TOWARDS THE DECIDABILITY OF THE THEORY OF MODULES OVER FINITE COMMUTATIVE RINGS

GENA PUNINSKI AND CARLO TOFFALORI

Abstract. On the basis of the Klingler–Levy classification of finitely generated modules over commutative noetherian rings we approach the old problem of classifying finite commutative rings R with decidable theory of modules. We prove that if R is (finite length) wild, then the theory of all R-modules is undecidable, and verify decidability of this theory for some classes of tame finite commutative rings.

1. Introduction

Int

Let A be a finite universal algebra in a language consisting of finitely many functional symbols and let V (A) denote the variety generated by A. As a far reaching consequence of the theory of decidable locally finite varieties the following result has been proved (see ^{\underline{M-V}} [17, p. 195, Corollary 14.2]). There is an algorithm that produces from A a finite (associative) ring R such that the theory of V (A) is decidable iff the theory T_{R} of all R-modules is decidable. Thus it appears to be important to classify finite rings with decidable (first order) theory of modules. This open problem was included in ^{\underline{M-V}} [17, p. 194, Problem 2] and then discussed at the beginning of ^{\underline{Poi2}} [19]. Also ^{\underline{Preb}} [21, Chapter 17, p. 350, Problem 1] treated it within the wider question of classifying all the sufficiently recursive rings with a decidable theory of modules.

Indeed both ^{\underline{Preb}} [21, Chapter 17] and ^{\underline{Poi2}} [19] are good accounts on what had been known on decidability of modules over finite (more generally sufficiently recursive countable) rings; not so much has happened since then. We will slightly update their account in the case of finite rings and clarify how recent developments could affect certain points of view.

The main conjecture is still the same as in ^{\underline{Preb}} [21, p. 332]: the theory of all modules over a finite ring R is decidable if and only if the category of finitely generated (hence finite) R-modules is tame. Unfortunately there is no generally accepted definition of tameness for finite rings, therefore one should stick with an informal one. We say that R is tame (or rather finitely generated tame) if one could ‘classify’ finitely generated R-modules, and R is wild otherwise. On the other hand there is a precise definition for R to be of finite representation type and one can prove (see ^{\underline{Preb}} [21, p. 333]) that in this case the theory of all R-modules is decidable.

2000 Mathematics Subject Classification. 03C60, 03B25, 16D50,16P10,16G60.

Key words and phrases. Theory of modules, decidability, finite commutative ring, Ziegler spectrum, Klingler–Levy classification.

If R is a finite dimensional algebra over a (finite) field then we can borrow a standard definition of tameness from representation theory (see ^{\underline{Sim}} [31, p. 290]). However it is not completely clear whether this definition corresponds to the intuitive one, its model theoretic meaning is obscure, and it is still unknown whether every finite dimensional algebra over a finite field is tame or wild (Drozd ^{\underline{Dz}} [7] proved a tame-wild dichotomy for finite dimensional algebras over algebraically closed fields).

Despite these drawbacks there is at least one advantage of using a functorial definition of wildness. Recall that a prototypical example of a wild algebra is given by the free algebra k⟨X, Y ⟩ (or even by some of its finite dimensional quotients) over a field k, and it is known that the theory of k⟨X, Y ⟩-modules (where k is an effectively given field) is undecidable. It is quite often possible to convert the existing proof of wildness of an algebra (or ring) R into an interpretation of the theory of k⟨X, Y ⟩-modules within the theory of R-modules showing that the latter is undecidable. Although this approach has been successfully utilized for many classes of wild algebras, the arguments still carry on ad hoc features and no general proof of ‘wildness implies undecidability’ has appeared yet. The best ‘uniform’ result is due to Prest ^{\underline{Pre-wild}} [22]: if R is a strictly wild finite dimensional algebra (in fact the assumptions are apparently weaker) then the theory of R-modules interprets the theory of k⟨X, Y ⟩-modules.

Concerning the second part of the conjecture — ‘tameness implies decidability’ for modules over finite rings — recent advances are quite modest. As follows from Baur ^{\underline{Bau}} [4] the theory of all quadruples of vector spaces over a finite (or sufficiently recursive countable) field is decidable. This implies that the theory of modules over the finite dimensional algebra kD̃_{4} (over an effectively given field k) is decidable. This algebra represents a well understood class of hereditary finite dimensional algebras over a field k. If k is effectively given and A is the path algebra of a Euclidean or Dynkin quiver, the proof of decidability of the theory of all A-modules (via Ziegler spectrum and pp-interpretations) was outlined by Prest ^{\underline{Pre-dec}} [20] and was eventually carried through by Geisler ^{\underline{Geis}} [10]. For instance, if k is algebraically closed, then every finite dimensional hereditary algebra A over k is a path algebra of a quiver without relations, and (using known results on wild hereditary finite dimensional algebras) one concludes (see ^{\underline{Preb}} [21, Theorem 17.22]) that the theory of A-modules is decidable iff A is tame.

If k is not algebraically closed, then (see ^{\underline{D-R}} [6]) there are more cases to consider, and Geisler’s long proof is difficult to generalize. For the Kronecker algebra kÃ_{1} below we will give a different proof of decidability based on a very clear description of the Ziegler topology in ^{\underline{Pre-zig}} [23] and ^{\underline{Rin-zig}} [30]. However some unexpected complications will make it quite lengthy.

Switching from algebras to general (finite) rings could also add complexity — the proof of the reduction from finite rings to finite dimensional algebras in ^{\underline{Preb}} [21, p. 345–346] appears to be incomplete. Thus our variant of the proof of decidability of modules over some classes of finite rings will also overcome this difficulty.

The situation for more complex classes of tame finite dimensional algebras is even less satisfactory. Indeed the progress on decidability of modules was limited to very few cases (for some classes of domestic string algebras described in ^{\underline{B-P}} [5], the decidability should follow from the description of the Ziegler topology).

To sum up, the conjecture that the theory of all modules over a (finite) ring R is decidable iff R is tame definitely still is consistent with what is known, but needs to be tested on a wider range of examples.

In this paper we investigate decidability of the theory of modules over finite commutative rings. One reduction is easy: every finite commutative ring R is a finite direct sum of local rings, therefore we may assume that R is local. A crucial ingredient for investigating in this case is the voluminous Klingler–Levy (KL for short) investigation of finitely generated modules over commutative noetherian rings (^{\underline{K-L1}} [15] and ^{\underline{K-L2}} [16] will be enough for our purposes) where the tame-wild dichotomy acquires a very precise meaning.

Suppose that R is a local commutative noetherian ring. Then ^{\underline{K-L1, K-L2}} [15, 16] show that either R has an artinian triad or Drozd ring as a homomorphic image, or R itself is a homomorphic image of a local Dedekind-like ring or is isomorphic to a Klein ring (see Section 3 for unexplained terminology). Furthermore in the former case R is wild (or rather finite length wild) where the term has precise meaning similar to the one in representation theory. On the other hand, with one small exception, in the latter case R is (informally) tame — ^{\underline{K-L2}} [16] provides a complete list of indecomposable finitely generated Rmodules. When R is finite, even this small deficiency in the KL-classification disappears, therefore we have a clear borderline between the wild and tame cases.

Here is the plan of our paper. We will recall in Section 2 the main facts about decidability of modules, and we will summarize in Section 3 the crucial points of the KL-analysis of modules over commutative noetherian rings.

After that we will deal with our main results. They are two-fold. Firstly we prove that wildness (as it is defined in ^{\underline{K-L1}} [15]) implies undecidability for modules over a finite commutative ring. As we have already mentioned there are two cases to consider here. If R is an artinian triad the undecidability of the theory of all R-modules has been known (at least in principle —see ^{\underline{J-L}} [14, Proposition 8.69 and Example 8.37]). To prove undecidability for modules over Drozd rings we employ a variant (see ^{\underline{K-L1}} [15, Section 4]) of Ringel’s proof of wildness of the category of finite dimensional modules over Drozd algebras. With some effort we will extract from this proof an interpretation of the theory of k⟨X, Y ⟩-modules (where k is the residue field of R) in the theory of modules over any Drozd ring R showing that the latter is undecidable.

Secondly we tackle in Sections 5–9 one tame case in the KL-classification. Namely we prove that the theory of all modules over a (finite commutative) Klein ring is decidable. As it is customary nowadays we will exploit an approach to decidability through the Ziegler spectrum (see ^{\underline{Pre-dec}} [20] for a detailed description of this approach). Thus to investigate decidability we have to classify the points of the Ziegler spectrum of a Klein ring (that is, indecomposable pure-injective modules) and also the topology of this space. The description of points follows from the known classification of indecomposable pure-injective kÃ_{1}-modules by applying reduction modulo the radical (see ^{\underline{J-L}} [14, p. 211] for the latter). The basis of open sets for the Ziegler topology can be similarly extracted from general results by Prest ^{\underline{Pre-zig}} [23] and Ringel ^{\underline{Rin-zig}} [30]. Unfortunately their description of the topology is not of the form that is required to prove decidability. Thus we will spend a great amount of time (and space) to overcome this difficulty. In detail, Section 5 will state the main decidability result as Theorem ^{\underline{Klein}} 5.1 and give the main steps of this proof. Section 6 illustrates the reduction from Klein rings to the Kronecker algebra kÃ_{1}, and Section 7 provides the aforementioned description of the Ziegler spectrum of kÃ_{1}. Section 8 transfers this analysis to an arbitrary Klein ring R via a suitable homeomorphism from the Ziegler spectrum of R to a large clopen subset of the spectrum of kÃ_{1} (actually consisting of all but one points). On this basis we will eventually show in Section 9 the decidability result for modules over finite Klein rings.

Finally we will briefly discuss in Section 10 decidability for the only remaining case of the KL-classification; in our setting this concerns finite factors of (complete local) Dedekind-like rings. It can be derived from general structure theory (see ^{\underline{K-L2}} [16]) that the decidability question can be reduced to the case when R is a pullback of a direct sum of (at most) two finite valuation rings and a field (see Section ^{\underline{S-conl}} 10 for precise description). A typical representative of this family is given by a Gelfand–Ponomarev algebra G_{2,3}(k) = k[X, Y : X^{2} = Y^{3} = XY = 0] (k is a finite field). According to the aforementioned conjecture, the theory of G_{2,3}(k)-modules has to be decidable, but we have very little to say (at least in this paper) to support it even for this particular algebra.

Because the main targets of this paper are logicians and experts in general (including commutative) ring and module theory and we can expect that their knowledge of representation theory might be sparse, we will be very scrupulous in explaining some instances of representations of hereditary finite dimensional algebras, that are certainly well known to experts. On the other hand those experts may find in this paper a new encouragement in applying representation theory to decidability of modules.

Before concluding this section let us introduce some notation that will be useful later. For any commutative ring R, mod-R denotes the class of finitely generated R-modules; L_{R} is the first order language of R-modules (as described, for instance, in ^{\underline{Preb}} [21, p. 2]) and T_{R}, as said, their first order theory.

2. Decidability. General Discussion

Dec-gen

Informally speaking, a theory T in an effectively given countable first order language L is decidable iff there is an algorithm producing for each sentence σ of L (as an input) the answer whether σ ∈ T or not (as an output), and undecidable otherwise. One can express this in a rigorous way via Church’s Thesis by saying that T is decidable iff the set of Gödel numbers of its theorems is recursive, and undecidable otherwise (see ^{\underline{Ra}} [25] for a discussion of this point and precise definitions).

When T is given by an effective (that is, recursively enumerable) set of axioms, then by applying deductions we can effectively produce a list of theorems of T, and thus (see again ^{\underline{Ra}} [25], or directly ^{\underline{Preb}} [21, Section 17.3]) T is decidable iff one can effectively produce a list of non-theorems of T, that is if also the complement of T is recursively enumerable. Taking negations this is the same as for every sentence σ in L to answer (uniformly) the question whether σ is true in some model of T. Of course if T is complete this difficulty disappears (hence every recursively axiomatizable complete theory is decidable), but in this paper we will mostly deal with incomplete theories.

In fact, we are going to consider the first order theory T_{R} of modules over a finite ring R. Of course, we can assume that R is given by a list of its elements 0, 1, r_{2}, …, r_{n} such that the operations +, −, × can be executed effectively, more generally that R is sufficiently recursive in the sense of \underline{Preb} [21, Section 17.1] and consequently that the decision problem of T_{R} makes sense. Furthermore T_{R} is recursively (indeed finitely) axiomatizable, and so to prove that T_{R} is decidable we have to list effectively the sentences in L_{R} which are true in some R-module. There is an elaborate way to do this using the Ziegler spectrum. This method was introduced by Ziegler ^{\underline{Zig}} [33] and developed by Prest (see ^{\underline{Pre-dec}} [20]). But first we have to give some definitions.

A positive-primitive formula (pp-formula) φ(x̄) in the free variables x̄ = (x_{1}, …, x_{n}) is an L_{R}-formula of the form ∃ ȳ(ȳA = x̄B), where ȳ = (y_{1}, …, y_{k}) is a tuple of bounded variables, A is a k × l and B is an n × l matrix over R. If M is a module and \overline{m} = (m_{1}, …, m_{n}) is a tuple of elements of M, then we write M ⊧ φ(\overline{m}), and we say that M satisfies φ(\overline{m}), if there is a tuple n̄ = (n_{1}, …, n_{k}) in M^{k} such that n̄A = \overline{m}B. The set φ(M) = \{\overline{m} ∈ M^{n} | M ⊧ φ(\overline{m})\} is easily seen to be a subgroup of the abelian group M^{n} (and is called the pp-subgroup defined by φ). Moreover φ(M) is a submodule of M over the ring S = End(M) of R-endomorphisms of M (via the diagonal action of S); for a commutative R, φ(M) is a submodule of M also over R. For instance if r ∈ R, then r | x ≐ ∃ y (yr = x) is a pp-formula such that (r | x)(M) = Mr for any R-module M.

