Bases of minimal elements of some partially ordered free abelian groups

Bases of minimal elements of some partially ordered free abelian groups

Pavel Pˇr´ıhoda

Abstract. In the present paper, we will show that the set of minimal elements of a full affine semigroupA ֒→N^k

0 contains a free basis of the group generated byAinZ^k. This will be applied to the study of the group K⁰(R) for a semilocal ringR.

Keywords: full affine semigroups, partially ordered abelian groups, semilocal rings, direct sum decompositions

Classification: 16D70, 20M14

1. Introduction

A subsemigroupA ofN^k

0 is called full affine if and only if for anya∈A and b∈N^k

0, ifa+b∈Athen b∈A. We may define a partial order onAbya≤cif and only if there isb∈Asuch thata+b=c. Clearly, this order onAcoincides with the order inherited fromN^k

0. In this paper, we will show that any full affine semigroup A contains in its set of minimal elements a free basis of the group generated byAinZ^k.

Following [4], we will denote byS⊕(P-ModR) the semigroup of isomorphism classes of finitely generated projective modules over an associative ring with unitR (see Section 3). The result proved in this paper was motivated by the purpose to find a weak form of Krull-Schmidt theorem for this class of modules over semilocal rings. Indeed, A. Facchini and D. Herbera in [3] and [4], proved that the natural semigroup homomorphismsS⊕(P-ModR)→ S⊕(P-ModR/J(R)), whereR is a semilocal ring, are precisely the order-unit embeddings of full affine semigroups intoN^k

0. We will use just the “only if” part of this statement to prove the weak form of Krull-Schmidt theorem over semilocal rings in Theorem 3.2.

2. Construction of a free basis

In the following, for an abelian partially ordered groupG,≤and≥will mean the given partial order and G⁺ will denote the positive cone of G, i.e. the set {g∈G; g≥0}.

Lemma 2.1. LetGbe a partially ordered abelian group such thatG⁺is a finitely generated semigroup. Then the order satisfies descending chain condition(d.c.c.) on the positive cone.

Proof: Letg1, . . . , gk be a set of non-zero generators for the monoidG⁺. One can easily see that in any infinite subset of N^k

0 there can be found two different elements (a₁, . . . , a_k), (b₁, . . . , b_k) satisfying ai≤bi, i= 1, . . . , k. It follows that every non-zero element of G⁺ can be expressed as a sum of g₁, . . . , g_k in only finitely many ways otherwise we would havea1g1+· · ·+a_kg_k=b1g1+· · ·+b_kg_k for some non-negative integers ai, bi such that ai ≤ bi for all 1 ≤ i ≤ k and aj < bj for some 1≤j≤k. But thengj≤0, a contradiction.

Suppose that the lemma does not hold. Then we can find an infinite strictly decreasing chainh0 > h1 > . . . of elements of G⁺. So there are some non-zero h^′₁, h^′₂, . . . inG⁺such thath_i=h_i+1+h^′_i+1, i= 0,1, . . . . Thenh₀ =h^′₁+h₁ = h^′₁+h^′₂+h₂ =· · ·=h^′₁+· · ·+h^′_i+h_i =. . . . We can express each h^′_i, h_i as a sum ofg₁, . . . , g_kand we see that h₀ can be expressed as a sum of generators in

infinitely many ways — a contradiction.

Lemma 2.2. LetGbe a partially ordered abelian group such that the order on G⁺ satisfies d.c.c. ThenG⁺ is generated(as a submonoid of G) by its minimal elements.

Proof: Suppose thatG⁺ is not generated by its minimal elements. LetH be the submonoid ofG⁺ generated by minimal elements ofG⁺. Since the order on G⁺satisfies d.c.c.,G⁺\H has a minimal elementh, which cannot be minimal in G⁺, so there is some 0< h^′ < h. This means that h=h^′+g for someg ∈G⁺ and, by minimality ofh, bothg andh^′ are inH, hence so ish, a contradiction.

Corollary 2.3. LetGbe a partially ordered abelian group. ThenG⁺is a finitely generated semigroup if and only if the order onG⁺satisfies d.c.c. andG⁺contains only finitely many minimal elements.

We will say that an element g∈G⁺ is anorder-unit ofG⁺ if for allh∈G⁺ there is a positive integernsuch thatng≥h.

Lemma 2.4. Let G be a partially ordered group such that G⁺ is a finitely generated semigroup and has at least two minimal elements. Suppose that for any distinct minimal elementsx, y of G⁺ and any positive integers m, n, mx6=ny.

ThenG⁺contains a non-zero element that is not an order-unit.

Proof: We proceed by induction on the number of minimal elements of G⁺. I.G⁺ has just two minimal elements. If mx > y for some m ∈N, then for the least suchm,mxwould be a multiple ofy.

II. Letg₁, . . . , g_k be all minimal elements ofG⁺ and k≥3. Suppose that all of them are order-units. For distinct integers 1< i, j ≤kwe can find non-negative

integersa_i, b₁, b₂, . . . , b_k,c₁, c₂. . . , c_k such that

aigi= Xk n=1

bngn, ai >0, b₁>0,

c₁g₁= Xk

n=2

cngn, c₁>0, c_j>0.

