Unidimensional modules: uniqueness of maximal non-modular submodels

Annals of Pure and Applied Logic 62 (1993) 175-181

North-Holland

17.5

Unidimensional modules: uniqueness of maximal non- modular submodels

Anand Pillay” Department of Mathematics, Notre Dame University, Notre Dame, IN 46556, USA

Philipp Rothmaler Institut fiir Logik, Christian-Albrechts-Universitiit zu Kiel, Ludewig-MeynStr. 4, D-24098 Kiel,

Germany

Communicated by A.H. Lachlan

Received 11 March 1992

Abstract

Pillay, A. and Ph. Rothmaler, Unidimensional modules: uniqueness of maximal non-modular

submodels, Annals of Pure and Applied Logic 62 (1993) 175-181.

We characterize the non-modular models of a unidimensional first-order theory of modules as

the elementary submodels of its prime pure-injective model. We show that in case the maximal

non-modular submodel of a given model splits off this is true for every such submodel, and we

thus obtain a cancellation result for this situation. Although the theories in question always

have models (in every big enough power) whose maximal non-modular submodel do split off,

they may as well have others where they don’t. We present a corresponding example.

We discuss some of the concepts from [6] in the context of unidimensional

theories of modules which are not totally transcendental (t.t.) and show that for

every non-modular model of such a theory there are models in every big enough

power having this-and up to isomorphism only this-non-modular model as its

maximal non-modular submodel. This extends a similar result (for arbitrary

weakly minimal unidimensional theories) from [8] in the context of modules.

Notice the effect this has on the spectrum function of such a theory: the number

of non-modular models constitutes a lower bound for big enough arguments.

We refer the reader to [6] for model-theoretic and to [7], [9], or [2] for

module-theoretic background.

Correspondence to: Ph. Rothmaler, Institut fiir Logik, Christian-Albrechts-UniversitZt zu Kiel,

Ludewig-Meyn-Str. 4, D-24098 Kiel, Germany. Email: [email protected]. dbp.de. *Supported by NSF grant DMS 90-06628.

0168~0072/93/$06.00 0 1993 - Elsevier Science Publishers B.V. All rights reserved

176 A. Pillay, P. Rothmaler

Just a few words about Teq: Let Leq+ be the fragment of Leq obtained by

adding only pp-definable imaginary sorts, i.e., a sort for every equivalence

relation of the form q(X - y), where $J is a pp formula (of corresponding length).

We use the same superscripts also for models and theories. The sort correspond-

ing to equality of elements is called the home sort. It is proved in [l] that for

modules, Meq+ suffices in the sense that every imaginary element is interalgebraic

with a pp-imaginary one (i.e., one from Meq+). Given an n-tuple in M and an

n-place pp formula q we write 6, for the equivalence class 6 + q(M”) of ti in the

sort of q(X - jj). The canonical image of a pp formula ~(2) in this sort is denoted

by Q,(X). Consult [4] for all this. Given a subset A of a model M, we also use

A ‘q+ to denote the subset {a,: ti E A’“, q a ppf of corresponding length} of

Meq+ ‘coming’ from elements of A. Note that A f~ B = 0 implies Aeq+ fl Beq+ = 0, or more exactly Oeq+ (and hence

AC9 17 Be4 c acleq(0) for any modules A and B such that A @ B <: M (a pure

submodule of M): For if QQ = 6,, where ti EA and 6 E B, we have 5 - 6 E v(M), hence on projecting (and by purity) Z E q(A) and 6 E v(B), and so 5, 6 E q(M),

i.e., 5, &* =ij* ,=0eq+.

Note also that Meq+ is pure injective (p.i.) in the obvious sense whenever M is.

Following [3] , a regular group is an m-definable connected group whose generic

is regular. Clearly, an m-definable weakly minimal connected group is regular. In

modules such groups have modular generics; this can be deduced either from the

general result [3, Theorem 3(b)] or directly from pp-elimination of quantifiers as

in [2, Claim, p. 1731. In our first lemma we show that this modularity extends to

regular types over arbitrary p.i. modules.

Lemma 1. Let N be an arbitrary pure injective module and T = Th(N).

All regular types over N in T as well as all regular types over Neq in Teq are modular.

Proof. Let N be as in the statement and suppose p = tp(a/N) is regular. By [7,

Corollary 6.251, this means that

p: = {V(& 0): V(X, 3) EP+)

defines a regular group (in any model).