For every integer n > 0, the pp-formulae φ(x̄) in n free variables x̄ (more precisely their classes with respect to the equivalence relation identifying two of them if and only if they define the same subgroup in each R-module M) form a modular lattice. The underlying order relation is defined as follows: for φ(x̄) and ψ(x̄) two (equivalence classes of ) pp-formulae, ψ(x̄) ≤ φ(x̄) if and only if ψ(M) ⊆ φ(M) for every R-module M, and hence if and only if ψ(x̄) implies φ(x̄) in T_{R}; accordingly ψ(x̄) < φ(x̄) means that ψ(M) ⊆ φ(M) for every R-module M but ψ(M) ≠ φ(M) for some Rmodule M. The meet of two pp-formulae φ(x̄) and ψ(x̄) is (the class of) their conjunction φ(x̄) ∧ ψ(x̄), while their join is (the class of) a pp-formula defining in every R-module M the sum φ(M) + ψ(M) (and is consequently denoted by φ(x̄) + ψ(x̄)).

An inclusion M ⊆ N of R-modules is said to be pure if for every \overline{m} ∈ M and every pp-formula φ(x̄), from N ⊧ φ(\overline{m}) it follows that M ⊧ φ(\overline{m}). A module M is pure-injective if it is injective with respect to pure embeddings. For instance every injective module is pure-injective, and so is every finite module over a finite ring.

The Ziegler spectrum Zg_{R} of a ring R is a topological space whose points are indecomposable pure-injective modules. The basis of open neighborhoods for Ziegler topology is given by the following sets (φ/ψ) = {M ∈ Zg_{R} | φ(M)/(φ ∧ ψ)(M) ≠ 0}, where φ and ψ are pp-formulae in the same number of free variables (actually pp-formulae with exactly one free variable suffices, see ^{\underline{Preb}} [21, Chapter 4]). By ^{\underline{Zig}} [33, Theorem 4.9] with respect to this topology Zg_{R} is a compact space.

If φ and ψ are pp-formulae and M is an R-module then Inv(M, φ, ψ) denotes the cardinality of the quotient group φ(M)/(φ ∧ ψ)(M) if it is finite, and ∞ otherwise. Clearly for every positive integer n the condition Inv(M, φ, ψ) ≥ n can be expressed by a ∀ ∃-sentence Inv(−, φ, ψ) ≥ n and consequently is preserved under elementary equivalence. Furthermore it is a consequence of the theorem of Baur and Monk (see ^{\underline{Zig}} [33, Cor. 1.5]) that every complete theory of modules can be axiomatized by sentences Inv(−, φ, ψ) ≥ n or their negations. Accordingly these sentences and their finite Boolean combinations are called invariant statements and, for every R-module M, the corresponding values Inv(M, φ, ψ) are called elementary invariants (as they characterize the elementary equivalence class of M).

Our main interest in the Ziegler spectrum is the following remarkable result (see ^{\underline{Zig}} [33, Thm. 9.4] or ^{\underline{Preb}} [21, Thm. 17.12]) which we adapt to finite rings.

dec-crit Fact 2.1. Let R be a finite ring with a countable Ziegler spectrum. Suppose further that there is an effective enumeration N_{1}, N_{2}, … of points of Zg_{R} and an effective enumeration of a basis (φ_{1}/ψ_{1}), (φ_{2}/ψ_{2}), … of open sets for Ziegler topology. Assume also that Inv(N_{i}, φ_{j}, ψ_{j}) can be effectively calculated for each i and j. Then the theory of R-modules is decidable.

Note that over a finite ring R an effective list of a basis for Ziegler topology is trivial to produce. Namely make a effective list of all pp-formulae φ_{1}, φ_{2}, … (say in one variable, which suffices — see ^{\underline{Preb}} [21, Chapter 4]) and then rearrange it into a list of pp-pairs (φ_{i}/φ_{j}) such that φ_{i} does not imply φ_{j} (hence this pair is non-trivial), where the last condition can be decided effectively. Unfortunately this brute force approach is not very effective — the main difficulty of calculating elementary invariants will become enormous; so the main point is to choose open basis for the Ziegler topology with a great care.

It could happen that a finite ring R has an uncountable Ziegler spectrum so the above result is not applicable — this is the case for Gelfand– Ponomarev algebras G_{n,m}(k) = k[X, Y : X^{n} = Y^{m} = XY = 0] (n, m ≥ 2, n + m ≥ 5) over a finite field k — see ^{\underline{P-P}} [24]. But to investigate decidability in this paper Fact ^{\underline{dec-crit}} 2.1 is all what we need.

Having fixed a means to prove decidability we turn now to undecidability. The main tool to prove undecidability of theories of modules is via interpretations. For a general definition of interpretation of classes of algebraic structures see ^{\underline{M-V}} [17, pp. 9–10], or also ^{\underline{Ra}} [25]. The main point for us will be the following. If one elementary class of algebraic structures interprets another elementary class with undecidable theory, then the the theory of the former class is also undecidable. We will use two particular instances of this construction (therefore two modeling examples of classes with an undecidable theory).

Recall that for every field k, k⟨X, Y ⟩ denotes the free algebra in two noncommuting variables over k. It is well known that, if k is sufficiently recursive, in particular finite, the theory of all k⟨X, Y ⟩-modules is undecidable (see ^{\underline{Preb}} [21, Thm. 17.13]). To interpret this theory in the theory of all modules over a ring R it suffices to find pp-formulas φ(ū), ψ(ū) and φ_{X}(ū, v̄), φ_{Y} (ū, v̄) such that the following holds. For every k⟨X, Y ⟩-module M there exists an R-module N such that the quotient group N′ = φ(N)/(φ ∧ ψ)(N) inherits a structure of k-vector space, φ_{X} and φ_{Y} define an action (of X and Y) on N′ and, equipped with these actions, N′ is isomorphic to M as a k⟨X, Y ⟩- module. Thus to prove undecidability of the theory of R-modules it suffices to interpret the theory of k⟨X, Y ⟩-modules in the theory of R-modules in the just described (that is, pp-definable) way.

Another modeling example for undecidability is given by (abelian) structures (V, W, f), where W ⊆ V are vector spaces over a field k, W is a subspace of V and f is a (k-linear) endomorphism of V. Again (see ^{\underline{Preb}} [21, Corollary 17.7]) if k is sufficiently recursive, then the theory of the class of these structures is undecidable. To interpret this class in the theory of Rmodules it suffices to find pp-formulae φ_{V} (ū), φ_{W} (v̄) and φ_{f}(ū, v̄) such that the following holds. For every triple (V, W, f) there is an R-module N such that φ_{W} (N) ⊆ φ_{V} (N) are k-vector spaces, φ_{f} defines a k-linear action f̄ on φ_{V} (N) and with respect to this action the structure (φ_{V} (N), φ_{W} (N), f̄) is isomorphic to (V, W, f). So the theory of triples (V, W, f) is interpreted in T_{R}, and even in a pp-definable way.

This is almost all we will need from logic (or recursion theory) to discuss decidability. As we will see the remaining part is purely algebraic. Indeed we are in a position to make the first (well known) reduction. If R is a finite commutative ring, then it is a (finite) direct sum of indecomposable, hence local rings. The following remark shows that decidability of the theory of R-modules can be decided componentwise.

comp Remark 2.2. Let R = R_{1} ⊕⋯⊕R_{n} be a decomposition of a (finite commutative) ring R into a direct sum of local rings. Then the theory of R-modules is decidable if and only if for each i the theory of all R_{i}-modules is decidable.

Proof. The direct product is a very particular case of a generalized product as it is defined in Feferman and Vaught ^{\underline{F-V}} [8, Section 2]. The result follows immediately from ^{\underline{F-V}} [8, Theorem 5.4]. □

3. Klingler–Levy classification

KL-class

In this section we summarize what we need for this paper from the remarkable KL-classification (see ^{\underline{K-L1}} [15], ^{\underline{K-L2}} [16]) of finitely generated modules over commutative noetherian rings. By Remark ^{\underline{comp}} 2.2 we may assume that R is a finite local commutative ring. But let us momentarily refer to the wider setting the KL-classification applies to, that is, to a local commutative noetherian ring R (for our purposes — see below — we may additionally assume that R is complete). Let J denote the Jacobson radical (that is, the unique maximal ideal) of R and let k = R/J be its residue field (again for our purposes we may assume that k is finite). If M is a finitely generated R-module then μ_{R}(M) will denote the minimal number of generators for M, hence (by Nakayama lemma) the dimension of M/MJ as a k-vector space.

We say that R is an artinian triad if μ_{R}(J) = 3 and J^{2} = 0. Thus every artinian triad is an artinian ring. A typical example of an artinian triad is given by a finite dimensional algebra k[X, Y, Z : X^{2} = Y^{2} = Z^{2} = XY = XZ = Y Z = 0].

Further R is said to be a Drozd ring if μ_{R}(J) = μ_{R}(J^{2}) = 2, J^{3} = 0 and x^{2} = 0 for some x ∈ J \ J^{2}. Again any Drozd ring is artinian. A prototype of Drozd rings is the Drozd algebra k[X, Y : X^{2} = Y^{3} = XY^{2} = 0]. A constructive description of Drozd rings as special subrings of principal ideal domains can be found in ^{\underline{K-L1}} [15, Thm. 6.5].

As it will be explained in few lines artinian triads and Drozd rings are ‘minimal wild’. In this perspective their counterpart are the so-called ‘maximal tame’ rings, that is, Klein rings and Dedekind-like rings.

R is a Klein ring if μ_{R}(J) = 2, μ_{R}(J^{2}) = 1, J^{3} = 0 and x^{2} = 0 for every x ∈ J. A typical example of Klein ring is given by the group algebra kG, where G is the Klein group and k is a field of characteristic 2.

Finally R is said to be a Dedekind-like ring if R is reduced (that is has no nilpotent elements) and, if Γ is the normalization of R (that is its integral closure in the ring of quotients), then Γ is a direct sum of (at most two) principal ideal domains, μ_{R}(Γ) = 2 and J equals the Jacobson radical of Γ. If R is complete then (see ^{\underline{K-L1}} [15, p. 351] this implies that either

i) Γ/J ≅ k that is, R = Γ is a discrete valuation domain; or

ii) Γ/J is a 2-dimensional extension of k in which case R is called unsplit; or

iii) Γ/J ≅ k×k and Γ is a direct sum of two noetherian valuation domains in which case R is said to be strictly split.

For various examples of Dedekind-like rings see ^{\underline{K-L2}} [16, Section 12]. Let us mention among them the algebras k[X, Y : XY = 0] (whose quotients include Gelfand–Ponomarev algebras) — they are strictly split Dedekindlike rings.

What will be essential for us is the following adaptation of Klingler–Levy dichotomy theorem (see ^{\underline{K-L1}} [15, Theorem 2.10]). Note that in our case k = R/J is finite, hence Γ/J as in ii) is a separable extension of k. This allows us to avoid the only unsettled case in KL-classification, the one just regarding unsplit rings and inseparable extensions (see a discussion in ^{\underline{K-L1}} [15, p. 357]).

Theorem 3.1. (Klingler–Levy dichotomy) Suppose that R is a finite local

dich

commutative ring. Then exactly one of the following holds.

1) R projects itself onto an artinian triad or a Drozd ring, or

2) R is either a Klein ring or a (proper) homomorphic image of some strictly split or unsplit complete local Dedekind-like ring with finite residue field.

Furthermore in case 1) the category of finite R-modules is wild, but in case 2) it is possible to classify finite R-modules.

We do not need a precise definition of wildness (see ^{\underline{K-L1}} [15, Definition 2.2]) in this paper, but rather some intermediate steps in the proof of that. Neither classification of finite R-modules in case 2) will play an essential role in this paper. However all this may be needed for the concluding discussion in Section 10.

Note that Theorem ^{\underline{dich}} 3.1 explains why artinian triads and Drozd rings can be meant as minimal wild, and Klein rings and Dedekind-like rings as maximal tame.

4. Undecidability

Undec

In this section we prove that the wild case in the KL analysis leads to undecidability. Namely if R is a finite commutative local ring that projects itself onto an artinian triad or a Drozd ring then the theory of all R-modules is undecidable. This confirms the ‘wild implies undecidable’ conjecture for finite commutative rings. Clearly it suffices to prove undecidability of the theory of modules for artinian triads and Drozd rings.

The case of an artinian triad has been already tackled. Namely the theory of R-modules interprets the theory of k⟨X, Y ⟩-modules in a pp-definable way. This is implicitly shown in ^{\underline{J-L}} [14, Proposition 8.69 and Example 8.37], but see also ^{\underline{GLW}} [11, Lemma 3] which in its turn refers to ^{\underline{War}} [32, Theorem 1].

art-triad Proposition 4.1. Let R be an artinian triad and let k = R/J be the residue field of R. Then the theory of R-modules interprets in a pp-definable way the theory of k⟨X, Y ⟩-modules. Consequently, if R is finite, then the theory of R-modules is undecidable.

Now let us treat Drozd rings. Suppose that R is a Drozd ring with Jacobson radical J and let k = R/J be its residue field. By the definition of Drozd ring there is x ∈ J \ J^{2} such that x^{2} = 0. It follows that J = ⟨x, y⟩ is generated by x and y for some y, therefore J^{2} = ⟨xy, y^{2}⟩. Consequently every element a ∈ J can be (nonuniquely) written as

(1)  a = a(x)x + a(y)y + a(xy)xy + a(y^{2})y^{2} ,

where a(x), a(y), a(xy) and a(y^{2}) are units in R or zero.

Drozd Theorem 4.2. Let R be a Drozd ring with the Jacobson radical J and residue field k = R/J. The theory of R-modules interprets in a pp-definable way the class (V, W, f), where V is a k-vector space, W is a subspace of V, and f is an endomorphism of V. Consequently if R is finite then the theory of all R-modules is undecidable.

Proof. The latter claim is a direct consequence of the former (see a discussion in Section ^{\underline{Dec-gen}} 2). So let us show how to interpret in R-modules the triples (V, W, f). We rearrange here the approach pursued for R-modules of finite length in ^{\underline{K-L1}} [15, Section 4] (see also ^{\underline{Rin-loc}} [26, p. 295]).

