THE TORSION-FREE PART OF THE ZIEGLER SPECTRUM OF THE KLEIN FOUR GROUP.

GENA PUNINSKI AND CARLO TOFFALORI

Abstract. We will describe the torsion-free part of the Ziegler spectrum, both the points and the topology, over the integral group ring of the Klein group. For instance we will show that the Cantor– Bendixson rank of this space is equal to 3.

1. Introduction

The Ziegler spectrum of a ring R, Zg_{R}, is a topological space whose points are (isomorphism types) of indecomposable pure injective (= algebraically compact) R-modules. The set is endowed with a (compact) topology whose basic open sets may be defined in a number of equivalent ways. The original definition [21] uses positive primitive formulas — a notion from model theory but one whose algebraic meaning is easily understood (such a formula defines the projection, to one or more specified components, of the solution set of a specified system of linear equations). Precisely, if ψ and φ are positive primitive formulas then the basic open set (φ/ψ) is defined to be the set of those M ∈ Zg_{R} which contain an element m which satisfies in M φ but not ψ. Many specific examples can be seen in Sections 3 and 5 of this paper. Alternative definitions of the basic (= compact) open sets of the topology can be given in terms of homomorphisms between modules (see [3] or [14, 5.1.3]) or in terms of finitely presented functors (see [14, 13.1.3]).

2000 Mathematics Subject Classification. 03C60, 16D50, 16G30, 16S34.

Key words and phrases. Ziegler spectrum, Klein four group, Cantor-Bendixson rank.

The paper was firstly discussed during the visit of the first author to the University of Camerino, supported by the Italian GNSAGA INDAM. The research of the first author is also partially supported by Russian Fund of Fundamental Research, grant number 10-01-00274-a.

Ziegler’s paper and subsequent work has shown the usefulness, for the model theory of modules, of describing this space but such description also has algebraic consequences. Indeed, the problem of describing the spectrum is an algebraic one, though concepts from model theory often are applied in solving it, as in this paper. Usually Zg_{R} is not so nice from the topological point of view: for instance it is rarely Hausdorff (see [11, 8.2.12] for some exotic examples of Ziegler spectra). However in many ‘natural’ settings this space is T_{0}, in particular this is the case when the Cantor–Bendixson rank of Zg_{R}, CB(Zg_{R}), is defined.

For example, if R = ℤ is the ring of integers, then the CB-rank of Zg_{R} is equal to 2 with a standard division of points into finite, Prüfer, adic and generic (= divisible torsion-free): in more detail, ℤ/p^{n}ℤ for p a prime and n a positive integer (these points have CB-rank 0), then, again for p a prime, ℤ_{p^{∞}} (p-Prüfer group) and ℤ̂_{p} (p-adic integers), both of CB-rank 1, finally ℚ (the only point with CB-rank 2). It follows that Zg_{ℤ} is T_{0} but not Hausdorff. To have an example of a basic open set, take φ ≐ (xp = 0) with p a prime, ψ ≐ (x = 0), then (φ/ψ) consists of the modules in the spectrum with (nontrivial) p-torsion, that is, the ℤ/p^{n}ℤ and ℤ_{p^{∞}}.

Due to results by Prest and Ringel (see [12], [18]) the picture is very similar for any tame hereditary finite dimensional algebra A over a field: the CB-rank of the Ziegler spectrum of A is 2 and the points of Zg_{A} are divided into finite length points, Prüfer and adic points (parameterized by the simple regular A-modules) and a unique (if A is connected) generic point. Note that for a finite dimensional algebra A the case of CB(Zg_{A}) = 1 is not possible (see Krause [9] and Herzog [6]), and CB(Zg_{A}) = 0 if and only if A is of finite representation type. Thus the simplest case of nontrivial behavior of Zg_{A} is when the CB-rank is equal to 2.