Realize p+ by b E N. Then, obviously, a - b realizes p:, hence

p: c tp’(a - b).

But then these types must be equal, for if II, is a pp formula satisfied by a - b, then ~(x - b) EP, hence v(x) EP:. Thus a - b is the generic of the (regular)

group defined by p Z, which implies that its type (over 0) is modular.

We claim that also 4 = tp(a - b/N) is modular, for which it suffices to show

that this type does not fork over 0. This, in turn, is equivalent to q: =p:. For the

non-trivial inclusion, let q(x) E q:, i.e., q(x) t, q(x, 0) for some ~(x, fi) E q+.

Then cp(x - b, li) EP, hence 97 EP:. Consequently, q is modular.

Unidimesional modules 111

Finally, as b E N, p and 4 are interalgebraic, whence p is also modular.

By [4, Section 5, (Aiii)], the same proof works for tp(aw/Neq) (by adding

subscripts v all over). 0

Recall from [9] or [7, Theorem 10.241 that a complete theory of modules with

m-dimension (in particular, a superstable theory of modules) has a prime pure

injective model M,, that is a p.i. model elementarily embeddable into every other

p.i. model. The next two lemmas establish a connection between this concept and

that of a non-modular model (of T or Teq), by which we mean a model M such

that for no a E Meg and A 5 Meq with tp(a/A) regular is stp(a/A) modular. Note

that in [6] we work entirely in Teq and define therefore non-modular models of

Teq only; by the definition just given a non-modular model of T is simply the

restriction to the home sort of a non-modular model of Teq.

Lemma 2. Let N be an infinite pure injective module with m-dimension. If N is non-modular, then it is isomorphic to the prime pure injective model of

its own theory.

Proof. Let N be a pure injective as in the statement. Then without loss i@,< N.

But every regular type over A& is modular by Lemma 1. So it remains to show

that there is a regular type over M, realized in N whenever fi, # N. As Lemma 1

holds also for the abelian structure Il;iEq, it’s enough to find one realized in

N”q\R

Clea:ly, && is a direct summand of N, whence there is some unlimited direct

summand V of N such that N = r;io @ V. Let U be a non-zero indecomposable

direct summand of V and p a type realized by some non-zero element of U. By [7,

Corollary 10.40a], p is domination equivalent to a regular type in Teq, which is

the type we are looking for. Cl

From now on T is a unidimensionul theory of a non-t.t. module, I@,, is its prime

pure injective and U its only unlimited pi. indecomposable (cf. [7, Lemma 7.71).

Fix also a rank 1 type p over 0 realized in U (by a non-zero element in a minimal

pp subgroup e(U) of U, which exists, since the unlimited part of T is t.t.).

In [6] it is shown that in this case Teq has non-modular models, even maximal

ones, over any set. Further, by coordinatization, one can replace ‘regular’ by

‘rank 1’ in the above definition of non-modular models; moreover, Me9 k Teq is

non-modular iff it omits the type p, [6, Fact 5.21, which is shown to be modular in

the next lemma. Since the type p is in the home sort, we thus see that a model of

T is non-modular iff it omits this type. We will also speak of maximal non-modular submodels of models of T which simply are the restrictions to the

home sort of maximal non-modular submodels of models of Teq. The type p is not realized in &, since for T as above we have Z = J = 0 and

K = {U} in [9, Theorem 9.11. Thus we know & is non-modular. The following

shows it is maximal such.


Lemma 3. The maximal non-modular model of T is isomorphic to the prime pure injectiue iI&.

Proof. Let M be a maximal non-modular model of T. By the preceding lemma

we must only show that M is pure injective, i.e., that it coincides with its own

pure injective hull n;i. For this, in turn, it suffices to show that k is still

non-modular. As mentioned, a model of T is non-modular iff it omits p. So it

remains to verify that &I omits this type.

Assume a E A? realizes p. As p has rank 1 and is not realized in M, a is

independent from M (over 0). Then, being unlimited, the p.i. hull H(a) of a splits

off in ti by [7, Theorem 5.291. On the other hand, a must be linked with some

tuple fi E M via some pp formula q, [7, Theorem 4.10(d)]:

cp(a, ti) A Ma, 0).