Let J = ⟨x, y⟩ as explained before. If r ∈ R then by r_{y} we denote the class r + Ry in the factor ring R/Ry; similarly r_{xy} denotes the class r + Rxy in the quotient R/Rxy. The following statement and diagram are taken from \underline{K-L1} [15, Lemma 4.3].

diag Fact 4.3. 1) All nonzero monomials x, xy, x_{xy}, … in diagram below are

nonzero.

2) soc(R1_{y}) = Rx_{y}, soc(R1_{xy}) = Rx_{xy} ⊕Ry_{xy}^{2} and soc(R) = Rxy ⊕Ry^{2} = J^{2}.

3) Rx ∩ Ry = Rxy and Rx_{xy} ∩ Ry_{xy} = 0.

4) Rx ≅ Ry_{xy} as R-modules via the multiplication map cx → cy_{xy}.

[figure: R1_{y}; R1_{xy}; R; Rx_{y}; Rx_{xy}; Ry_{xy}; Rx; ≅; mult; Ry; Ry_{xy}^{2}; Rxy; Ry^{2} .]

Let us choose a k-basis of W and extend it to a k-basis of V. Thus

W = k^{(β)}, V = k^{(α)}

for some cardinals β ≤ α. Construct an R-module M as follows. First form the direct sum

N = \frac{R^{(β)}}{R^{(β)}y} ⊕ \frac{R^{(α)}}{R^{(α)}xy} ⊕ R^{(α)}

[figure: R^{(β)}1_{y}; R^{(α)}1_{xy}; R^{(α)}; R^{(β)}x_{y}; R^{(α)}x_{xy}; R^{(α)}y_{xy}; R^{(α)}x; ≅; mult; R^{(α)}y; R^{(α)}y_{xy}^{2}; R^{(α)}xy; R^{(α)}y^{2}; ι; f]

fig1

Figure 1

and put for simplicity R_{0} = R^{(β)}, R_{1} = R^{(α)}. Note that R_{0} embeds into R_{1} in the natural way; for every a ∈ R_{0} let a′ denote its image via this embedding (so the juxtaposition of a and a suitably long zero sequence). Also note that R_{1}/R_{1}J is isomorphic to k^{(α)} so to V as an R-module, and hence as a k-vector space. Under this point of view, for c ∈ R_{1}, c^{∗} ∈ R_{1} is determined by the condition

f(c + R_{1}J) = c^{∗} + R_{1}J .

Now we want to make some identifications in N hence factor N by some submodule N(0). To see this factorization clearly let us reproduce one more diagram from ^{\underline{K-L1}} [15, p. 368] (see Figure 1).

Thus we make the identifications shown by the 3 arrows on the diagram. So let N(0) be a submodule of N generated by {ax_{y} − a′y_{xy}^{2} | a ∈ R_{0}}, {by_{xy} − bx | b ∈ R_{1}} and {cx_{xy} − c^{∗}y^{2} | c ∈ R_{1}}. Thus a generic element of N(0) is of the form

(2)  (ax + R_{0}y, −a′y^{2} + by + cx + R_{1}xy, −bx − c^{∗}y^{2})

with a ∈ R_{0}, b, c ∈ R_{1}.

Set M = N/N(0). Looking at the representation of elements of J in the form (1) we may assume the following.

(∗) Each component a_{i}, i < β of a and c_{i}, i < α of c in N(0) is either zero or invertible, and each component b_{i}, i < α of b is either invertible or of the form b_{i}(y)y, where b_{i}(y) ∈ R in its turn is either zero or invertible.

Now we show that V, W and f can be recovered in M by some suitable pp-formulae. More precisely, V lives in M as R_{1}/R_{1}J ≅ k^{(α)} at the place R^{(α)}y_{xy}^{2} (see Figure 1) and W as R_{0}/R_{0}J ≅ k^{(β)} at the place R^{(β)}x_{y} (hence W is identified with a subspace of V via ι).

Thus we define V by the following formula

φ_{V} (v) ≐ ∃ u (uxy = 0 ∧ v = uy^{2}) ,

while W is given by

φ_{W} (v) ≐ ∃ u (uy = 0 ∧ v = ux) .

The definition of f is more complex.

φ_{f}(v, v̄) ≐ ∃ u ū t (uxy = 0 ∧ v = uy^{2}∧
∧ūxy = 0 ∧ v̄ = ūy^{2} ∧ tx = ūy ∧ ty^{2} = ux) .

To understand the definition of φ_{f} we suggest to look at the following diagram.

[figure: u,ū; t; ux; ūy; tx; mult; ty; v̄=ūy^{2}, v=uy^{2}; ty^{2}; f]

Thus we use t to ‘twist’ v via f to get v̄.

Here are the details of this interpretation. First let us deal with φ_{V} (v). Let us calculate the kernel of xy in M, so the image of this kernel in y^{2} will give us φ_{V} (M). Take an element of M, so the N(0) class of the triple in N

(d + R_{0}y, e + R_{1}xy, g)

with d ∈ R_{0}, e, g ∈ R_{1}. Its image via xy is

(R_{0}y, R_{1}xy, gxy) ,

and it is zero in M if and only if it equals in N an element (2) from N(0)

(ax + R_{0}y, −a′y^{2} + by + cx + R_{1}xy, −bx − c^{∗}y^{2}) ,

where a ∈ R_{0} and b, c ∈ R_{1} satisfy (∗). Let us compare the elements of these triples componentwise. Equating the first coordinates we obtain ax ∈ R_{0}y. If a_{i} is a unit for some i < β, then a_{i}x ∈ Ry implies x ∈ Ry, whence J = Ry is principal which is impossible for Drozd ring. By (∗) it follows that a_{i} = 0 for any i, whence a itself is 0, as well as a′.

Now comparing the second coordinates we obtain by + cx ∈ R_{1}xy, hence b_{i}y + c_{i}x ∈ Rxy for each i < α. If b_{i} is a unit then y ∈ Rx, a contradiction. So by (∗) b_{i} is of the form b_{i}(y)y where b_{i}(y) is either 0 or invertible, therefore b_{i}(y)y^{2} + c_{i}x ∈ Rxy. But then c_{i} has to be 0 otherwise by (∗) c_{i} invertible, hence x ∈ ⟨y^{2}, xy⟩ = J^{2}, a contradiction. Because the sum of Ry^{2} and Rxy is direct (see Fact ^{\underline{diag}} 4.3), also b_{i}(y) is zero. In conclusion b = c = 0 and hence we can assume c^{∗} = 0 as well.

Equating the third coordinates we obtain gxy = 0. It follows that g_{i}xy = 0, hence g_{i} ∈ J for each i < α. To summarize, the generic element in the kernel of xy in M is the N(0)-class of a triple

(d + R_{0}y, e + R_{1}xy, g) ,

where no component of g ∈ R_{1} is a unit. Hence its image via y^{2} is the class in M of

(R_{0}y, ey^{2} + R_{1}xy, 0) .

Conversely, every element of this form can be seen as the y^{2}-image of the N(0)-class of the triple (R_{0}y, e + R_{1}xy, 0) taken by xy to (R_{0}y, R_{1}xy, 0), that is to zero.

Now observe that, for every e as before, ey^{2} ∈ R_{1}xy iff e ∈ R_{1}⟨x, y⟩ = R_{1}J whence the map

ey^{2} + R_{1}xy ↦ e + R_{1}J

yields an isomorphism between {ey^{2}+R_{1}xy | e ∈ R_{1}} and (R/J)^{(α)} ≅ k^{(α)} ≅ V. In conclusion φ_{V} (v) defines in M a k-vector space R_{1}/R_{1}J isomorphic to V.

Now let us consider W and φ_{W} (v). First we calculate the kernel of y in M, so the image of this kernel in x will give us φ_{W} (M). Take a generic element of M, so the N(0)-class of a triple (d+R_{0}y, e+R_{1}xy, g) with d ∈ R_{0} and e, g ∈ R_{1}. It image via y

(3)  (R_{0}y, ey + R_{1}xy, gy) .

is equal to 0 in M. Thus this element coincides in N with some element of the form (2)

(ax + R_{0}y, −a′y^{2} + by + cx + R_{1}xy, −bx − c^{∗}y^{2}) ,

where a ∈ R_{0} and b, c ∈ R_{1} satisfy (∗). Let us compare this element with element (3) componentwise. As above equating the first coordinates one sees that a and a′ must be 0. Comparing the two remaining coordinates we obtain for each i < α,

(a) −e_{i}y + b_{i}y + c_{i}x ∈ Rxy and

(b) g_{i}y = −b_{i}x − c_{i}^{∗}y^{2}.

If c_{i} is a unit then (a) implies x ∈ Ry, which is impossible. Otherwise by (∗) c_{i} = 0 for every i < α, hence c = 0 and then we may assume that c^{∗} = 0.

In particular c_{i}^{∗} = 0 for every i, hence (b) becomes g_{i}y = −b_{i}x. If b_{i} is invertible, this implies x ∈ Ry, a contradiction. Otherwise by (∗) again b_{i} = b_{i}(y)y, where b_{i}(y) is either zero or invertible, hence (b) can be written as g_{i}y = −b_{i}(y)xy. Since y ∉ Rxy we conclude that g_{i} is not invertible. Decomposing g_{i} ∈ J as in (1) we obtain g_{i}y = g_{i}(x)xy + g_{i}(y)y^{2}. Thus g_{i}(x)xy + g_{i}(y)y^{2} = −b_{i}(y)xy yields g_{i}(y) = 0 (since y^{2} ∉ Rxy by Lemma ^{\underline{diag}} 4.3). Because each g_{i}, i < α, is not invertible and g_{i}(y) = 0 we conclude that g_{i}x = 0, hence gx = 0.

Now (a) can be written as

−e_{i}y + b_{i}(y)y^{2} ∈ Rxy .

It follows that e_{i} is not a unit hence (see representation (1)) e_{i}x ∈ Rxy for every i < α and then ex ∈ Rxy. Thus the image in x of our generic triple is

(d + R_{0}y, e + R_{1}xy, g) · x = (dx + R_{0}y, R_{1}xy, 0) ,

which is equivalent in M to

(R_{0}y, −d′y^{2} + R_{1}xy, 0) .

Conversely every such element is an image in x of some element in the kernel of y. Thus the set of N(0)-classes of these triples yields the subspace W inside V isomorphic to R_{0}/R_{0}J ≅ k^{(β)}.

Finally let us deal with φ_{f}(v, v̄). We claim that this formula defines f when V ≅ R_{1}/R_{1}J is recovered in M in the way shown before.

First we prove that any two elements v, v̄ ∈ V such that f(v) = v̄ satisfy φ_{f}(v, v̄). We can write v = (R_{0}y, ey^{2} + R_{1}xy, 0), v̄ = (R_{0}y, e^{∗}y^{2} + R_{1}xy, 0), where

f(e + R_{1}J) = e^{∗} + R_{1}J .

Choose u = (R_{0}y, e + R_{1}xy, 0), ū = (R_{0}y, e^{∗} + R_{1}xy, 0), and put t = (R_{0}y, R_{1}xy, −e^{∗}). Clearly uxy = 0, uy^{2} = v and ūxy = 0, ūy^{2} = v̄ held in N and consequently in M. It remains to check that tx = ūy and ty^{2} = ux. Indeed

tx = (R_{0}y, R_{1}xy, −e^{∗}x) and ūy = (R_{0}y, e^{∗}y + R_{1}xy, 0)

represent the same class in M. Furthermore

ty^{2} = (R_{0}y, R_{1}xy, −e^{∗}y^{2}) and ux = (R_{0}y, ex + R_{1}xy, 0)

are equal in M because f(e + R_{1}J) = e^{∗} + R_{1}J.

Conversely, take two elements v, v̄ ∈ N such that their classes in M satisfy φ_{f}(v, v̄). In particular there are u, ū ∈ M such that uxy = 0, uy^{2} = v and ūxy = 0, ūy^{2} = v̄. By the first part of the proof (the description of the kernel of xy)

u = (d + R_{0}y, e + R_{1}xy, g) and ū = (d̄+ R_{0}y, ē + R_{1}xy, ḡ)

for some d, d̄∈ R_{0}, e, ē ∈ R_{1} and g, ḡ ∈ R_{1} such that no component of g and ḡ is a unit. It follows that

v = (R_{0}y, ey^{2} + R_{1}xy, 0) and v̄ = (R_{0}y, ēy^{2} + R_{1}xy, 0) .

Furthermore there exists t ∈ M such that ty^{2} = ux and tx = ūy. Suppose that t = (r + R_{0}y, s + R_{1}xy, h) for some r ∈ R_{0} and s, h ∈ R_{1}. Thus

ty^{2} = (R_{0}y, sy^{2} + R_{1}xy, hy^{2})

is equal in M to ux = (dx+R_{0}y, ex+R_{1}xy, gx) which clearly coincides with

(dx + R_{0}y, R_{1}xy, gx − e^{∗}y^{2}) .

Comparing ty^{2} with ux we obtain that, for some a ∈ R_{0} and b, c ∈ R_{1} satisfying (∗), the following holds

(a) dx − ax ∈ R_{0}y,

(b) sy^{2} + a′y^{2} − by − cx ∈ R_{1}xy, and

(c) hy^{2} = gx − e^{∗}y^{2} − bx − c^{∗}y^{2}. If some component of c is a unit, then (b) implies x ∈ Ry, a contradiction.

Otherwise by (∗) c = 0, therefore we can assume c^{∗} = 0 as well.

Recall that no component g_{i} of g is a unit and then observe that in (c) g_{i}x has the form g_{i}(y)xy, where g_{i}(y) is either 0 or invertible. Furthermore by (c) again, no component of b is a unit, whence (∗) yields b_{i} = b_{i}(y)y where b_{i}(y) is either 0 or invertible. So in (c) for every i < α we have

h_{i}y^{2} = −e_{i}^{∗}y^{2} + (g_{i}(y) − b_{i}(y))xy .

