decidability for theories of modules - Semantic Scholar

DECIDABILITY FOR THEORIES OF MODULES FRANCHISE POINT AND MIKE PREST

Introduction We are interested here in the first-order theory of /?-modules: in when it is decidable and in when it differs from the theory of finitely generated /^-modules. It is an open problem to characterise those rings over which the common theory TR of /^-modules is decidable, although an answer has been given for some classes of algebras (see [8,11,12]). Here we show that decidability of 7^ is an algebraic property, in that it is invariant under 'effective Morita equivalence' (so depends only on the category of /^-modules)—at least if R is sufficiently decidable. If a ring is of finite representation type then every module over it is a direct sum of finitely presented modules. It follows easily that the theory TfRp of the finitely presented modules coincides with TR (if a sentence is satisfied in some module then it is satisfied in a finitely presented module) and is decidable (again, provided the ring is sufficiently decidable). We suspect that the rings of finite representation type are the only artinian rings over which these two theories coincide: as it is, we have been able to show only that over certain algebras not of finite representation type these theories differ. That TR and TfRp are unequal outside finite representation type is not unreasonable since TK[X] and Tf£[X] are different and one should expect that, for R a A-algebra not of finite representation type, there will be at least one embedding which induces an interpretation of the category of A^J-modules into the category of /^-modules. There are rings not of finite representation type over which the theory of all modules coincides with the theory of finitely presented modules. For instance, let R be a commutative regular ring with the set of principal maximal ideals dense in speci?. It is not difficult to show (cf. §3) that every sentence satisfied in an /^-module is satisfied in a finitely presented i?-module. Morita invariance of decidability actually holds in contexts more general than that of module categories (for an application see [11]). The paper ends with some results on decidability over von Neumann regular rings. As for the background required: we suspect that not many readers will have background both in model theory and in the representation theory of algebras, but considerations of length preclude our including the relevant material here. We refer the reader to [12] or to survey papers and the like. We wish to thank Angus Macintyre, discussions with whom have contributed significantly to the results in this paper.

Received 3 July 1986; revised 7 November 1987. 1980 Mathematics Subject Classification (1985 Revision) 03C60. J. London Math. Soc. (2) 38 (1988) 193-206 7

JLM 38

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Preliminaries At each point in the discussion R is a fixed ring (associative with a 1). By an algebra we mean an algebra over a (fixed) field K (usually, but not invariably, the algebra will be finite-dimensional as a ^-vector space). We denote the category of /^-modules by MR, and mod-/? denotes the full subcategory whose objects are the finitely presented modules (notice that if the ring is artinian or even noetherian then a module is finitely presented if and only if it is finitely generated). The canonical first-order language for /^-modules has symbols for addition in a module, for the zero element of a module and for the multiplications by the various elements of the ring. A great deal is now known about the model theory of modules and we shall have occasion to call on some results in the area, but usually our techniques are fairly direct. Perhaps the main obstacle to the general reader is the idea of interpreting one theory in another: for discussion of this one may consult [6, Chapter 6] for example. By TR we denote the common theory of all (right) /^-modules: TR = {a: a is a sentence in the language for /^-modules and a is true in every /^-module}. By TfRp we denote the common theory of all finitely presented ^-modules. Following [7, p. 157], we take R to be the ring Z of integers, fix a prime p, and consider the sentence a: 3v(vp = 0 A M 0 ) A Vy3w(u = wp). If an abelian group Mz satisfies a (we write M\=o) then M must be infinitely generated: observe, also, that a is satisfied by the Pnifer module Zp*. Thus Tz # Tfzp. Since we shall be concerned mainly with AT-algebras we should consider the ring K[X] rather than Z: but there we have TK{X] ^ Tf,[(X] for essentially the same reason—take any a e K and, replacing the prime peZby the prime (X— a)sK[X], consider 3v(v(X-tx) = 0 A v # 0) A Vv3w(v = w(A'-a)) to see a sentence whose negation is in Tf£[X] but not in TK{XY The example just given depends on an epi- but not monic morphism: dually one may obtain an example of a sentence satisfied only in (certain) infinitely generated modules, by using a monomorphism which is not epi; specifically multiplication by the prime p (respectively X—tx) in the localised Z-module Z (p) (respectively K[X]module K[X]{ v _ a) ). It follows that if R is a tf-algebra then TR ^ TfRp provided we can arrange the following. There is a finitely axiomatisable (by fi say) subclass ^ of J(R, within every member of # one may define (and define uniformly in # ) a Apfj-module; every K[X]module can be got in this way; an interpreted Ap^-module is infinite-dimensional if and only if the /^-module in which it is interpreted is infinite-dimensional. If we have this then, given a sentence a true in every finitely presented A![Ar]-module, but false in some infinitely presented ApfJ-module, we let T be the corresponding sentence in the language of i?-modules such that, for each M in # one has that M satisfies T if and only if the XpfJ-module interpreted in M satisfies a. Then the sentence / / A T lies in TfRp\TR. For concrete examples of such interpretations, see [11]. In fact, experience shows that each time we have the situation described above, it will have arisen from a functor from XjXI-modules to /^-modules, and the interpretation of a Apf]-module inside an /^-module will be inverse to the functor. Is this a general phenomenon? For our results on decidability we must impose some conditions on the ring R: for it makes little sense to talk about decidability of the theory of i?-modules if, for