Let us multiply the first equation byc₁ and the second byb₁ and add together.

We get dgi = P_k

n=2dngn, d > 0 and dj > 0. Now let us equip G with order

≤^′ whose positive cone is the monoid generated by g₂, . . . , g_k. It follows that all (minimal) elements of (G,≤^′) are order-units. This contradicts the induction

hypothesis.

A partial order is calledunperforated if for allg∈Gand all positive integersn, ifng∈G⁺ theng∈G⁺.

Corollary 2.5. LetG be an unperforated partially ordered abelian group such thatG⁺is a finitely generated semigroup and has at least two minimal elements.

ThenG⁺contains a non-zero element that is not an order-unit.

Proof: LetG be an unperforated partially ordered abelian group. Letx, y be two distinct minimal elements ofG⁺. Suppose that there are positive integers m, nsuch thatmx=ny. For instance, letm≤n. Then 0 =mx−ny≤n(x−y).

SinceGis unperforated, we havex−y∈G⁺, and soy≤x, a contradiction. Now

apply Lemma 2.4.

The given partial order onGis calleddirected if for allg, h∈Gthere isk∈G such thatk≥g andk≥h, or, equivalently,G=G⁺−G⁺.

IfI is a directed convex subgroup ofGthenI is called anideal ofG.

Notation. LetG be a partially ordered abelian group. Let us denote by M the set of minimal elements of G⁺, by Mthe power set of M and by I the set of ideals ofG.

Lemma 2.6. LetGbe a partially ordered abelian group such thatG⁺is a finitely generated semigroup. Then the assignmentϕ:I → Mdefined byϕ(I) =I⁺∩M is an injective map. Moreover every idealI is generated byP

m∈ϕ(I)m.

Proof: Let us define the map from Mto the set of subgroups ofG defined by ψ(A) = hAi, the subgroup generated by A, for any A ⊂M. For any I ∈ I we haveψϕ(I)⊂ I. Now, leti ∈ I⁺. By assumption iis a sum of some minimal elements ofG⁺ sayi=m1+· · ·+mk. Everymj for 1≤j ≤kis in I because I is convex. Thus I⁺ ⊂ψϕ(I). Since I is directed we have I =hI⁺i ⊂ψϕ(I).

ThusI=ψϕ(I) andϕis injective.

For the rest of the proof, let J be the ideal generated by i = P

m∈ϕ(I)m.

ObviouslyJ ⊂I. SinceJ is convex we haveϕ(I) =ϕ(J) and soJ =ψϕ(J) =

ψϕ(I) =I.

We have seen that ifG⁺is finitely generated, then every ideal ofGis generated by some element ofG⁺. For anyg∈G⁺ letIg denote an ideal generated by g.

It can be easily seen thatIg is a group generated by the set{h∈G⁺; h≤ngfor somen∈N}. Hence, if the order onGis directed, thenIg=Gif and only ifgis an order-unit ofG⁺.

Lemma 2.7. LetGbe an unperforated partially ordered abelian group, and let Ibe an ideal of G. ThenG/Iwith the factor order is also an unperforated group.

Proof: See [5, Proposition 1.20].

Theorem 2.8. LetGbe a directed unperforated partially ordered abelian group such that G⁺ is a finitely generated semigroup. Then G is free and has a free basis of minimal elements of G⁺.

Proof: Gis obviously a finitely generated torsion free group and hence it is free.

Let us proceed by induction on the number of minimal elements ofG⁺. If G⁺ has only one minimal element, this element is a generator ofG.

Let us suppose thatG⁺has minimal elementsg1, . . . , gk. By Corollary 2.5 the setS ={Ih; h∈G⁺, h6= 0, Ih 6=G}is nonempty. According to Lemma 2.6 the cardinality ofSis bounded by the cardinality ofMand so the systemSis finite.

Let Ig be the maximal element of S with respect to inclusion. Since ϕ(Ig) 6=

{g₁, . . . , g_k} we can apply the induction step. Thus there areh₁, . . . , h_l ∈ϕ(Ig) which form some free basis ofIg.

Now we claim that∀h∈G⁺\Ig we haveZh∩Ig = 0. Supposenh=h₁−h₂ for someh₁, h₂∈I_g⁺ and npositive integer. Then 0≤h≤nh≤h₁ andh∈Ig, a contradiction.

By Lemma 2.7,G/Igwith the factor order is directed and unperforated and has finitely generated positive cone. Suppose that the cone has at least two minimal elements. By Corollary 2.5 there is someg^′∈G⁺ such thatg^′+Ig is a non-zero non-order-unit element of (G/Ig)⁺. SoIg⊂I_g+g′ andI_g+g′ 6=Gsince otherwise g+g^′would be an order-unit inG⁺andg^′+Ig = (g+g^′)+Igwould be an order-unit in (G/Ig)⁺. ButIg was the maximal element ofS, a contradiction. So the cone (G/Ig)⁺has only one minimal element, sayh+Ig. By Lemma 2.1 and Lemma 2.2 we see thatG/Igis generated byh+Ig. Let us setA={x∈G⁺; x+Ig=h+Ig}.