However, projecting onto the direct summand H(a) of &f, q(a, m) implies

q(a, o), a contradiction. 0

The referee pointed out a shorter albeit less elementary proof of this: we know

that tiO is non-modular; since T is superstable, there is a regular type over &

realized in every proper extension M of &I0 (cf. [7, Proposition 6.26]), which

therefore cannot be non-modular by Lemma 1.

Thus for models of T ‘non-modular’ is synonymous with ‘elementarily

embeddable in iI&‘.

Lemma 4. Let V be a direct sum of copies of U. Then any rank 1 type (over 0) realized in Veq is modular. Hence, if M is a

non-modular model, then

Me9 fl Veq G acleq 0

(and Meq+ F-I Veq+ G Oeq+).

Proof. The second assertion follows from the first by coordinatization.

For the first assertion, let a E Veq be of rank 1 (over 0). Then it is interalgebraic

with some b, E Veq+. Now tp(b) is unlimited, hence the generic of an w-definable

connected subgroup. Then b, is the generic of a weakly minimal such group and

so is modular. 0

We are ready to prove the main result.

Theorem 1. Let M be a non-modular model of T and N = M CB V (where, as above, V is a direct sum of copies of U).

M is maximal non-modular in N and if M’ is any other maximal non-modular submodel in N, then M’ is isomorphic to M; moreover, N = M’ @ V.


Proof. First we verify that M is maximal non-modular in N, for which it suffices

to show that every rank 1 type over Me9 realized in NC4 is modular, cf. [6,

Theorem 5.51. Being a direct summand of N, M is relatively p.i. in N (in the

obvious sense) and literally the same proof as for Lemma 1 goes through (where

a E N to begin with). So M is maxima1 non-modular in N.

Let M’ be another maximal non-modular mode1 in N. Consider the projection

n of N onto M that corresponds to the decomposition N = M @ V.

Claim. n restricted to M’ is a pp-type preserving map, i.e., M’ k ~(5) iff M F ~J(JC(~)) for all pp formulas Q, and tuples 2 E M’.

To prove the claim, let cp be a pp formula. As M’ and M are elementary in N

and Q, commutes with @, all we are left with is the implication

for all 2 EM and V E V. But

(M’)eq+ 3 (rn + cqm = 6, E vq+,

hence b, = &,, by Lemma 4, i.e., 0 E q(N). The claim is proved.

Thus JG embeds M’ purely into M. Since M’ = M, this embedding is even

elementary, i.e., setting Q = n(M’) we have Q KM. If Q = M, then first of all,

by the claim, we have M = M’. Secondly, for every m E M there is m’ E M’ with

m = n(m’) = m’ + (n(m’) -m’) EM’ + V, hence N = M’ + V. But now the sum

is direct by Lemma 4 and the non-modularity of M’, whence the theorem is

proved once we have shown that Q = M.

Assuming the contrary, a pp formula 0 as mentioned after Lemma 2 yields an

a EM of rank 1 over Q: Namely, as T has no Vaughtian pairs, there is an

a E O(M)\Q; but 8 has rank 1, so tp(alQ) also does. Choose a pp formula

~(x, Z) of rank 1 in tp(a/Q) and let a’ E Q satisfy it. Then cp(a -a’, o), hence

also a has rank 1 over a’ and so tp(a/Q) does not fork over a’. Let a’ + u’ EM’ be a preimage of a’ under Z, where U’ E V. Clearly cp((a + u’) - (a’ + u’), fi), whereby R(a + u’la’ + u’) s 1. In particular, R(a + u’/M’)s 1. But this rank

cannot be 0, for if a +u’EM’ then a =TC(~ +u’)E Q. Thus we know that

R(a + u’/M’) = 1.

We will finish by showing that, moreover, tp(a + u’/M’) is non-modular (as

a + u’ EN, this contradicts the maximality of M’ in N, by [6, Lemma 6.21).

Since

M’s n(M’) + V E acl(Q U V),

it suffices to show that stp(a + u’/Q U V) is non-modular of rank 1, equivalently

(as these types are interalgebraic) that so is stp(a/Q U V). Since also stp(a/Q) is

non-modular of rank 1 (remember, M is non-modular!), this amounts, by

soundness (cf. [5, Proposition 3.13]), to showing that this latter strong type is not

realized in acl(Q U V).