Since xy ∉ Ry^{2} we conclude that g_{i}(y) − b_{i}(y) is not a unit, therefore (g_{i}(y) − b_{i}(y))xy = 0 for any i. In conclusion hy^{2} = −e^{∗}y^{2}.

Now let us examine the further equality tx = ūy in M. Note that

ūy = (R_{0}y, ēy + R_{1}xy, ḡy)

and tx = (rx+R_{0}y, sx+R_{1}xy, hx) which equals modulo N(0) (hence in M)

(rx + R_{0}y, sx − hy + R_{1}xy, 0) .

Comparing these triples, we get, for a suitable choice of a ∈ R_{0} and b, c ∈ R_{1} satisfying (∗), the following.

(a′) rx − ax ∈ R_{0}y, (b′) sx − hy − ēy + a′y^{2} − by − cx ∈ R_{1}xy, (c′) ḡy = −bx − c^{∗}y^{2}. As above, (c′) implies that no component b_{i} of b can be a unit, hence

by (∗) b_{i} = b_{i}(y)y, where b_{i}(y) is either 0 or a unit. Similarly by (b′) no component of s − c in R is a unit, whence (s − c)x ∈ R_{1}xy and (b′) becomes

−hy − ēy + a′y^{2} − b(y)y^{2} ∈ R_{1}xy ,

where b(y) = (b_{i}(y))_{i<α}. Multiply by y and get ēy^{2} = −hy^{2}, hence

ēy^{2} = e^{∗}y^{2} ,

in other words f(e + R_{1}J) = ē + R_{1}J, as claimed. □

At this point it is straightforward to deduce (from Proposition ^{\underline{art-triad}} 4.1 and Theorem ^{\underline{Drozd}} 4.2) the following general result.

undec-final Corollary 4.4. Suppose R is a finite commutative ring that is wild in the Klingler–Levy classification (that is, R projects itself onto an artinian triad or a Drozd ring). Then the theory of all R-modules is undecidable.

5. Klein rings. The main result

S-Decid

The aim of the next five sections is to prove the following.

Klein Theorem 5.1. Let R be a finite commutative Klein ring. Then the theory of all R-modules is decidable.

Actually this result will be shown only in Section 9, but the intermediate sections will prepare its proof and clarify several preliminaries.

First let us recall once again the definition of Klein ring (see Section ^{\underline{KL-class}} 3): A finite local commutative ring R with Jacobson radical J (and residue field k = R/J) is said to be a Klein ring if J is 2-generated, J^{2} is a principal ideal of R, J^{3} = 0 and x^{2} = 0 for every x ∈ J. In particular any Klein ring is an artinian ring of length 4.

Furthermore, it is easily shown (see ^{\underline{K-L1}} [15, Lemma 2.9]) that the residue field of R has characteristic 2, and R itself has characteristic 2 or 4. As we have already mentioned, a typical example of a Klein ring is given by the group ring kG, where k is a field of characteristic 2 (finite in our case) and G = C(2)^{2} is the Klein group. Another example, which is not an algebra over a field, is the ring ℤ/4ℤ[x : x^{2} = 0].

It follows from ^{\underline{K-L2}} [16, Theorem 1.13] that every Klein ring is quasi-Frobenius with the socle J^{2}, therefore ^{\underline{J-L}} [14, Proposition 8.69] yields that every R-module is a direct sum of a free R-module and a module over the ring R′ = R/J^{2}. Note also that, because R is artinian, the class of free R-modules is axiomatizable (see ^{\underline{Preb}} [21, Theorem 14.28]). Thus by ^{\underline{F-V}} [8, Theorem 5.4] to prove that the theory of all R-modules is decidable it suffices to show that the theory of free R-modules is decidable, and the theory of all R′-modules is decidable. The first case is easy (in fact it works for any finite ring).

free-Klein Lemma 5.2. The theory of free modules over a finite Klein ring is decidable.

Proof. Every free R-module is of the form R^{(I)} for some set I. By ^{\underline{F-V}} [8, Theorem 6.4] again the theory of this class is decidable if the theory of R_{R} is decidable. Because R is finite, all the elementary invariants Inv(R_{R}, φ, ψ) can be calculated effectively. Thus the theory T of R_{R} can be recursively axiomatized by invariant sentences. Since T is complete, it is decidable. □

Thus to verify Theorem ^{\underline{Klein}} 5.1 it suffices to prove the following.

J2 Proposition 5.3. Suppose that R is a finite commutative ring whose Jacobson radical is 2-generated and satisfies J^{2} = 0. Then the theory of all R-modules is decidable.

Note that any further assumption on R, that is, R being a homomorphic image of a Klein ring, or having characteristic 2 or 4, is not essential for this proposition.

Incidentally it may be useful to observe that the characteristic 2 case is easy to treat. In other words the following holds.

char2 Remark 5.4. Let R be a finite (or even sufficiently recursive) Klein ring of characteristic 2. Then the theory of R-modules is decidable.

In fact, by I. S. Cohen’s Theorem, R, as a local commutative noetherian complete equicharacteristic ring, is an algebra over some field k of characteristic 2. Indeed R can be regarded as a quotient algebra of the algebra k[X, Y : X^{2} = Y^{2} = 0] through a finitely generated ideal, and even equals this algebra itself or its quotient through the socle k[X, Y : X^{2} = Y^{2} = XY = 0] ^{\underline{Poi}} [18], where k is finite (or sufficiently recursive). Hence R-modules, as a finitely axiomatizable subclass of the class of modules over k[X, Y : X^{2} = Y^{2} = 0], have a decidable theory.

Before approaching the proof of Proposition ^{\underline{J2}} 5.3 we should first explain quite a lot, as said. One technical tool we need is the so-called ‘reduction modulo radical’ functor (see ^{\underline{J-L}} [14, p. 221]). This will be the matter of the next section.

6. From Klein rings to kÃ_{1}

ss-red

For the definition and basic properties of path algebras of quivers the reader can consult ^{\underline{ARS}} [2, Section III.1] or ^{\underline{ASS}} [1, Section II.1]. In this paper we need just one particular example of a path algebra. Let k be a field (with an intention to use R/J for k) and let kÃ_{1} denote the Kronecker algebra over k. Thus kÃ_{1} is the path algebra of the following quiver

[figure: ∘; 1; α; β; ∘; 2]

In particular, kÃ_{1} is a hereditary 4-dimensional algebra with the following basis: e_{1} (the idempotent corresponding to the vertex 1), e_{2} (the idempotent corresponding to the vertex 2), and α, β such that e_{1} · α = α · e_{2} = α and e_{1} · β = β · e_{2} = β (all undefined products are set to be zero). Every kÃ_{1}- module can be seen as the direct sum of two vector spaces V and W over k with two linear maps α and β from V to W (see ^{\underline{ASS}} [1, Section III.1] for this way of defining modules over path algebras). We will sometimes denote it by (V, W).

One of the main tools in the oncoming proof of decidability will be, as already said, a functor F, called ‘reduction modulo radical’, from the category of R-modules to the category of kÃ_{1}-modules. In this section we gather some well known properties of F, mostly when it is restricted to the category of pure-injective modules.

First let us introduce F in detail. Choose generators x, y for J and recall that k = R/J is a finite field.

Let M be a (right) R-module. Then M/MJ is an R/J-module, hence a vector space over k. From J^{2} = 0 it follows that J annihilates MJ, hence MJ is also an R/J-module, therefore a vector space over k. We assign to M the following kÃ_{1}-module F (M)

\xymatrix{M/MJ \ar[r]^{α= ×x} \ar[r]_{β= ×y} & MJ ,}

where α : M/MJ → MJ is given by multiplication by x, and β is given by multiplication by y. Because x, y ∈ J and J^{2} = 0, both maps are well defined and, since R is commutative, k-linear. Thus we have defined F on objects.

To complete the definition of F, let N be another R-module and let f : M → N be a morphism (of R-modules). Thus, the kÃ_{1}-module F (N) is given by the following diagram

\xymatrix{N/NJ \ar[r]^{α= ×x} \ar[r]_{β= ×y} & NJ .}

Then (see ^{\underline{ASS}} [1, Section III.1] again) F (f) has to be given by a pair of klinear mappings f_{1} : M/MJ → N/NJ and f_{2} : MJ → NJ such that the two corresponding diagrams commute. One of them is

\xymatrix{M/MJ \ar[r]^{α= ×x} \ar[d]_{f_{1}} & MJ \ar[d]^{f_{2}} \\ N/NJ \ar[r]_{α= ×x} & NJ ,}

and the other is similar with α replaced by β.

Since f : M → N is a morphism, we have f(MJ) ⊆ NJ, hence define f_{2} to be the restriction of f to MJ, and let f_{1} : M/MJ → N/NJ denote the map induced by f.

It is easily verified that F is an additive covariant functor from the category of R-modules to the category of kÃ_{1}-modules. The properties of F we will list below are taken (or can be easily derived) from ^{\underline{J-L}} [14, Section at p. 211] and ^{\underline{ARS}} [2, Section X.2].

Clearly F preserves direct limits (hence direct sums) and direct products. Furthermore F is full, but not faithful. Namely, if M and N are Rmodules, then the kernel of the mapping Hom(M, N) → Hom(F (M), F (N)) is Hom(M, NJ), that is, consists of morphisms f : M → N whose image is contained in NJ. Since J^{2} = 0, it follows easily that, for every R-module M, the kernel of the surjection End(M) → End(F (M)) belongs to the Jacobson radical of End(M). It follows that F reflects (and preserves) the property of being isomorphic.

By ^{\underline{J-L}} [14, Proposition 8.61], F preserves and reflects pure-injectivity, that is if M is an R-module, then M is pure-injective iff F (M) is a pure-injective kÃ_{1}-module. Furthermore if M is pure-injective, then M is indecomposable iff F (M) is indecomposable. Note that every finite R-module is pureinjective so as every finite kÃ_{1}-module, and F preserves and reflects the property of being finitely generated (hence finite). Thus the above applies to finite modules.

Note that F is not a dense functor. Indeed, for every R-module M, its image F (M) = (V, W) clearly has the property V J = V α + V β = W. For instance, the simple (projective) kÃ_{1}-module S_{2} = (0, k) is not isomorphic to F (N), for any R-module N. However using projective covers, it can be shown that an kÃ_{1}-module K = (V, W) is isomorphic to F (M) for some R-module M iff V J = W; and every kÃ_{1}-module L is of the form L = F (N) ⊕ L′, where N is an R-module and L′ is a direct sum of copies of S_{2}.

Note that kÃ_{1} is isomorphic to the k-algebra ( \begin{matrix}k & k⊕k \\ 0 & k \end{matrix} ), where k ⊕ k is considered as a k-k-bimodule via the diagonal action of k. Thus we can apply the results of ^{\underline{ARS}} [2, Chapter X]. For instance (by ^{\underline{ARS}} [2, Theorem X.2.4]) the category mod-R of finite R-modules is stably equivalent to the category mod-kÃ_{1} of finite kÃ_{1}-modules. Although it is beyond the scope of this paper, note that by ^{\underline{Gei}} [9, Corollary 3.9] this fact implies that the Krull–Gabriel dimension of R is 2.

Furthermore using a well known description of irreducible morphisms and almost split sequences in the category of finite dimensional kÃ_{1}-modules (see some examples below), we can derive a similar description of irreducible morphisms and almost split sequences in the category of finite R-modules (see \underline{ARS} [2, Chapter V] for definitions). Namely by ^{\underline{ARS}} [2, Proposition 2.5] a short exact sequence of finite R-modules 0 → A → B → C → 0, where A and C are indecomposable and A is not simple, is almost split in mod-R iff 0 → F (A) → F (B) → F (C) → 0 is an almost split sequence in mod-kÃ_{1}. Thus evaluating by F we can find ‘almost all’ almost split sequences in mod-R. Furthermore by ^{\underline{ARS}} [2, Lemma X.1.2] we have the following property of irreducible maps: if M and N are indecomposable finite non-projective R-modules, then the mapping Hom(M, N) → Hom(F (M), F (N)) induces an isomorphism of kvector spaces of irreducible maps Irr(M, N) → Irr(F (M), F (N)).

Suppose that S is a simple R-module, that is, S ≅ R/J. Then F (S) = (R/J, 0) ≅ S_{1} = (k, 0) = e_{1}kÃ_{1}/ Jac(e_{1}kÃ_{1}) is a simple injective kÃ_{1}- module, in particular S_{1} is not a source of an almost split sequence in mod-kÃ_{1}. However, since S is not injective, by ^{\underline{ARS}} [2, Proposition V.3.5] it is a source of an almost split sequence 0 → S → B → C → 0 in mod-R, where B is a projective cover of C. It easily follows that, if f_{x} : S → R is given by multiplication by x, f_{y} : S → R is defined similarly, and

f = (f_{x}, f_{y}) : S → R^{2}, then 0 → S ⟶^{f} R^{2} ⟶ R^{2}/ im(f) → 0 is an al-

most split sequence in mod-R. Therefore f_{x} : S → R is an irreducible morphism whose image F (f_{x}) : S_{1} → e_{1}kÃ_{1} is zero, hence not irreducible (but left almost split). This is because we cannot apply the aforementioned Auslander’s result on irreducible maps when one of modules is projective.

Note also that F (R) = (R/J, J) = (k, k⊕k) ≅ e_{1}kÃ_{1} and every projective R-module is free, hence (see also ^{\underline{ARS}} [2, Lemma X.2.2]) F preserves (and reflects) projectives.

7. The Kronecker algebra kÃ_{1}

S-kronecker

ss-fd

7.1. Finite dimensional modules over kÃ_{1}. In this section we recall the classification of indecomposable finite dimensional (hence finite, in our case) modules over the Kronecker algebra kÃ_{1}. Since kÃ_{1} is a path algebra of the quiver without relations, it is hereditary, therefore the structure of finite dimensional kÃ_{1}-modules is well understood (see ^{\underline{ARS}} [2, Chapter VIII] or ^{\underline{Rin-inf}} [27]). In particular indecomposable finite dimensional kÃ_{1}-modules can be divided in 3 subclasses

• the preprojectives,

• the regulars,

• the preinjectives.