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instance, the word problem for the ring R is unsolvable (for terms a,/? define equal ring elements if and only if Vi>(ua = v/1) is in 7^). As to just how much of the theory of the ring we should ask to be decidable—we offer a tentative solution, and we work under the corresponding hypothesis. Whether or not this solution is a good one remains to be seen but for 'concrete' rings there should be no problem. Another problem is that even 'well-known' rings can be presented in perverse ways such that the theory of modules over them is undecidable for that reason (an

example follows). As an example of the sort of pathology which can occur, let Q be the field of rationals, indexed by the natural numbers N in any ' reasonable' way. Then certainly the theory of Q-modules is decidable. Now let n be any non-recursive permutation of f^J. If the theory T of modules over the rationals with this new ordering were decidable, then the permutation n could be recovered as follows. Let neN and let a/b be the «th element of Q in the first ordering. Then nn may be identified (recursively) as the integer m such that the sentence Vi>(i> + v + ... + v = (v + ... + v)-m) lies in 7", where v arises a times on the left-hand side and b times on the right (we thank the referee for pointing out this example). It is even possible to have recursive presentations of rings R, S such that, abstractly, R and S are isomorphic, yet the theory of modules over R is decidable and the theory of S-modules is undecidable (see the example given at the end of the third section). So, when we discuss decidability, there will always be an assumed presentation, or listing, of the ring in the background. We make the assumption that, when discussing decidability, our rings are recursively presented with decidable word problem. That is, the elements of the ring are recursively listed (say as words in a certain set of generators), and the sum and product of any two elements may be recursively identified as members of this list. For instance, a finite-dimensional algebra over, say, the rational field or algebraic closure thereof, would normally be presented in such a way. We must assume more: questions such as 'aebRV are effectively 'encoded' in the theory of the modules (in this case by Vv(vb = 0 -> va = 0)) and, unless they are effectively answerable, the theory of i?-modules cannot be decidable. The maximum assumption is that the entire first-order theory of the ring, in the language for rings augmented by constants for its elements, is decidable. Let us test the reasonableness of this by taking the ring to be a field K. It is a classical result that if AT is finite or is countable and algebraically closed, then the theory of Aîs decidable. On the other hand, the theory of the rational field Q is undecidable (since it interprets that of the ring of integers Z), yet the theory of Q-vector spaces is decidable (directly, or since Q is of finite representation type) (and, come to that, the theory of Z-modules is decidable). So it certainly may be that the theory of R as a ring is undecidable, yet the theory of .K-modules is decidable. It seems then that we must require some restricted part of the theory of the ring to be decidable. By the observations above, this restricted part must include the quantifier-free part of the theory, together with at least some existential sentences (3wa = bw). The following suggestion is made in [12] and we adopt it here. Whenever the question of the decidability of the theory of R-modules is raised, it will be (implicitly) assumed that the largest complete theory of R-modules is decidable. Let us call this condition (D). At first sight, this may not look like a condition on the theory of the ring R, but

