of modules over henselian rings, Comm. Algebra 18Ž5. Ž1990., 1323᎐1335. wVx P. Vamos, Decomposition problems for modules over valuation domains, J. London ´ Math. Soc. Ž 2 . 41 Ž1990., 10᎐26. wWx R. Warfield, Large modules over Artinian rings, in ‘‘Representation Theory of Algebras’’ ŽR. Gordon, Ed.., pp. 451᎐463, Dekker, New York, 1978. wZx P. Zanardo, Indecomposable finitely generated modules over valuation domains, Ann. Mat. Uni¨ . Ferrara 31 Ž1985., 71᎐89. wZ1x P. Zanardo, Quasi-isomorphisms of finitely generated modules over valuation domains, Ann. Mat. Pura Appl. Ž 4 . 151 Ž1988., 109᎐123.