One way to distinguish among modules in these classes is to look at the so called dimension vectors. If M = (V, W) is a finite dimensional kÃ_{1}- module, then the dimension of M, dim(M), is defined as the ordered pair (dim(V), dim(W)).

Then every indecomposable preprojective kÃ_{1}-module has dimension (n, n + 1) for some integer n ≥ 0, and there is one isomorphism type for each dimension. Here is the shape of an indecomposable preprojective module of dimension (2, 3)

\xymatrix{α \ar[d] & ∘ & α \ar[d] & ∘ & β \ar[d] \\ ∘ & β \ar[r] & ∘ &  & ∘}

which is a typical string module. The morphisms between preprojective modules are also well understood (so as, more generally, the morphisms between string modules). The following diagram of the category of preprojective kÃ_{1}-modules is taken from ^{\underline{Rin-tame}} [28, p. 124].

[figure: (1, 2); P_{2}; (3, 4); P_{4}; τ; …; (0, 1); P_{1}; (2, 3); P_{3}; τ; …]

where solid arrows denote irreducible morphisms (so the space Irr(P_{1}, P_{2}) has dimension 2) and the dashed arrows show Auslander–Reiten (AR-) translates. For instance all nonzero morphisms between indecomposable preprojectives go from the left to the right, and each such morphism is mono.

Dually, every indecomposable preinjective kÃ_{1}-module has dimension (m+ 1, m), m ≥ 0, one isomorphism type for each dimension, and they are also string modules. Here is a diagram for an indecomposable preinjective module of dimension (3, 2)

[figure: ∘; α; ∘; β; α; ∘; β; ∘; ∘]

Furthermore the following diagram represents the category of preinjective kÃ_{1}-modules.

[figure: …; (4, 3); I_{4}; τ; (2, 1); I_{2}; …; (3, 2); I_{3}; τ; (1, 0) ,; I_{1}]

where solid arrows stand for irreducible morphisms and dashed arrows show AR-translates. For instance two linearly independent irreducible epimorphisms from I_{3} to I_{2} are given by either factoring I_{3} by utmost left part shown by bullets, or by utmost right part shown by diamonds.

[figure: •; α; ∘; β; α; ⋄; β; •; ⋄]

Note that the kernels of these two epimorphisms are nonisomorphic (simple regular) kÃ_{1}-modules.

Again, nonzero morphisms between indecomposable preinjective modules go from the left to the right in the above diagram, and each such morphism is epi. If k is finite, then indecomposable preprojective and preinjective kÃ_{1}- modules can be effectively given by generators and relations, so as irreducible maps between them.

The remaining indecomposable finite dimensional kÃ_{1}-modules are called regular. Each regular module has dimension (n, n), n > 0, but with many non-isomorphic modules of the same dimension. The easiest way to comprehend regular modules is to use a special functor G from the category of k[X]-modules to the category of kÃ_{1}-modules. Namely every k[X]-module M can be considered as a k-vector space V, where X acts as an endomorphism. Define G(M) to be the following kÃ_{1}-module:

\xymatrix{V \ar[r]^{α= 1} \ar[r]_{β= X} & V ,}

where α acts as identity, and β acts as multiplication by X. The definition of G(f), where f is a morphism of k[X]-modules, is obvious. Then G is a full and faithful additive covariant functor commuting with direct limits and (arbitrary) products. In particular (by ^{\underline{J-L}} [14, Proposition 8.61]) G preserves and reflects pure-injectivity and the property of being indecomposable (within the class of pure-injective modules). However G is not dense — its image consists of kÃ_{1}-modules (V, W) such that α : V → W is an isomorphism.

Suppose that H is an indecomposable finitely generated torsion k[X]- module, therefore H ≅ k[X]/p^{n}(X)k[X], where p(X) is an irreducible polynomial over k and n ≥ 0. Then G(H), the image of H, will be a regular indecomposable finite dimensional kÃ_{1}-module. For instance, if p = X and n = 1, then G(H) is the following string module

[figure: ∘; α; ∘]

and, if p = X − 1 and n = 1, we obtain the following band module (see ^{\underline{B-R}} [3] for more on string and band terminology)

\xymatrix{∘ \ar[d]_{α=1} \ar[d]^{β=1} \\ ∘}

where α and β act as identity maps.

The functor G covers ‘almost all’ regular kÃ_{1}-modules, except a few. In fact the only regular kÃ_{1}-modules that are not covered are string modules like

\xymatrix{β \ar[d] & ∘ & α \ar[d] & ∘ \\ ∘ &  & ∘ & β \ar[l]}

on which α does not act as an isomorphism (from V to W). One possibility to cure this situation is to consider the second functor G_{1} similar to G, but where β acts as identity, and α acts as multiplication by X.

If k is finite, then we can produce a list of irreducible polynomials over k, hence (applying G or G_{1}) an effective list of regular kÃ_{1}-modules given by generators and relations. When drawing a diagram for the category of finite dimensional kÃ_{1}-modules, one should put regular modules on the right of preprojectives and on the left of preinjectives — then any nonzero morphism goes from the left to the right. For instance (see ^{\underline{Rin-tame}} [28, p. 331]) there is no nonzero morphism from regular to preprojective modules.

Recall (see ^{\underline{ARS}} [2, Chapter VII]) that using irreducible maps we can arrange finite dimensional kÃ_{1}-modules into an Auslander–Reiten (AR-) quiver. We have already seen the connected components of this quiver consisting of preinjectives and preprojectives modules. The remaining components consist of regular modules and are homogeneous tubes.

[figure: ∘; M_{3}; .; .; .; ∘; M_{2}; ∘; M_{1} = M]

The regular kÃ_{1}-module M on the mouth of the tube is called a quasisimple, or simple regular module. Note (see ^{\underline{Rin-inf}} [27]) that regular kÃ_{1}-modules form an abelian category, and simple regular modules are exactly simple objects in this category. Furthermore each quasi-simple module is isomorphic to the module G(H) or G_{1}(H), where H is a simple k[X]-module, therefore H ≅ k[X]/p(X)k[X] for an irreducible polynomial p(X) of degree n. It follows that End(H) ≅ End(G(H)) is a field K which is a simple extension of k of degree n, hence |K| = |k|^{n}. Clearly (since k is finite) this endomorphism ring, and hence the cardinality of |K| can be calculated effectively.

The irreducible mappings in each tube are the images of irreducible morphisms between finitely generated torsion k[X]-modules, so they also can be calculated effectively. For example the multiplication by X defines an irreducible monomorphism f : k[X]/Xk[X] → k[X]/X^{2}k[X], hence its image G(f) will be an irreducible monomorphisms M_{1} → M_{2} in the corresponding tube of the AR-quiver.

Another important tool in the representation theory of kÃ_{1} (more generally, any hereditary finite dimensional algebra) is a (non-symmetric) bilinear form ⟨−, −⟩ : K_{0}(kÃ_{1}) → ℤ defined by ⟨M, N⟩ = dim Hom(M, N) − dim Ext(M, N) with the corresponding quadratic form q(M) = ⟨M, M⟩ (see ^{\underline{ARS}} [2, Section VIII.3]). The value of this bilinear form depends only on dimensions of M and N. Namely if dim(M) = x̄ = (x_{1}, x_{2}) and dim(N) = ȳ = (y_{1}, y_{2}), then ⟨M, N⟩ = ⟨x̄, ȳ⟩ = x_{1}y_{1} − 2x_{1}y_{2} + x_{2}y_{2}, therefore q(M) = x_{1}^{2} − 2x_{1}x_{2} + x_{2}^{2}. For instance q(M) = 1 for any preprojective or preinjective module M, and q(N) = 0 if N is a regular module.

Using this bilinear form it is not difficult to calculate Hom’s and Ext’s between finite dimensional kÃ_{1}-modules. Another possibility to do that is to use the AR-translate τ. Because kÃ_{1} is hereditary, by ^{\underline{ARS}} [2, p. 258], τ provides an equivalence from the full subcategory of mod-kÃ_{1} consisting of modules without projective direct summands onto the full subcategory of mod-kÃ_{1} consisting of modules without injective direct summands. For instance, if M and N are indecomposable non-projective and non-injective kÃ_{1}-modules, then Hom(M, N) ≅ Hom(τM, τN) and Ext(M, N) ≅ Ext(τM, τN) (kvector spaces isomorphisms), and the same is true with τ replaced by τ^{−1}.

Remark 7.1. (cp. ^{\underline{Rin-zig}} [30, p. 114]) If M is an indecomposable regular kÃ_{1}-

hom-I

module, then Hom(M, I) ≠ 0 for every preinjective kÃ_{1}-module I.

Proof. Since I is preinjective, I = τ^{k}I′ for some injective module I′ and some k ≥ 0. Because every tube over kÃ_{1} is homogeneous, we conclude M = τ^{k}M. Then Ext(M, I) = Ext(τ^{k}M, τ^{k}I′) ≅ Ext(M, I′) = 0, since I′ is an injective module. It follows that ⟨M, I⟩ = dim Hom(M, I). From dim(M) = (n, n) and dim(I) = (m, m + 1) for some n > 0, m ≥ 0 we obtain ⟨M, I⟩ = ⟨(n, n), (m + 1, m)⟩ = n(m + 1) − 2nm + nm = n > 0, as desired. □

ss-pi

7.2. Pure-injective modules over kÃ_{1}. Recall (from Section ^{\underline{Dec-gen}} 2) that the Ziegler spectrum of a ring R, Zg_{R}, is a topological space whose points are indecomposable pure-injective R-modules, and whose topology is given by the basic open sets (φ/ψ) = {N ∈ Zg_{R} | φ(N)/(φ ∧ ψ)(N) ≠ 0}, where φ and ψ are pp-formulae (and restricting to formulae with only one free variable is sufficient). In this section we recall a description of the Ziegler spectrum of kÃ_{1}. All the results are taken from Ringel ^{\underline{Rin-zig}} [30] and Prest ^{\underline{Pre-zig}} [23], we only add some explanations.

First of all each indecomposable finite dimensional kÃ_{1}-module is pure injective, therefore such modules are points of Zg_{kÃ_{1}}. It follows from ^{\underline{Pre-zig}} [23, Theorem 2.5] that every infinite dimensional point of Zg_{kÃ_{1}} is the image under G or G_{1} of a non-finitely generated point of Zg_{k[X]}. On the other hand recall (see ^{\underline{Pre-dec}} [20]) that infinitely generated indecomposable pure-injective modules over k[X] are classified as follows.

For every irreducible polynomial p(X) there is a unique p-Prüfer module P_{p} and a unique p-adic module A_{p}. Furthermore there exists a unique generic module Q = k(X). We will call the images of these points under G or G_{1} the Prüfer, adic and generic points of Zg_{kÃ_{1}} (note that G(Q) ≅ G_{1}(Q), hence we have just one generic point Q). Let us say more precisely what those images are.

Let p(X) be an irreducible polynomial over k. Then we have the following ray of irreducible monomorphisms in the category mod-k[X] of finitely generated k[X]-modules:

k[X]/p(X)k[X] ⟶^{×p} k[X]/p^{2}(X)k[X] ⟶^{×p} ….

The direct limit along this ray is the p-Prüfer module P_{p}. Applying G or G_{1} to this ray we obtain a ray of irreducible monomorphisms M_{1} → M_{2} → …, where M = M_{1} is the simple regular kÃ_{1}-module corresponding to k[X]/p(X)k[X]. The direct limit of this directed system will give us an indecomposable pure-injective (M-Prüfer) kÃ_{1}-module P_{M}. To grasp the shape of this module let us consider one example. Suppose that M is the following simple regular string module

[figure: •; α; •]

Then M_{2} has the following diagram

[figure: ∘; α; β; •; α; ∘; •]

where the irreducible monomorphism M_{1} → M_{2} is shown by identifying the bullets. Taking the direct limit along this ray, we obtain that P_{M} has the following diagram

[figure: ∘; α; β; ∘; α; β; •; α; …; ∘; ∘; •]

Then (see ^{\underline{Rin-some}} [29, Section 4]) P_{M} is the so-called direct sum module — the underlying vector space for M is a direct sum of 1-dimensional spaces corresponding to vertices. Furthermore (see ^{\underline{Rin-some}} [29] again) this module is contracting — the shift by 1 to the right is a morphism of P_{M} which is epi but not mono (its kernel consists of the two utmost right vertices, therefore isomorphic to M). The construction of the M-adic module A_{M} is just dual. Namely we should start with a coray of irreducible epimorphisms

k[X]/p(X)k[X] ⟵ k[X]/p^{2}(X)k[X] ⟵ …

given by factoring out the socle, and apply G (or G_{1}) to get a coray of irreducible epimorphisms M = M_{1} ⟵ M_{2} ⟵ … in mod-kÃ_{1}. The inverse limit along this coray will give us the M-adic module A_{M} ∈ Zg_{kÃ_{1}}. For instance, if M is the simple regular string kÃ_{1}-module as above, then A_{M} has the following diagram:

[figure: •; α; β; ∘; α; β; •; ∘; ∘; …]

Now A_{M} is a direct product module — its underlying vector space is the direct product of 1-dimensional spaces corresponding to the vertices. Furthermore this module is expanding — the shift by 1 to the right is a monomorphism f of A_{M} that is not epi, and whose cokernel (shown by bullets) is isomorphic to M.

Note that most Prüfer and adic modules will be counted twice (when applying G or G_{1}), but this does not disturb our proof of decidability.

Finally the generic kÃ_{1}-module Q has the following presentation

\xymatrix{k(X) \ar[r]^{α= 1} \ar[r]_{β= ×X} & k(X) ,}

where k(X) is the field of quotients of k[X]. By ^{\underline{Rin-zig}} [30, Proposition 4] if f : A_{M} → A_{M} is the aforementioned monomorphism, then the direct limit of the chain A_{M} ⟶^{f} A_{M} ⟶^{f} … is isomorphic to a direct sum of copies of the generic module Q.