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it is so. It amounts to solvability of certain systems of linear equations being algorithmically determinable (see [12]). Since such solvability may be expressed as an existential sentence (of a rather simple sort) in the theory of the ring R with constants from R, this hypothesis is indeed the requirement that a certain part of the positive existential theory of the ring, with constants for its elements, be decidable. It should also be pointed out that this condition is the exact generalisation to pp-types of the condition that all questions 'aebRV be effectively answerable. Examples of rings which satisfy condition (D) are (cf. [12]): any recursively presented field with decidable word problem, such as finite fields, the rationals, finite extensions of these and their algebraic closures (Gaussian elimination gives a decision procedure!); any algebra A, finite-dimensional over such a field K, which is recursively presented over K in the sense that there is given a (finite) A>basis {av..., an} of A over K, together with the multiplication constants cijk which say how the basis elements multiply together—a t a t = J ] ^ cijlcak; the ring of integers Z, or K[X] where K is such a field; any recursively presented finite ring.

1. Invariance under Morita equivalence Recall that rings R and S are said to be Morita equivalent if their categories of modules JlR and Jis are naturally equivalent (that is, 'essentially the same category'). A property is said to be Morita invariant if whenever R has that property and S is Morita equivalent to R then S has that property also—thus the property is really associated to the category of modules rather than to the ring. We show here that, for rings with sufficiently decidable theory, decidability of the theory of modules is invariant under effective Morita equivalence. This makes the goal of providing an algebraic classification of those ('small') rings with decidable theory of modules seem more likely to be attainable. Now ([1], for example) if R and S are Morita equivalent then there exists P, which is a finitely generated projective right /^-module, such that S ^ End (PR) and R ~ End( s P): moreover, every S-module M* has the form HomR(P,M) for some MEMR (where as for asM* and seS is defined to be the composition of functions as—note that, since S is the endomorphism ring of PR, this does define a right action of S on Hom fl (P, M)). Since, as has been observed above, even isomorphisms between rings need not preserve decidability, we must be careful to specify what is meant by 'giving' a Morita equivalence. Fix a ring R satisfying condition (D). A Morita equivalence is specified by 'giving' a finitely generated projective generator. We may do this by generators and relations—specifying a submodule K of some free module Rm by giving a finite set of elements of Rm which are to be generators for K. Abstractly, this defines P by the short exact sequence 0

>K

>Rm

>P

>0:

since P is projective, this sequence splits, so P may be thought of as a direct summand of Rm. But we are then faced by the problem: can (generators for) P be found computably from the generators of Kl We do not know whether there is a general algorithm for doing this. But it turns out that generators for K suffice, since it is not actually P that we need to know; rather we need to know the endomorphism ring S of P. Now, S may be characterised as the ring of endomorphisms of Rm (that is, m x m