As already said, the general structure of the Ziegler spectrum of a ring R is often too involved, but it gets clearer when restricted to more manageable subcategories of R-modules. For instance, when R = D is a commutative noetherian domain, or an order over a commutative Dedekind domain D, one could try to describe the closed subset Zg_{tf}(R) of Zg_{R} consisting of D-torsion-free R-modules. There are just a few papers addressing this question. For instance Herzog and Puninskaya [7] proved that, if D is a complete local commutative noetherian domain of Krull dimension 1, then the category of finitely generated torsionfree D-modules is of finite representation type iff Zg_{tf}(D) is a discrete (finite) space.

For our purposes more relevant is a paper by Marcja, Prest and Toffalori [10] that investigates the (D-) torsion-free part of the Ziegler spectrum over the group ring DG, where D is a commutative Dedekind domain of characteristic zero and G is a finite group. For instance they proved that every D-torsion-free indecomposable pure injective Rmodule is either divisible or reduced, therefore carries a natural structure of an (indecomposable pure injective) module over a completion D̂_{P} G for some maximal ideal P of D. Furthermore they showed that the topology on Zg_{tf}(DG) is obtained by gluing together ‘P -patches’ topological spaces associated to maximal ideals P of D. Despite [10] contains a lot of information on Zg_{tf}(DG), the context seems to be too general to give a detailed description of this space.

In this note we will refine the analysis of [10] when D is the ring ℤ of integers and G = C(2)^{2} is the Klein group, so for R = ℤC(2)^{2}, to give a very explicit description of Zg_{tf}(R). For instance we will show that the CB-rank of this space is equal to 3, the only points of maximal rank being the modules ℚ Ge_{i} where the e_{i} are the indecomposable idempotents of ℚ G; moreover those are the only closed points of Zg_{tf}(ℤG).

Using the existence of almost split sequences (for lattices over orders over complete discrete valuation domains) we will also prove that the isolated points in Zg_{tf}(ℤG) are exactly the indecomposable lattices over ℤ̂_{p}G, but in contrast to the case of finite dimensional algebras none of them is closed (or finitely generated) as a ℤG-module.

The crucial point of our proofs is to show that a large locally closed subset of Zg_{tf}(ℤG) is homeomorphic to a cofinite (clopen) subset of the Ziegler spectrum of the 4-subspace quiver kD̃_{4} (for k a field). This will be proved via a functor Δ providing a representation equivalence between certain categories of torsion-free ℤ̂_{2}G-modules and kD̃_{4}-modules for k = GF (2) the field with 2 elements, first exploited by Butler [2] to classify ℤ̂_{2}G-lattices. We will show that Δ behaves well (in particular remains full) when restricted to a suitable category of pure injective torsion-free modules over ℤ̂_{2}G. This allows us to recover pure injective modules in this category from corresponding pure injective kD̃_{4}-modules. Thus Zg_{tf}(ℤG) contains a large subspace homeomorphic to a clopen subset of Zg_{kD̃_{4}}. Therefore in addition to ℤ̂_{2}G-lattices it contains Prüfer and adic points (parameterized by simple regular kD̃_{4}-modules) each having CB-rank 1, and a unique point G′ of CBrank 2 (corresponding via Δ to the generic point over kD̃_{4}). Note that G′ itself is not of finite endolength as an R-module. The only generic (= closed) points of Zg_{tf}(ℤG) are the modules ℚGe (for e an indecomposable idempotent of ℚG) and these have CB-rank 3.

Note that reductions (via functors) of categories of lattices over orders into categories of finite dimensional modules over finite dimensional algebras is an important tool in classifying lattices. As we will show, in our particular case, that is, R = ℤC(2)^{2}, the Ziegler spectrum is quite rigid with respect to such a functor. Recall that a similar (even more transparent) effect on the Ziegler spectrum has been observed by Puninski and Toffalori [16] for the so-called Klein rings (a special class of commutative artinian rings), where as the modeling example the Kronecker algebra kÃ_{1} was used. In this paper we will exploit a very general result of Prest [14, 18.2.5] on definable functors between definable categories.