As we have already noticed in Remark ^{\underline{hom-I}} 7.1, Hom(M, I) ≠ 0 for any regular kÃ_{1}-module M and any preinjective module I. Therefore by ^{\underline{Rin-zig}} [30, Proposition 1] the Prüfer module P_{M} is a direct summand of a direct product of any infinite set of non-isomorphic preinjective modules.

Now we proceed describing the topology on Zg_{kÃ_{1}}. It follows from ^{\underline{Preb}} [21, Proposition 13.4] that the isolated points of this space are exactly the finite dimensional ones. Furthermore because every finite dimensional kÃ_{1}- module has finite endolength, every such point is closed.

The next level of isolation is represented by Prüfer and adic points: those are exactly the points of Cantor–Bendixson (CB for short) rank 1. In fact Ringel ^{\underline{Rin-zig}} [30] gives a nice basis of open neighborhoods of Prüfer and adic points.

Let M be a simple regular kÃ_{1}-modules and let O_{M} consist of P_{M}, all preinjective points, and all regular points on the tube with M on the mouth (often called M-regular points). Then O_{M} is a Ziegler open set containing P_{M}. In fact O_{M} consists of the points N ∈ Zg_{kÃ_{1}} such that Hom(M, N) ≠ 0. Furthermore every open subset of O_{M} containing P_{M} is cofinite, that is, of the form O_{M} \ {N_{1}, …, N_{k}}, where the N_{i} are finite dimensional points. Moreover, because every finite dimensional point is closed, every such set is open.

Similarly, let J_{M} consist of A_{M}, all preprojective points, and all M-regular points. Then J_{M} is a Ziegler open set containing A_{M}. Moreover, every open subset of J_{M} containing A_{M} is of the form J_{M} \ {N_{1}, …, N_{k}}, where the N_{i} are finite dimensional points. Note also that J_{M} consists of the N ∈ Zg_{kÃ_{1}} such that Ext(M, N) ≠ 0.

Finally the only remaining point of Zg_{kÃ_{1}} is the generic point Q. It has CB-rank 2, and the basis of open neighborhoods of Q is given by the collection of open sets Zg_{kÃ_{1}} \{N_{1}, …, N_{k}}, where N_{i} are finite dimensional points. What follows is that Zg_{kÃ_{1}} has CB-rank 2, in particular it is a T_{0}-space. Then the elementary duality (see ^{\underline{Her-dual}} [12]) can be defined pointwise, hence provides a homeomorphism between the left and the right Ziegler spectra of kÃ_{1}. For instance the image of the right Prüfer point P_{M} is the left adic point A_{M′}, where M′ is a left quasi-simple kÃ_{1}-module such that both M and M′ are the images of the same simple k[X]-module. Note also (see ^{\underline{Pre-dec}} [20]) that the functors G and G_{1} induce homeomorphisms onto closed subsets of Zg_{kÃ_{1}}.

ss-more

7.3. kÃ_{1}. More on the Ziegler spectrum. In this section by refining the information on Zg_{kÃ_{1}} we prove the following.

dec0 Theorem 7.2. The theory of all kÃ_{1}-modules (over a finite field k) is decidable.

The result is definitely not new and our proof is hardly shorter. Indeed it is not difficult (see ^{\underline{Poi}} [18] for similar arguments) to interpret the theory of kÃ_{1}- modules in the theory of quadruples of k-vector spaces, hence decidability (over an effectively given field) follows from Baur’s Theorem in ^{\underline{Bau}} [4]. But our main goal is to make a pattern which will be used in the next section to prove decidability in an essentially less friendly environment.

In the previous section we described the topology of Zg_{kÃ_{1}}. However this description is not of the form that is required to prove decidability. For this purpose (see Fact ^{\underline{dec-crit}} 2.1), for each point N ∈ Zg_{kÃ_{1}}, we should effectively produce a basis of open neighborhoods of N of the form (φ/ψ), where φ and ψ are pp-formulae, and calculate elementary invariants Inv(N, φ, ψ). Clearly this is the same as to produce a recursive set of axioms for the theory of N proving that this theory is decidable. The aim of this section is to obtain such a basis for every point of Zg_{kÃ_{1}}.

Note that the basis for the Ziegler topology for N that can be extracted from Prest and Ringel (see previous section) is consisting of open sets of the form ∩_{i=1}^{n}(φ_{i}/ψ_{i}), where φ_{i} and ψ_{i} are pp-formulae. Indeed, if P_{M} is a Prüfer point, then we can take (Hom(M, −)/x = 0) as (φ_{1}/ψ_{1}), and use the remaining (φ_{i}/ψ_{i}) to isolate, hence throw away any finite number of finite dimensional points. By ^{\underline{Zig}} [33, Theorem 4.9] there are pp-formulae φ and ψ such that N ∈ (φ/ψ) and (φ/ψ) ⊆ (φ_{i}/ψ_{i}) for every i. However it is not clear how to find such a pp-pair effectively and how to calculate the corresponding elementary invariant. Thus our proof will overcome this difficulty.

First we introduce some general notation. Suppose that M = ⟨x̄ | x̄A = 0⟩ is a finitely presented module over a ring R given by generators and relations (here x̄ = (x_{1}, …, x_{n}) is a finite tuple of generators of M and A is a rectangular R-matrix). We will assign to this representation the ppformula φ ≐ x̄A = 0 (depending on the choice of representation). If N is an arbitrary R-module, then φ(N) = {n̄ ∈ N^{n} | n̄A = 0} can be identified with Hom(M, N) (both viewed as abelian groups, or even as modules over S = End(N)). Although this identification depends on the representation of M, the condition Hom(M, N) ≠ 0 does not, and this is true exactly when φ(N) ≠ 0. A more rigorous explanation is that the functors Hom(M, −) and φ from the category of R-modules to the category of abelian groups are isomorphic. Thus (φ/x = 0) is an open set in the Ziegler spectrum of R consisting of all N ∈ Zg_{R} such that Hom(M, N) ≠ 0. In fact this is a part of a more general construction.

Suppose that N = ⟨ȳ | ȳB = 0⟩ is another finitely presented R-module and let f : M → N be a morphism such that f(x̄) = ȳC (hence CA = BD for some matrix D with entries in R). Let ψ denote the pp-formula ∃ ȳ(ȳB = 0 ∧ ȳC = x̄), thus ψ generates the pp-type of f(x̄) in N (see \underline{Preb} [21] for unexplained terminology). Then for any R-module L, we have that φ(L)/(φ ∧ ψ)(L) ≠ 0 (that is, L ∈ (φ/ψ)) if and only if there is a morphism g : M → L that cannot be factored through f. Again, both the pp-formulae φ and ψ depend on the choice of representations, but the condition L ∈ (φ/ψ) does not, because (φ/ψ) is isomorphic to the functor Hom(M, −)/I, where I consists of the morphisms from M that can be factored through f. Considering the zero morphism M → 0, we see that this construction includes the previous one.

If f : M → N is a left almost split morphism in the category of finitely presented R-modules, then (φ/ψ) is a minimal pair in the theory of R-modules (again see ^{\underline{Preb}} [21, Section 9.2] for a definition and properties of minimal pairs). If M has a local endomorphism ring, then (see ^{\underline{Her-zig}} [13, Proposition 5.3]) the pure-injective envelope of M is an indecomposable pure-injective module, hence a point of Zg_{R}, and this point is isolated by (φ/ψ).

That being said, let us come back to the points M in Zg_{kÃ_{1}}. Our aim is to equip each of them with a basis as described before.

First suppose that M is a finite dimensional, hence an isolated point of Zg_{kÃ_{1}}. As we have already mentioned, M can be effectively given by generators and relations. Assume first that M is regular, hence M = M_{n} is the image under G (or G_{1}) of the k[X]-module k[X]/p^{n}(X)k[X], where p(X) is an irreducible polynomial. This module is the source of the left almost split monomorphism k[X]/p^{n}(X)k[X] → k[X]/p^{n+1}(X)k[X] ⊕ k[X]/p^{n−1}(X)k[X], where the first coordinate mapping is given by multiplication by p, and the second coordinate one is factoring by the socle. The image of this morphism (under G or G_{1}) is an almost split monomorphism f : M_{n} → M_{n+1}⊕M_{n−1} that can be calculated effectively. Then the minimal pair (φ/ψ) corresponding to f can be calculated effectively, and this pair isolates M_{n}. What remains to do is to calculate Inv(M_{n}, φ, ψ) effectively.

Since (φ/ψ) is a minimal pair in the theory of kÃ_{1}-modules and M_{n} ∈ (φ/ψ), it is a minimal pair in the theory of M_{n}. By ^{\underline{Preb}} [21, Proposition 9.6], Inv(M_{n}, φ, ψ) is the cardinality of End(M_{n})/ Jac End(M_{n}). Since M_{n} has regular length n and is uniserial in the category of regular modules, it is easily seen that End(M_{n})/ Jac End(M_{n}) is isomorphic to the field End(M_{1}) = K (see Section ^{\underline{ss-fd}} 7.1), and (since k is finite) the cardinality of K can be calculated effectively.

Similarly if M is a finite dimensional indecomposable preprojective or preinjective kÃ_{1}-module, then from the description of morphisms between such modules (see Section ^{\underline{ss-fd}} 7.1) we can effectively construct a left almost split morphisms f : M → N in mod-kÃ_{1}, and this mapping defines a minimal pp-pair (φ/ψ) that isolates M. It is easily seen that End(M)/ Jac End(M) is isomorphic to k, hence Inv(M, φ, ψ) is the cardinality of k.

Now consider the case of an M-Prüfer point P_{M} ∈ Zg_{kÃ_{1}}, where M is a simple regular kÃ_{1}-module. Recall that O_{M} is a subset of Zg_{kÃ_{1}} consisting of P_{M}, all M-regular points, and all preinjective points. As we have already mentioned, Hom(M, N) ≠ 0 for N ∈ Zg_{kÃ_{1}} iff N ∈ O_{M}, hence O_{M} is an open subset of Zg_{kÃ_{1}} defined by an effectively given pair (φ/x = 0). We will refine this pair to obtain a basis of open neighborhoods for P_{M}. The problem is to throw away any finite number of M-regular and preinjective points.

But first we address the problem of elementary invariants.

min Proposition 7.3. (Hom(M, −)/x = 0) is a minimal pair in the theory of P_{M}. If k is finite, then the corresponding elementary invariant of P_{M} is equal to |K|.

Proof. Choose a representation M = ⟨x̄ | x̄A = 0⟩, therefore a pp-formula φ ≐ x̄A = 0 that represents Hom(M, −). Thus φ(P_{M}) can be identified with {n̄ ∈ P_{M} | n̄A = 0}.

Note (see ^{\underline{Rin-inf}} [27]) that P_{M} is uniserial in the category of regular modules, and its regular socle is isomorphic to M. It follows that for every morphism f : M → P_{M}, the image f(x̄) is in the regular socle of P_{M}. Since End(M) = K is a (finite) field, every two such nonzero images, say f(x̄) and f′(x̄), can be identified by an automorphism of the socle. Thus φ(P_{M}) has cardinality |K|.

It remains to prove that (φ/x = 0) is a minimal pair in the theory of P_{M}. By ^{\underline{Preb}} [21, Proposition 9.6], it suffices to show that End(P_{M})/ Jac End(P_{M}) ≅ K. Since M is a regular socle of P_{M}, there is a natural mapping (given by restriction) π : End(P_{M}) → End(M) = K. By ^{\underline{Rin-inf}} [27, Proposition 4.7], P_{M} is injective in the category of kÃ_{1}-modules without preinjective direct summands. It follows that π is onto. Since End(P_{M}) is a local ring and K is a field, π is an isomorphism. □

Now for any n, r ≥ 1 we define a refinement (φ_{n}/ψ_{r}) of (φ/x = 0) (in this case refinement means just that each φ_{n} implies φ). Let g_{n} be the composition of irreducible monomorphisms M ⟶^{f_{1}} M_{2} ⟶^{f_{2}} … ⟶^{f_{n−1}} M_{n}, and let φ_{n} generate the pp-type of the image g_{n}(x̄) in M_{n}. Clearly φ_{n} implies φ for every n. In fact, the ‘top’ of the interval (φ/x = 0) in the lattice of pp-formulae over kÃ_{1} is a chain φ > φ_{1} > φ_{2} > …, that is, if any pp-formula ψ satisfies ψ < φ, then either ψ is equivalent to φ_{s} for some s, or ψ ≤ φ_{t} for every t. Indeed, if we consider the image of g_{n}(x̄) under the almost split monomorphism h : M_{n} → M_{n+1} ⊕ M_{n−1}, only the first coordinate of h(g_{n}(x̄)) is nonzero.

Now we define ψ_{r}. It is easily calculated that if dim(M) = (l, l), then dim Hom(M, I_{r}) = l. Let h_{1}, …, h_{l} be a basis of Hom(M, I_{r}), and for every i = 1, …, l let θ_{i} generate the pp-type of h_{i}(x̄) in I_{r}. Then we set ψ_{r} = ∑_{i=1}^{l} θ_{i}. Clearly all this can be calculated effectively.

We claim that (φ_{n}/ψ_{r}) is an open subset of O_{M} containing P_{M} but neither M_{1}, …, M_{n−1} nor I_{r−1}, …, I_{1}. It would clearly follow that the (φ_{n}/ψ_{r}) form a basis of open neighborhoods for P_{M}.

First we show that ψ_{r} ≤ φ_{n} for every n and r, therefore every neighborhood (φ_{n}/ψ_{r}) is nontrivial. Because ψ_{r} = ∑_{i=1}^{l} θ_{i}, it suffices to show that θ_{i} implies φ_{n} for every n (and i). By the definition of φ_{n}, it is enough to prove that the following diagram can be completed

[figure: 0; M; h_{i}; g_{n}; M_{n}; h; I_{r}]

The cokernel of g_{n} is a regular module isomorphic to M_{n−1} and Ext(M_{n−1}, I_{r}) = 0 by Lemma ^{\underline{hom-I}} 7.1, hence h exists.