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matrices) which send K to K, modulo those which send Rm to K (we are grateful to Michael Butler for pointing this out). Since we have explicit generators for K, the endomorphisms of Rm fixing K are recursively identifiable and the condition 'f—g sends Rm to K1 also is recursive (both by condition (D)). Thus one may find a recursive presentation of S under which S has solvable word problem and also (as is easily checked) satisfies condition (D). This is all that we need, but it is perhaps worth pointing out that in general it is unlikely that we would know that K is a direct summand of Rm without being given that information in a computable fashion (for example, by being given generators for a complement P). The fact that P is a generator is, in fact, irrelevant to the proof below: even that P is projective is only needed to give us that every S-module is isomorphic to one of the form M* = Horn (P,M) for some /^-module M. Since the S-action on M* is defined by setting as (aeM*,seS = EndP) to be the composition of functions as, we may think of an element of the S-module M * as being the same as a morphism from P to the corresponding /?-module M. That, in turn, is just a morphism from Rm to M which factors through R/K. Suppose that the given generators of K are bl,...,bs, where bk = Y^-\eilik-> t n e ej being canonical generators for Rm. Then a morphism / : Rm —* M with fe} = a} factors through R/K if and only if the images of the bk are all zero; that is, if and only if Yi?-\ ai ljk = 0 f ° r e a c n ^ = 1> • • • >s- Given an element a* of M*, let a = (alt..., am) E Mm be the corresponding m-tuple of elements of M. LEMMA 1.1. Let notation be as above. Let 0(v15 ...,vn)be a formula in the language of S-modules. Then there is a formula ^(vv ...,vn) in the language of R-modules such that for any elements a1,...,an in any S-module M*, one has M* t= 0(a 1 5 ..., an) if and onlyifM\=$(a1,...,an). IfR satisfies condition (D) and if a finite generating set for K is given explicitly, then 0 may be found effectively from cj>. Proof. The proof goes by induction on the complexity of the formula (j>. 1. Suppose that where the st are in S. We are supposing that each st is given as an endomorphism of Rm: that is, as an m x m matrix (rjH)n—so, thinking of 5 acting on the right, we have s{(et) = Y,?-i ej rnv Therefore the equation £ «< st = 0 says that the morphism £ at st: PR -*• MR is the zero morphism: that is, s n

Substituting for siel this becomes

The formula A ? - I E ? - I E £ I U « ' V H = ° i s $ = 0(yn>--->yi,»>y2i> •••^nm) and it follows from the discussion above that M* t=0(a 19 ...,an) if and only if 2. The other non-trivial case is that 0 has the form 3vQ t//(v0, vx,..., vn), where we may assume by induction that the result holds for y/. Recall that an element of M* = Horn (P, M) is given by an m-tuple (c 1 5 ..., cm) of elements of M, the entries of which satisfy /\£ = 1 X £ i c ^ * = 0, where the tneR are defined above.

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Define 0(y 15 ...,iJ n ) to be 3vo(y/(vo,... ,vn) A A U X ^ i % tik = 0), where v0 is (yol,...,yOTO) (and where y) has been inductively defined). If Af*t=0(fl 15 ...,a n ) then there exists a0 in M such that M * N y/(ao,...,an). By the induction hypothesis, this implies that M satisfies y/(a0,..., an) and hence M1= 0(a 1 5 ..., an) (since the entries of a0 satisfy the required condition). Conversely, if M f= 0(a l 5 ..., a n ), say M satisfies y/(6,«!,..., an) for some £, then the condition on the entries of b ensures that b 'is' an element ' 6 ' of M*. So, by induction, one has that is, M*\= {aXi...,an), as required. M*\= y/(b,av...,an); 3. Define (y/) = ->ip: it is immediate that this preserves correctness of the translation. The statement regarding effectivity follows since the hypotheses imply that, given (f>, we may write down 0. 1.2. Suppose that R is a recursive ring which satisfies condition (D). Suppose also that a finite set of generators for a submodule K ofRn is given, and assume that Rn/K is a projective generator for JlR. Let S = End PR be the corresponding ring Morita equivalent to R. Then S has a recursive presentation and satisfies condition (D). If the theory of R-modules is decidable then so is the theory of S-modules. COROLLARY