If it were, we would have some a” E acl(Q U V) non-modular and of rank 1

over Q and depending on V over Q. Then a”Q and V would fork with each other,

which together with 1-basedness would imply

acP(u”Q) fl acleq V $ acleq 0,

cf. [5, Lemma 3.21. This, however, would contradict Lemma 4, as u”Q is

contained in some non-modular model.

This proves the theorem. 0

As a consequence we get the following cancellation result.

Corollary 2. Suppose M and M’ are non-modular models of T and V and V’ are direct sums of copies of U.

Then MG3V=M’$V’ implies M=M’ and V=V’.

Proof. Consider an isomorphism bringing M’ CD V’ to M CT3 V and denote the

corresponding images of M’ and V’ by M” and V” respectively. By the theorem,

M and M” are both maximal non-modular inside N = M @ V = M” @ V”; further

N = M” 43 V and M = M”, hence also M = M’. Finally,

I/’ g I/” s (,” @ V”) J,” = N/M” = (,” @ V)/M” z V. q

Finally we present an example showing that, in contrast to the rank 1 case, in

general T can have models whose maximal non-modular submodel does not split

off (but whenever one of them does, by the theorem, all of them do). In fact the

example given, though an abelian structure, is not a module. However, it is not

hard to turn it into one.

Consider the abelian structure M on the direct sum ZiWw, of countably many

copies of the cyclic group with four elements equipped with infinitely many

predicates P,, defining a descending chain of subgroups of index 4”, respectively,

in M. Let T be the theory of this structure. This theory has quantifier elimination,

is unidimensional (and not t.t.) and its only unlimited p.i. indecomposable is

U = & with every element satisfying each of the P,,. Let p be the type

{2x=O~x#O}U{P,(x):n<w} (realized by 2 E U)

and q the type

(2x # 0) U {Pn(x): n < o} (realized by 1 and 3 in U).

Then U = H(p) = H(q). M oreover, U is generated by a realization of q. Note the

following consequences of quantifier elimination in T: T has few types (thus also

prime models over finite sets), p has rank 1 (and is the only candidate for what we

called p above) and q has rank 2, p and q are both non-isolated and, if Q realizes

p and b realizes q, then the only way in which a and b can fork over 0 is for

2b = a, moroever, in this case tp(b/a) is still non-isolated.


Let N be the prime model over a realization of p. By what we said above, 9 is

not realized in N, hence N contains no copy of U. As N realizes p, the maximal

non-modular model M of N is proper, thus does not split off.

Note that any model of T is atomic over I U J, where I is a Morley sequence in

q and J is a Morley sequence in p with I and J independent. So T has countably

many countable models.

Notice, still N has, up to isomorphism, a unique maximal non-modular

submodel: for, being atomic, the prime pure injective model has, up to

isomorphism, only one countable elementary submodel.

References

[l] D. Evans, A. Pillay and B. Poizat, Gruppa v gruppe (in French), Algebra i logika 29 (1990)

368-377 (English translation: A group in a group, Algebra and Logic 29 (1990) 244-252.

[2] I. Herzog, Ph. Rothmaler, Modules with regular generic types, in: A. Nesin and A. Pillay, eds.,

The Model Theory of Groups (Notre Dame Univ. Press, Notre Dame, 1989) 138-176.

[3] E. Hrushovski, Locally modular regular types, Lecture Notes in Math. 1292 (Springer, Berlin,

1987) 132-164.

[4] Th. Kucera and M. Prest, Imaginary modules, J. Symbolic Logic 57 (1992) 698-723.

[5] M. Ch. Laskowski, Uncountable theories that are categorical in a higher power, J. Symbolic Logic

53 (1988) 512-530.

[6] A. Pillay and Ph. Rothmaler, Non-totally transcendental unidimensional theories, Arch. Math.

Logic 30 (1990) 93- 111.

[7] M. Prest, Model theory and Modules, London Math. Sot. Lecture Notes Series 130 (Cambridge

Univ. Press, Cambridge, 1988).

[8] Ph. Rothmaler, The spectrum function of an almost weakly minimal unidimensional theory is

eventually constant, Travaux du Colloque Sovieto-Francais sur la ThCorie des Modeles

(Karaganda, 1990) 181-184.

[9] M. Ziegler, Model theory of modules, Ann. Pure Appl. Logic 26 (1984) 149-213.

Unidimensional modules: uniqueness of maximal non-modular submodels

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