Now we check that M_{1}, …, M_{n−1} do not belong to (φ_{n}/ψ_{r}). Indeed, otherwise a nonzero \overline{m} ∈ M_{s}, s < n satisfies φ_{n}. In particular there exists a morphism f : M_{1} → M_{s} such that f(x̄) = \overline{m}. Since φ_{n} generates the pp-type of g_{n}(x̄) in M_{n}, there is a morphism h : M_{n} → M_{s} such that the following diagram commutes

[figure: M_{s}; M; f; g_{n}; M_{n}; h]

Since M_{n} has regular length n > s and is uniserial in the category of regular modules, h kills its regular socle M, hence f(x̄) = 0, a contradiction.

Now we claim that M_{s} ∈ (φ_{n}/ψ_{r}) for any s ≥ n. Indeed by the construction, g_{s} factors through g_{n}, hence g_{s}(x̄) satisfies φ_{n} in M_{s}. Suppose that this tuple satisfies ψ_{r}, hence g_{s}(x̄) = \overline{m}_{1} + ⋯ + \overline{m}_{l} for some \overline{m}_{j} ∈ M_{l} such that M_{s} ⊧ θ_{j}(\overline{m}_{j}) (j = 1, …, l). Since h_{j}(x̄) is a free realization of θ_{j}(x̄) in I_{r}, there is a morphism g : I_{r} → M_{s} sending h_{j}(x̄) to \overline{m}_{j}. But (see Section ^{\underline{ss-fd}} 7.1) every morphism from a preinjective to a regular module is zero, therefore \overline{m}_{j} = 0, and then g_{s}(x̄) = 0, a contradiction.

Since P_{M} is a union of the M_{s}, s ≥ n, it follows easily that g_{n}(x̄) ∈ M_{n} ⊆ P_{M} satisfies φ_{n} but not ψ_{r} in P_{M}, hence P_{M} ∈ (φ_{n}/ψ_{r}).

It remains to prove that I_{s} ∉ (φ_{n}/ψ_{r}) for every s < r. Suppose that 0 ≠ n̄ ∈ I_{s} satisfies φ_{n}. In particular, there exists a map f : M → I_{s} such that f(x̄) = n̄. Since s < r, there is an epimorphism g : I_{r} → I_{s} which is a composition of irreducible epimorphisms whose kernels are not isomorphic to M. This is possible because (see Section ^{\underline{ss-fd}} 7.1) at each step there are two irreducible morphisms with non-isomorphic regular kernels. We claim that the following diagram can be completed

[figure: M; h; f; I_{r}; g; I_{s}; 0]

Arguing by induction we may assume that r = s + 1. Then the kernel of g is a simple regular string module S of dimension (1, 1). Since M is not isomorphic to S, therefore Ext(M, S) = 0, which yields an extension as desired.

If \overline{m} = h(\overline{x}) ∈ I_{r}, then I_{r} ⊧ ψ_{r}(\overline{m}) by the definition of that. Applying g we obtain I_{s} ⊧ ψ_{r}(\overline{n}), hence I_{s} ∉ (φ_{n}/ψ_{r}).

Note that there is no problem in calculating the elementary invariants Inv(P_{M}, φ_{n}, ψ_{r}). Indeed, by Proposition ^{\underline{min}} 7.3, (φ/x = 0) is a minimal pair in the theory of P_{M}, therefore all these invariants are equal to |K|.

A similar construction is possible for adic points. But to avoid technicalities we will use the following trick. Construct a basis of open neighborhoods for a left Prüfer point P_{M} as above, and then apply the elementary duality. Since A_{M} is dual to P_{M}, we obtain an effective basis of open neighborhoods for A_{M}. Moreover, by construction (or by the general theory — see ^{\underline{Her-dual}} [12]) the elementary invariants will be the same.

What remains is to construct an effective basis of open neighborhoods for the generic point Q of Zg_{kÃ_{1}} and calculate elementary invariants. Recall that Q has the following diagram

\xymatrix{k(X) \ar[r]^{α= 1} \ar[r]_{β= ×X} & k(X) ,}

Since Q is the image (via G) of the k[X]-module k(X), it follows that End(Q) ≅ k(X) is an infinite field. Hence every elementary invariant in Q is either 0 or ∞. Clearly Q = Qe_{1}⊕Qe_{2} is a decomposition of Q as an End(Q)- module, and both Qe_{1} and Qe_{2} are 1-dimensional vector spaces over End(Q) (recall the e_{1} and e_{2} denote the idempotents in the basis of kÃ_{1} introduced at the beginning of Section ^{\underline{ss-red}} 6). In particular (by ^{\underline{Preb}} [21, Proposition 9.6]) (e_{1} | x/x = 0) is a minimal pair in the theory of Q. Thus for every pp-formula φ implying e_{1} | x we have either φ(Q) = 0 or φ(Q) = Qe_{1}. It follows from \underline{Zig} [33, Theorem 4.9] that a basis of open neighborhoods for Q can be chosen among the pairs (φ/ψ) such that ψ < φ ≤ e_{1} | x. Thus the collection {(φ/ψ) | ψ < φ ≤ e_{1} | x and ψ(Q) = 0, φ(Q) ≠ 0} forms a basis of open neighborhoods for Q.

As T_{kÃ_{1}} is recursively axiomatizable, we can effectively list the pairs (φ/ψ) such that ψ < φ ≤ e_{1} | x. So the only thing to check is whether φ(Q) = 0 or not (and the same for ψ). We will reduce this question to a similar question about an adic module A_{M}. As we have already proved that the theory of A_{M} is decidable, this question can be answered effectively.

Choose any regular simple module M. Let A_{M} be the corresponding adic module, and let f be a monomorphism of A_{M} which is not epi, and whose cokernel is isomorphic to M (see an example in Section ^{\underline{ss-pi}} 7.2). Then, by ^{\underline{Rin-zig}} [30, Proposition 4], the direct limit A_{M} ⟶^{f} A_{M} ⟶^{f} … is isomorphic to a direct sum of copies of Q. Now it is easily seen that φ(Q) = 0 iff φ(A_{M}) = 0.

Thus we have constructed an effective basis of open neighborhoods for each point of Zg_{kÃ_{1}} and calculated the corresponding elementary invariants. By Fact ^{\underline{dec-crit}} 2.1 it follows that the theory of all kÃ_{1}-modules is decidable.

8. Klein rings again. The Ziegler spectrum

ss-Klein

We deal here again with Klein rings. More precisely, we consider a finite (or countable effectively given noetherian) local commutative ring R whose Jacobson radical J is 2-generated and J^{2} = 0. For instance every quotient of a Klein ring modulo its socle is such (see the beginning of Section ^{\underline{S-Decid}} 5). In this section we describe the Ziegler spectrum of R. Recall that in Section ^{\underline{ss-red}} 6 we introduced a functor F, the reduction modulo radical, from the category of R-modules to the category of kÃ_{1}-modules, where kÃ_{1} is the Kronecker algebra over the residue field k = R/J. In particular, this functor preserves and reflects pure-injectivity and the property of being isomorphic. Furthermore when restricted to the category of pure-injective modules F preserves and reflects the property of being indecomposable. Also F sends finite R-modules to finite kÃ_{1}-modules and vice-versa.

Recall (see Section ^{\underline{ss-pi}} 7.2) that every indecomposable pure-injective kÃ_{1}- module is preprojective, regular, preinjective, or Prüfer, adic, or generic. We say that an indecomposable pure-injective R-module is preprojective (regular, …) if its image F (M) is such. Of course, there are some drawbacks in this terminology. For instance if S = R/J is the unique simple R-module, then F (S) = I_{1} is a simple injective kÃ_{1}-module, although S is not injective.

One more caution should be taken. Recall that the image of F consists of kÃ_{1}-modules (V, W) such that V J = W, and every kÃ_{1}-module is of the form F (M) ⊕ P_{2}^{(α)}, where P_{2} is a simple projective kÃ_{1}-module. Thus every indecomposable pure-injective kÃ_{1}-module but P_{2} is the image via F of the unique indecomposable pure-injective R-module. Since {P_{2}} is clopen in Zg_{kÃ_{1}}, therefore F induces a bijection from Zg_{R} onto the clopen set Zg_{kÃ_{1}} \{P_{2}}. In fact this map respects topology.

top Proposition 8.1. F induces a homeomorphism from Zg_{R} onto Zg_{kÃ_{1}} \{P_{2}}.

Proof. Because R is an Artin algebra, by ^{\underline{ARS}} [2, Theorem V.1.15], the category of finite R-modules has almost split sequences. From ^{\underline{Preb}} [21, Proposition 13.1] it follows that every finite point of Zg_{R} (that is, every indecomposable finite R-module) is isolated, and those are the only isolated points of Zg_{R}. Thus F behaves well on isolated points.

Let P_{M} be an M-Prüfer point of Zg_{R} associated with a simple regular module M — this rather means that F (P_{M}) is an F (M)-Prüfer point of Zg_{kÃ_{1}} associated with the simple regular module F (M). Let O_{M} be a subset of Zg_{R} consisting of P_{M}, all M-regular points, and all preinjective points. Recall (see Section ^{\underline{ss-pi}} 7.2) that F (O_{M}) is open in Zg_{kÃ_{1}} and every open subset of F (O_{M}) containing F (P_{M}) is of the form F (O_{M}) \ {N_{1}, …, N_{k}}, where the N_{i} are finite points. Furthermore F (O_{M}) is defined by a pair (φ/x = 0), where φ corresponds to Hom(F (M), −) (see Section ^{\underline{ss-more}} 7.3).

We will show that O_{M} is an open set in Zg_{R}, and every its open subset containing P_{M} is of the form O_{M} \{M_{1}, …, M_{k}}, where the M_{i} are finite points. Consider the functor Hom(M, −)/ Hom(M, −)J whose value at an R-module N is (the abelian group) Hom(M, N)/ Hom(M, NJ) ≅ Hom(F (M), F (N)). With respect to a given representation M = ⟨x̄ | x̄A = 0⟩ of M this functor is isomorphic to the pp-pair (φ/ψ), where φ ≐ x̄A = 0 and ψ ≐ J | x̄, viewed as a functor. Therefore O_{M} = (φ/ψ) is an open set which contains P_{M}.

Since every finite point in Zg_{R} is closed (being of finite endolength), each set O_{M} \ {M_{1}, …, M_{k}} is open. Thus a restriction of F to O_{M} is continuous. To prove that this restriction is a homeomorphism it remains to check the following. If T = {M_{1}, M_{2}, … } is an infinitely set of (pairwise non-isomorphic) finite points in O_{M}, then P_{M} belongs to \overline{T}, the closure of T. Clearly we may assume that either each M_{i} is M-regular, or each M_{i} is preinjective.

Suppose that each M_{i} is M-regular. Then lifting irreducible mappings from kÃ_{1}-modules, we can arrange the M_{i} into a ray M_{n_{1}} → M_{n_{2}} → … by compositions of irreducible morphisms. Because F preserves direct limits, the direct limit along this ray is isomorphic to P_{M}, hence P_{M} is in the closure of T.

Assume now that every M_{i} is preinjective. By Ringel ^{\underline{Rin-zig}} [30, Proposition 1], F (P_{M}) is a direct summand of a direct product of the F (M_{i}). Then P_{M} is a direct summand of a direct product of the M_{i}, hence P_{M} ∈ \overline{T}.

Thus F is a homeomorphism when restricted to the open neighborhood O_{M} of P_{M}. By dual arguments (or elementary duality) F preserves and reflects the basis of open neighborhoods of any adic point A_{M}.

It remains to look at the generic point Q. Recall that a basis of open neighborhoods for F (Q) is given by the sets Zg_{kÃ_{1}} \{N_{1}, …, N_{k}}, where N_{i} are finite points. Since every finite point in Zg_{R} is closed, the preimage of this set in Zg_{R} is open.

Thus it suffices to prove that for every set T = {M_{1}, M_{2}, … } of (different) points of Zg_{R} \{Q} which either is infinite or contains an infinite point, Q is in the closure of T.

First we prove that \overline{T} contains an infinite point. Indeed if T contains infinitely many preinjective points, then, as we have already seen, \overline{T} contains (any) Prüfer point. By (elementary) duality, if T contains infinitely many preprojective points, its closure will contain all adic points.

If T contains infinitely many M-regular points for a given simple regular module M, then (as above, by using direct limits) it is easily shown that P_{M} ∈ \overline{T}. Thus it remains to consider the case when T contains infinitely many regular points M_{1}, M_{2}, … with non-isomorphic regular socles. But in this case by ^{\underline{Rin-zig}} [30, Proposition 5], Q is a direct summand of the module ∏_{i} M_{i}/ ⊕_{i} M_{i}, hence Q ∈ \overline{T}.

Thus we may assume that either P_{M} ∈ T or A_{M} ∈ T. Again by elementary duality it suffices to consider the latter case. As we have already mentioned, Q is a direct summand of a direct limit of copies of A_{M}, hence Q ∈ \overline{T}. □

Note that the functor F clearly defines a pp-interpretation (see ^{\underline{Geis}} [10, Definiton 1.1]) of the theory of kÃ_{1}-modules without P_{2} as a direct summand in the theory of R-modules.

ques Question 8.2. Is the theory of R-modules interpretable in the theory of kÃ_{1}-modules?

If this were the case then most results of this section (on Zg_{R}) and of the next section (on decidability) would become trivial (modulo the corresponding results for kÃ_{1}-modules). However we believe that this not true, because too much ‘individuality’ of R is lost when applying F.

9. Klein rings. Decidability

ss-Kdec

In this section we prove Proposition ^{\underline{J2}} 5.3, therefore Theorem ^{\underline{Klein}} 5.1. Recall that R is a finite commutative ring with Jacobson radical J such J^{2} = 0 and J is 2-generated. By Fact ^{\underline{dec-crit}} 2.1 to prove decidability of the theory of all R-modules it suffices to equip each point of Zg_{R} with an effective basis of open neighborhoods and calculate the corresponding elementary invariants.

Recall (from the previous section) that each point of Zg_{R} is preprojective, regular, preinjective, or Prüfer, adic, or generic.