Proof. Let a be a sentence in the language of S-modules. By 1.1, a is true in some S-module if and only if a is true in some i?-module, in other words, if and only if -^6 is not in the theory of /^-modules. The theorem follows. One may therefore say that 'effective Morita equivalence' preserves decidability of the theory of modules. It has been conjectured, and the conjecture verified in some cases [8, 11], that if A is an algebra, finite-dimensional over a field, then the theory of ^-modules is decidable if and only if A is of tame or finite representation type. The work that has been done on this so far has used results from the representation theory of algebras. Now, especially in the case when the underlying field is algebraically closed, it is a commonly used reduction to replace the algebra A with the basic algebra which is Morita equivalent to it. So it is useful to be able to make the same reduction for the decidability problem. In the case of any particular algebra A, it will be easy enough (assuming that A has been given in a reasonable way) to find a presentation for the relevant projective ' P ' (it may be found as a direct summand of A which contains just one copy of each indecomposable projective). So this gives one kind of application of the above. At least one application which we have in mind actually requires the above ideas to be set in a somewhat more general context. One should note that nowhere in the proof of 1.1 did we require that P was projective (this was part of the data of the category equivalence). What we did need was that P was finitely presented and generated the category (in that case J/R) being considered. The following is a useful perspective on the above. Given any Grothendieck abelian category with a generating set of finitely presented objects (such as JiR with {/?}, JlR with {Rn: nsco} or (mod-/?, Ab) with {(M, - ) : Me mod-/?}), one may define a language for that category, with one sort for each element of the chosen generating set (see [10]). Given an object of the category, its 'elements' of a given sort are the morphisms from the corresponding generator to that object. For example, an element of an /^-module is 'really just' a morphism from RR to that module: similarly, with


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R, P, S as before 1.1, an element of an S-module is really just a morphism from the finitely presented generator P of MR to a certain i?-module. For any given such category # , there is a good deal of choice in the generating set and hence in the language (it is not the category of modules which changes under Morita equivalence, but rather the way in which it is presented). What 1.1 and its obvious generalisations say is that, so long as we confine ourselves to generating sets (always of finitely presented objects) which are recursively equivalent, we shall have recursively equivalent languages: hence properties such as decidability are invariant. An example of this more general situation is provided by tilting functors. In [11] it is shown that the path algebra of an extended Dynkin quiver has decidable theory of modules. Use of tilting functors allows us to extend this result to certain other algebras; indeed their use permits us to consider only one orientation of each quiver. A tilting functor of the sort we have in mind is not a Morita equivalence, but it is almost so: the projective P is replaced by a preprojective or preinjective module and one has that the two relevant module categories have very large (finitely axiomatisable) equivalent subcategories, the equivalence being induced by Hom(P, —). Thus (the proof of) 1.1 applies to these subcategories. The modules excluded from the subcategories are those which have a summand isomorphic to one of a fixed finite set of finitely presented indecomposables, and they may be treated separately. This is enough (see [11]) to allow decidability to be transferred, provided the tilting (preprojective or preinjective) module is given explicitly. 2. The theory of finitely presented modules We shall denote by TRV the common theory of the finitely presented /^-modules. We suspect that, among right artinian rings, it is only in the case of finite representation type that this coincides with the theory, TR, common to all /^-modules. We can at least prove that, for certain kinds of algebras not of finite representation type, the theories differ and, also, that for algebras of finite representation type they do coincide. This last point follows quite easily from the fact [3] that every sentence in the theory of modules is equivalent to a boolean combination of statements of the form Inv ( —, 0, y/) ^ n, where 0, y are pp formulas (this statement says that the index of the subgroup defined by y/ in that defined by 0 is at least «). It follows that if a sentence is true in a direct sum of modules then it is true in some finite subsum. If R is of finite representation type then every /?-module is a direct sum of finitely presented submodules: so it follows that if a sentence is true in some module then it is true in a finitely presented module. Therefore TfRp = TR: the direction 2 is always true and, if we had aeTfRv\TR, then ->a would be true in some module but in no finitely presented module—in contradiction to the above. A couple of remarks are in order: that every sentence true in some module is true in a finite direct sum of indecomposable pure-injective modules (this for any ring), is a consequence of [15, 6.9]; the property that we used above is characteristic of the right pure-semisimple rings, and so we have the following (but let us note that it is still an open question whether there are any right pure-semisimple rings which are not of finite representation type—certainly there are none among finite-dimensional Kalgebras [2]). PROPOSITION 2.1. If R is a ring of finite representation type {or more generally (?) if R is right pure-semisimple) then TfRv = TR.