The setAis a nonempty subset ofG⁺\Igand, by Lemma 2.1, it has some minimal elementh₀. Suppose that there is 0< x < h₀. We have 0≤x+Ig≤h₀+Ig in the factor, hence by minimality ofh₀andh+Ig we havex∈Ig. We see thath₀ has to be minimal inG⁺, otherwise it would be a sum of elements strictly smaller thenh0and thus it would be an element ofIg. Now it can be easily seen that the minimal elementsh0, h1, . . . , hl form a free basis ofG.

3. Application

By the following we shall see that Theorem 2.8 indeed concerns full affine semigroups.

Lemma 3.1. LetG be a directed partially ordered group. If G⁺ is isomorphic to a full affine semigroup thenGis unperforated and G⁺ is a finitely generated semigroup.

Proof: Gis generated (as a group) byG⁺and so we may assume thatG⁺ is a full affine subsemigroup ofN^k

0 and Gis the subgroup ofZ^k generated byG⁺. If ng∈G⁺ forg∈Gandnpositive, theng∈N^k

0, henceg∈G∩N^k

0 =G⁺. It remains to show, due to Corollary 2.3, that the order onG⁺ satisfies d.c.c.

and that it has only finitely many minimal elements. But the order onG⁺ coin- cides with the product order inherited fromN^k

0 and, in this order, any subset of N^k

0 has the required properties.

On the other hand, if a partially ordered groupGis unperforated and G⁺ is a finitely generated semigroup, thenG⁺is isomorphic to a full affine semigroup.

For details see [1] where such semigroups are also called normal semigroups.

Recall that the ringR is called semilocal if R/J(R) is a semisimple ring. Let us briefly recall the construction of the semigroup S⊕(P-Mod R) for a ring R (for details see [4]). The elements ofS⊕(P-Mod R) are the isomorphism classes of finitely generated projective modules overR. Now we will define a semigroup structure on this set by [P]+[Q] = [P⊕Q]. For a semilocal ringR,S⊕(P-ModR) is a cancellative semigroup and so it can be embedded into its group of fractions K₀(R). The order on K₀(R) is given by K₀(R)⁺=S⊕(P-ModR). Clearly, [R] is an order-unit of K₀(R)⁺and [P] is a minimal element of K₀(R)⁺if and only ifP is indecomposable. It can be shown (see e.g. [3], [4]) that ifRis semilocal, then the partially ordered group K₀(R) can be embedded into the partially ordered group (Z^k,N^k

0), where k is the cardinality of the representative set of simpleR/J(R) modules.

Hence, using Theorem 2.8 and Lemma 3.1, ifRis a semilocal ring then K₀(R) is a free group and the set of minimal elements of K₀(R)⁺ contains a free basis of K₀(R), in other words:

Theorem 3.2. Let R be a semilocal ring. There are non-zero finitely gene- rated indecomposable projective R-modules, say P₁, . . . , P_k, such that for any finitely generated projective module Q there exist unique numbers n₁, . . . , n_k, m₁, . . . , m_k∈N₀ such thatnimi = 0fori= 1, . . . , k, and

Q⊕P₁⁽ⁿ¹⁾⊕ · · · ⊕P_k^(nk)≃P₁^(m1)⊕ · · · ⊕P_k^(mk).

LetM be a module (over any ring) and letS be the endomorphism ring ofM. It is a well known fact (see e.g. [2]) that the categoriesproj−Sof finitely generated

projective modules overS andadd M of direct summands of finite direct sums of M are equivalent. Thus using Theorem 3.2 we obtain:

Corollary 3.3. Let M be a module with semilocal endomorphism ring. There exist non-zero indecomposable modules in add M, say M₁, . . . , M_k, such that if Qis any module inadd M then there are unique non-negative integersn₁, . . . , n_k, m₁, . . . , m_k such thatnimi = 0fori= 1, . . . , k, and

Q⊕M₁⁽ⁿ¹⁾⊕ · · · ⊕M_k^(nk)≃M₁^(m1)⊕ · · · ⊕M_k^(mk).

References

[1] Bruns W., Herzog J.,Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, 1993.

[2] Facchini A.,Module theory. Endomorphism rings and direct sum decompositions in some classes of modules, Progress in Mathematics 197, Birkh¨auser, 1998.

[3] Facchini A., Herbera D., K0 of a semilocal ring, J. Algebra225(2000), no. 1, 47–69.

[4] Facchini A., Herbera D.,Projective modules over semilocal rings, in: D.V. Huynh (ed.) et al., Algebra and its Applications: Proceedings of the International Conference, Contemp.

Math.259, 2000, 181–198.

[5] Goodearl K.R.,Partially ordered abelian groups with interpolation, Mathematical Surveys and Monographs no. 20, Amer. Math. Soc., 1986.

Department of Algebra, Faculty of Mathematics and Physics, Charles University, Sokolovsk´a 83, 186 75 Praha 8, Czech Republic

(Received February 4, 2003)