Suppose first that M is a finite, hence isolated point of Zg_{R}. If M is not injective, then it is a source of an almost split sequence that can be calculated effectively (say, using ^{\underline{ARS}} [2, Proposition V.2.2]). Let M → N be the corresponding left almost split morphism, and let (φ/ψ) be a pp-pair associated to this morphism as in Section ^{\underline{ss-more}} 7.3. Clearly this pair can be found effectively and isolates M (see ^{\underline{Preb}} [21, Proposition 13.1]).

Similarly if M is an injective finite R-module, then the pp-pair (φ/ψ) corresponding to the (left almost split) mapping M → M/ soc(M) isolates M. Since M is finite, the elementary invariant Inv(M, φ, ψ) can be also calculated effectively.

Now we should do the same for an M-Prüfer point P_{M} (again this rather means that F (P_{M}) is an F (M)-Prüfer point of Zg_{kÃ_{1}}). As in Section ^{\underline{ss-Klein}} 8 we can choose a representation ⟨x̄ | x̄A = 0⟩ of M, and put φ ≐ x̄A = 0 and ψ ≐ J | x̄. Then for every R-module N we obtain

(∗∗) φ(N)/(φ ∧ ψ)(N) = Hom(M, N)/ Hom(M, NJ) ≅ Hom(F (M), F (N)) .

Since F (M) is a simple regular kÃ_{1}-module it follows (taking into account the description of Zg_{kÃ_{1}} in Section ^{\underline{ss-pi}} 7.2) that (φ/ψ) = O_{M} is an open set of Zg_{R} consisting of P_{M}, all M-regular points, and all preinjective points. Using (∗∗) again we obtain φ(P_{M})/(φ ∧ ψ)(P_{M}) ≅ Hom(F (M), F (P_{M})), therefore, by Proposition ^{\underline{min}} 7.3, Inv(P_{M}, φ, ψ) = |K| which can be calculated effectively. Furthermore End(P_{M})/ Jac End(P_{M}) is clearly isomorphic to the ring End(F (P_{M}))/ Jac End(F (P_{M})) ≅ K (see the proof of Proposition ^{\underline{min}} 7.3). It follows from ^{\underline{Preb}} [21, Proposition 9.6] that (φ/ψ) is a minimal pair in the theory of P_{M}.

Thus (arguing as in Section ^{\underline{ss-more}} 7.3) it suffices to refine this pp-pair to a pppair (φ_{n}/ψ_{r}) throwing away any finite number of M-regular points M_{1}, …, M_{n−1} and any finite number of preinjective points I_{r−1}, …, I_{1}.

We employ the same idea as in Section ^{\underline{ss-more}} 7.3. Recall (see Section ^{\underline{ss-red}} 6) that F restricted to the category of regular R-modules preserves and reflects irreducible morphisms. Let g_{n} : M → M_{n} be a composition of irreducible morphisms corresponding (via F) to irreducible monomorphisms in modkÃ_{1}, and let φ_{n} generate the pp-type of g_{n}(x̄) in M_{n}. Clearly φ_{n} implies φ for every n. Before even defining ψ_{r} let us notice that already (φ_{n}/ψ) does not contain M_{1}, …, M_{n−1}. Indeed, otherwise there exists a tuple \overline{m} ∈ M_{s}, s < n such that M_{s} ⊧ φ_{n}(\overline{m}) ∧ ¬ ψ(\overline{m}), in particular \overline{m} ∉ M_{s}J. It readily follows that there are morphisms f : M → M_{s} and h : M_{n} → M_{s} such that f(x̄) = \overline{m} and the following diagram commutes

[figure: M_{s}; M; f; g_{n}; M_{n}; h]

Applying F to M ⟶^{g_{n}} M_{n} ⟶^{h} M_{s}, as in Section ^{\underline{ss-more}} 7.3 we conclude that F (f) = F (hg_{n}) = 0, hence f(M) = hg_{n}(M) ⊆ M_{s}J, a contradiction.

Note also that each M_{s}, s ≥ n satisfies φ on g_{s}(x̄), therefore P_{M} satisfies φ on the direct image of this element (as we will see below g_{s}(x̄) ≠ 0 in P_{M}).

Now we are ready to define ψ_{r}. Let Hom(M, I_{r}) = {h_{1}, …, h_{l}} and for every i = 1, …, l let θ_{i} generate the pp-type of h_{i}(x̄) in I_{r}. Set ψ_{r} = ∑_{i=1}^{l} θ_{i}. We claim that the interval (φ_{n}/ψ_{r} + J | x) contains P_{M} but not I_{r−1}, …, I_{1} (neither M_{1}, …, M_{n−1} as we have already proved).

First let us check that each M_{s}, s ≥ n opens this pair on g_{s}(x̄). We have already seen that g_{s}(x̄) satisfies φ_{n} in M_{s}. Suppose that it satisfies ψ_{k}+J | x. Then g_{s}(x̄) = \overline{m}_{1} + ⋯ + \overline{m}_{l} + \overline{m} where M_{s} ⊧ θ_{j}(\overline{m}_{j}) for every j = 1, …, l and \overline{m} ∈ M_{s}J. Since h_{j}(x̄) is a free realization of θ_{j}, there is a morphism g : I_{r} → M_{s} such that g(h_{j}(x̄)) = \overline{m}_{j}. Since F (I_{r}) is a preinjective and F (M_{s}) is a regular kÃ_{1}-module, it follows that Hom(F (I_{r}), F (M_{s})) = 0, in particular F (g) = 0. We conclude that \overline{m}_{j} ∈ M_{s}J for each j, hence g_{s}(x̄) ∈ M_{s}J. Since x̄ generates M, it follows that g_{s}(M) ⊆ M_{s}J. Applying F we obtain F (g_{s})(F (M)) = 0, a contradiction. As in Section ^{\underline{ss-more}} 7.3 it follows that P_{M} ∈ (φ_{n}/ψ_{r} + J | x̄) for all n and r.

Before proving that I_{s} ∉ (φ_{n}/ψ_{r} + J | x̄) for each s < r, let us make a short remark. Suppose that M and N are regular R-modules. It follows from ^{\underline{ARS}} [2, Proposition IV.4.5 and Lemma X.2.3] that the modules Ext(N, τM) and Hom(M, NJ) have the same length. If M and N are simple regular Rmodules from different tubes then using F we conclude that f(M) ⊆ NJ for every morphism f : M → N. However it is quite possible that f ≠ 0, hence Ext(N, M) ≠ 0 (because τ(M) = M). Thus at this stage we cannot directly repeat the arguments from Section ^{\underline{ss-more}} 7.3, but we can proceed as follows.

To check that I_{s} ∉ (φ_{n}/ψ_{r} + J | x̄), it suffices to prove that if n̄ ∈ I_{s}, with s < r, satisfies φ_{n}, then n̄ satisfies ψ_{r} + J | x̄. Indeed, clearly there exists f : M → I_{s} such that f(x̄) = n̄; choose g : I_{r} → I_{s} as in Section ^{\underline{ss-more}} 7.3. Thus we have the following diagram.

\xymatrix{ &  & M \\  &  & f \ar[d] \\ I_{r} & g \ar[r] & I_{s}}

Applying F we can complete the image of this diagram in mod-kÃ_{1} by a map from F (M) to F (I_{r}) (see Section ^{\underline{ss-more}} 7.3 again). It follows that there exists a morphism h : M → I_{r} such that n̄ − gh(x̄) ∈ I_{s}J. But by the definition of that, gh(x̄) satisfies ψ_{r} = ∑_{i=1}^{l} θ_{i}, hence n̄ satisfies ψ_{r} + J | x̄.

Thus we have constructed an effective basis of open neighborhoods for P_{M}. Since each (φ_{n}/ψ_{r} + J | x) is a refinement of the minimal pair (φ/ψ), it follows that Inv(P_{M}, φ_{n}, ψ_{r} + J | x) = |K|. A similar basis of open neighborhoods for any adic point of Zg_{R} can be constructed using elementary duality (note that elementary duality preserves elementary invariants).

It remains to consider the generic point Q of Zg_{R}. First we will find a minimal pair in the theory of Q.

min2 Lemma 9.1. Let φ ≐ x = x and let ψ ≐ J | x. Then (φ/ψ) is a minimal pair in the theory of Q and Inv(Q, φ, ψ) = ∞.

Proof. The functor (φ/ψ) is isomorphic to the functor Hom(R, −) / Hom(R, −)J. Since R goes to P_{1} via F, for every R-module N we have φ(N)/ψ(N) ≅ Hom_{R}(R, N)/ Hom(R, NJ) ≅ Hom_{kÃ_{1}}(F (R), F (N)) = Hom(e_{1}kÃ_{1}, F (N)) ≅ F (N)e_{1}.

For instance, φ(Q)/ψ(Q) ≅ F (Q)e_{1} ≅ k(x) is 1-dimensional over End(F (Q)) = k(X). Because End(Q)/ Jac End(Q) is isomorphic to End F (Q), it follows that φ(Q)/ψ(Q) is 1-dimensional over End(Q)/ Jac End(Q). By ^{\underline{Preb}} [21, Proposition 9.6], (φ/ψ) is a minimal pair in the theory of Q. □

By Ziegler ^{\underline{Zig}} [33, Theorem 4.9] the pp-pair (φ/ψ) can be refined to a basis of open neighborhoods of Q. Arguing as in Section ^{\underline{ss-more}} 7.3 it suffices to decide effectively, for every pp-formula θ such that J | x implies θ, whether θ(Q) = QJ or not. Applying F and ^{\underline{Rin-zig}} [30, Proposition 4] we see that the direct limit A_{M} ⟶^{f} A_{M} ⟶^{f} … of M-adic R-modules is isomorphic to a direct sum Q^{(I)} of copies of Q. We claim that QJ ⊂ θ(Q) if and only if A_{M} ∈ (θ/J | x), and the last question can be answered effectively (as we have already proved the theory of A_{M} is recursively axiomatized, hence decidable).

Indeed, suppose that A_{M} ⊧ θ(m) for some m ∉ A_{M}J. Then the image of m in the above direct limit satisfies θ. If m ∉ Q^{(I)}J, we are done. Otherwise m ∈ A_{M}J for some copy of A_{M} with a large enough index l. Using F we obtain the following diagram.

[figure: A_{M}/A_{M}J; F (f^{l}); A_{M}J; F (f^{l}); A_{M}/A_{M}J; A_{M}J]

But then m ≠ 0 in the top copy of A_{M}/A_{M}J while F (f^{l})(m) = 0 in the bottom copy of A_{M}/A_{M}J, a contradiction, since F (f^{l}) is mono.

Conversely, if Q ⊧ θ(m) for some m ∈ Q \ QJ, then we obtain m ∈ A_{M} \A_{M}J and A_{M} ⊧ θ(m) for some copy of A_{M} with a large enough index.

10. Conclusions

S-conl

Recall that our ultimate goal is to classify finite commutative rings with decidable theories of modules. Thus let R be a finite commutative ring, we would like to know whether the theory T_{R} of all R-modules is decidable (or undecidable). We decompose R into a direct sum of local rings (this clearly can be done effectively) and then (see Remark ^{\underline{comp}} 2.2) reduce the decidability problem to the local case. Thus we may assume that R is a finite local commutative ring with Jacobson radical J and residue field k = R/J.

First we control whether R has an artinian triad or a Drozd ring as a factor (clearly this can be checked effectively). If this is the case then by Corollary ^{\underline{undec-final}} 4.4 the theory of all R-modules is undecidable.

Otherwise by the KL-dichotomy (see Theorem ^{\underline{dich}} 3.1) R is a Klein ring or a homomorphic image of a complete local Dedekind-like ring. If R is a (finite) Klein ring then (by Theorem ^{\underline{Klein}} 5.1)the theory of all R-modules is decidable.

Otherwise R is a homomorphic image of a complete local (infinite) Dedekindlike ring with a finite residue field (the same as R). Since R is finite, R is a proper quotient of such a ring. Then it follows from ^{\underline{K-L2}} [16, Section 11] that one of the following holds.

1) R is a finite valuation ring (hence R is of finite representation type and the theory of all R-modules is decidable);

2) there is a finite commutative valuation ring V whose residue field K = V/ Jac(V) is a 2-dimensional extension of k, and R is isomorphic to the pullback

\xymatrix{R \ar@{-->}[rr] \ar@{-->}[dd] &  & V \\  &  & π \ar[d] \\ k & ⊂ & K}

3) there are two finite commutative valuation rings V_{1} and V_{2} with common residue field k such that R is isomorphic to the pullback

[figure: R; V_{1} ⊕ V_{2}; (π_{1}, π_{2}); k; k ⊕ k ,]

where k → k ⊕ k is the diagonal embedding.

4) R has a simple socle (hence is quasi-Frobenius) and R/ soc(R) is as in 1), 2) or 3).

Since all the properties of finite rings 1)–4) can be recognized effectively, we will eventually find a representation of R of the form 1)–3), or such a representation for R/ soc(R). As we have already noticed in case 4) R is quasi-Frobenius, hence every R-module is a direct sum of a free module and R/ soc(R)-module. Thus arguing as after Theorem ^{\underline{Klein}} 5.1 (and case 1) being trivial) we may assume that R is of the form 2) or 3).

For instance a Gelfand–Ponomarev algebras G_{2,3}(k) is in the class 3) with V_{1} = k[X : X^{2} = 0] and V_{2} = k[Y : Y^{3} = 0]. Note that G_{2,3}(k) is a nondomestic string algebra. Although some information on the Ziegler spectrum of this algebra has been obtained in ^{\underline{P-P}} [24], it is still far from being complete. Therefore the decidability of modules over this particular algebra is still a very much open problem.

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Department of Mathematics, University of Manchester, Alan Turing Building, Oxford Road, M13 9PL, Manchester, UK

E-mail address: gpuninski@maths.man.ac.uk

Department of Mathematics and Informatics, University di Camerino, Via Madonna delle Carceri 9, 62032, Camerino, Italy

E-mail address: carlo.toffalori@unicam.it