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For proving the converse, we come up against the problem of having an appropriate definition for tame and wild representation type. A tame or wild category of modules embeds a subcategory which is representation-equivalent to modK[X]—the category of finitely presented Ap^-modules (a wild category even 'contains' mod-AX A", y » . For our purposes, we need rather more: we need infinitely generated Ap^-modules to witness that TfRp ^ TR (see the preliminaries section), and we need an interpretation defined in a finitely axiomatisable subcategory of the category that we are considering. Now, experience suggests that, in any particular case, the algebraic proof of tameness or wildness actually provides an interpretation of JfK[X] (or some localisation thereof). But how may we prove the general result? The general definitions of tame and wild seem just a little weak for what we want. So we content ourselves, for now, with proving that TfRp # TR within some particular classes of algebras. 2.2. Let R be the path algebra over a quiver {without relations) of infinite representation type {the base field K being arbitrary). Then TfRv ^ TR. THEOREM

Proof. We may suppose that the quiver under consideration is connected. For the category of modules over a quiver is equivalent to the product of the module categories over its connected components, and each of these 'component' subcategories is clearly finitely axiomatisable (by an annihilator condition): so we may restrict to some connected subquiver of infinite representation type. Now, any connected quiver of infinite representation type contains (up to orientation) one of the tame connected quivers shown in Figure 1. For instance the category of modules over the quiver in Figure 2 has, as a finitely axiomatisable subcategory, the category of modules over Di: an axiom is Vy(ve5 = 0), where eh is the idempotent corresponding to vertex 5. In a similar way, any quiver of infinite representation type has, as a finitely axiomatisable subcategory of its category of modules, the category of modules over at least one quiver on the above list. Let us consider An. Let ex be that idempotent of the path algebra which corresponds to vertex 1. Let a be the sentence which states that for each e(j ihj) # («+1,1) its kernel in Mei is zero and its image is all of Me} (Ma module). Then any module M satisfying this condition has all but one (viz. en+l x) of the etj 'acting as isomorphisms'. We may then interpret a ^[A^-module in M by taking it to be Mex with Abaction given by the composition of the 'inverses' of el2,e23,...,en n+l together with c n + l i l . It is not difficult to see that any AÂ'j-module may arise in this way. Thus we have an interpretation (actually a functor) of the category of K[X]modules in the category of yfn-modules (and hence in the category of modules over any quiver which contains An). Of course, under these interpretations, finitedimensional (over K) modules do correspond to finite-dimensional modules and, in particular, Priifer modules over K[X] do go to infinitely presented modules. Hence distinctness of Tfp and T is preserved. For more detail on these interpretations see for example [11]. One may note that these interpretations are always given by pp formulas. The other diagrams are dealt with in analogous ways (for example consult [11], or for Z)4 see [4]). In the case of Di one may use 'quadruples' to interpret modules over the localisation K[X,X-l,{l-Xy1]. For the other diagrams one may refer to [11, Section 1], where interpretations of ApT]-modules are given for the corresponding


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O An (n+1 points)

•—•—•—•

•

FIG. 1

subspace problems (for what is wanted here, ignore the references there to the extra specified subspace). Of course, in all this, we have considered only certain orientations. But it has already been remarked that changes of orientation may be effected by tilting functors and that these preserve decidability (the reader who wants more detail may consult [11])COROLLARY 2.3. Let K be an algebraically closed field and let R be a finitedimensional hereditary K-algebra which is not of finite representation type. Then 1

R ^

2

R-

The corollary is immediate, since the category of modules over such a ring is Morita equivalent to the category of modules over the path algebra of a quiver without relations (of infinite representation type). Now we turn to local algebras. Here we shall assume only the weaker finiteness hypothesis of being noetherian. THEOREM 2.5. Let Rbea noetherian local K-algebra. Suppose that R is not of finite representation type. Then TfRv # TR.

Proof. Let / be the Jacobson radical of R. By ' local' we mean that, as a AT-space, R = K\+J. If J/J2 has dimension 1 over K then each quotient Jn/Jn+1 has dimension 1. So either R ~ A [ J ] / ( I n > for some nsco—a possibility excluded by requiring that R be not of finite representation type—or else we argue as follows. Let M be the inverse limit of the directed system —»...—»{R/P) R

—»(R/J)R.

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So M is also a module over the commutative ring R, which is the inverse limit of the 'same' diagram, regarded as a diagram of rings. Let reR be such that J = rR + P: so (using induction on n) Jn = rJn~l + Jn+1 for each n ^ 1. Then multiplication by r is an endomorphism of MR which is clearly not epi. On the other hand, if one had mr = 0 for some m e M, m # 0 then (using the representation of the element m as a sequence of elements of the R/Jn) one quickly derives that rneJm for some m> n. Hence Jn = •/'" and so, by Nakayama's lemma, Jn = 0—contrary to hypothesis. Thus, multiplication by r is monic but not epi and so (see the preliminaries section) 1

R

r

J

ft •

Suppose now that J/P has dimension at least 2: then R/I ~ K[X, Y]/(X, Y)2 for some ideal / (go to R/P first, then use the fact that R = K-1 + / ) . Since / is finitely generated, the class of /^-modules which annihilate / is finitely axiomatisable. So it will be enough to prove the result for S = K[X, Y]/(X, 7> 2 . Let a be the sentence in the language of S-modules which states that (i) \mX ^ im Y and (ii) kerA'c ker Y. We claim that any module satisfying a must be infinite-dimensional. For, choose by (i) some a o eker y\ker X. By (i), there is ax with a1 Y = a0 X 7* 0. By (ii), ax X ^ 0: so there is a2 with a2 Y = ax X\ and so on. The an may be seen to be linearly independent (think of the 5-module generated by the an as a Priifer-type module under the action of ' Y~*X': with this picture in mind it is not difficult to prove (by induction) that any AMinear combination of the an which is equal to zero has all coefficients equal to zero). Thus any S-module satisfying a is infinite-dimensional. Moreover, there does exist an S'-module which satisfies a\—take the S'-module with generators an (new) which are related as above. We would now be finished if our original ring R were finite-dimensional: for then, any module satisfying a above would be infinitely presented (would even be infinitely generated). As it is, we finish by noting that, since such a module is infinitely generated as a natural /^//-module, it is indeed infinitely generated as an Rmodule. Now suppose that R is a commutative noetherian A^-algebra. If 7^ is of infinite representation type it follows that some localisation R(P) of/? is of infinite representation type (otherwise each localisation would be artinian (being of finite representation type) and so R would be artinian, hence (R would then have only finitely many maximal ideals) R would be a product of rings (its localisations) of finite representation type, and so would itself be of this sort). The class of /?(P)-modules forms an axiomatisable subclass of the class of Rmodules [9]: unfortunately it need not be finitely axiomatisable—yet we need this for constructing a sentence in the language of /^-modules (from one in the language of i?(P)-modules) which is satisfied only by infinite-dimensional modules. If, however, R actually is artinian then (as in [8]) this class will be finitely axiomatisable and so we obtain the following. COROLLARY 2.6. Let R be a finite-dimensional commutative K-algebra which is not of finite representation type. Then Tf£ # TH.

3. Modules over regular rings In this section we consider decidability of the theory of /^-modules where R is a commutative regular ring (the results could be extended easily at least to abelian regular rings—these have 'enough' central idempotents).


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For such a ring R let B(R) denote the boolean algebra of idempotents of R and let B*(R) be the Stone space (space of ultrafilters in B(R)). Recall that there is a homeomorphism between B*(R) and the space Spec (R) of maximal ideals of R with the usual topology, given by Ft-* £ {eR: eeB(R)\F}, where B(R)\Fis the ideal dual to Fe B*(R), and /H-> B(R)\(I n B(R)) for 7a maximal ideal of R. We shall usually work in terms of the space of maximal ideals. It is easy to see that for each / e Spec (R) the quotient R/I is a field and for any module MR, the quotient M/MI is a vector space over R/I. For each prime power pn define Q(pn) to be the set of those points / of Spec (R) such that the ring R/I is a field with pm elements for some m^n:

Q(pn) = {/e Spec (R): R/I - F(/>ffl) for some m \ n). To Q(pn) we may associate F(/?n) = {eeB(R): 0{e) n Q(/?n) ^ 0 } , where