It is well known that Butler-like functors exist in the wider context of ℤ-orders; so one may expect a similar description of Zg_{tf}(…) in this broader setting. For instance our results may suggest that the least positive value of the CB-rank of Zg_{tf}(…) for various similar orders over Dedekind domains is 3. However the arguments we apply in this paper seem to be too ‘ad hoc’ to admit an easy generalization.

In the remainder of the paper, unless otherwise stated, G denotes the Klein group C(2)^{2} and R is the integral group ring ℤG; moreover Zg_{tf} abbreviates Zg_{tf}(R). Modules are assumed to be right modules.

Here is the plan of the next sections. After recalling in Section 2 some basic facts about G and modules over R = ℤG, we begin in Section 3 our analysis of Zg_{tf}; in particular we deal with those points of the spectrum that are ℚG-modules or ℤ̂_{p}G-modules for some odd prime p. This reduces our investigation to the key case p = 2. In Section 4 we summarize the picture of the Ziegler spectrum of the quiver algebra kD̃_{4} where k is any field. In Section 5 Butler’s functor Δ is introduced in the wider setting linking certain indecomposable pure injective representations of kD̃_{4} (over k = GF (2)) and certain indecomposable pure injective ℤ̂_{2}G-modules. This will ultimately provide the expected description of the 2-patch of Zg_{tf}. The topology of Zg_{tf} will be treated in the final Section 6.

We assume some basic familiarity with the model theory of modules, just regarding positive primitive (pp for short) formulas and types, pure injective modules, pure injective hulls and so on. Classical sources for these matter include [8], [11] or [21], but [13] provides a shorter and more fitting introduction for algebraists.

We thank the anonymous referee for her/his careful reading of a previous version of this work and her/his valuable suggestions.

2. The Klein four group

Recall that G = C(2)^{2} denotes the Klein four group, that is, the non-cyclic group of order 4 (hence the direct product of two copies of the group C(2) with 2 elements). If a, b are generators of these copies of C(2), then a and b generate G subject to relations a^{2} = b^{2} = 1 and ab = ba.

Since G is a finite abelian group, the integral group ring R = ℤG is commutative and noetherian (but, say, (1 − a)(1 + a) = 0, hence R contains zero divisors). Furthermore the set of nonzero divisors of R contains ℤ \ {0}, therefore inverting integers we obtain that ℚG is a classical quotient ring of R. Since the characteristic of ℚ is zero this ring is semisimple artinian with the following list of indecomposable idempotents: e_{1} = 1/4 · (1 + a)(1 + b), e_{2} = 1/4 · (1 + a)(1 − b), e_{3} = 1/4 · (1 − a)(1 + b) and e_{4} = 1/4 · (1 − a)(1 − b). Observe that 4e_{i} ∈ R for every i, but R has no nontrivial idempotents.

For each prime p let ℤ_{p} be the localization of ℤ with respect to the prime ideal pℤ and let ℤ̂_{p} be the completion of ℤ_{p} in the topology defined by powers of pℤ (thus ℤ̂_{p} is a complete noetherian valuation domain). Let us put for simplicity R_{p} = ℤ̂_{p}G.

Recall that an R-module M is said to be (ℤ-) torsion-free if, whenever mk = 0 for some m ∈ M and 0 ≠ k ∈ ℤ, then m = 0. Most R-modules considered in this paper will be torsion-free. A finitely generated torsion-free R-module M is said to be an R-lattice. A similar terminology will be used for modules over R_{p} = ℤ̂_{p}G (clearly ℤ̂_{p}- torsion-free is the same as ℤ-torsion-free). However R_{p}-lattices are not finitely generated as R-modules.

If M is a torsion-free module over R, then M is a submodule of M ⊗_{ℤ} ℚ = Mℚ, in particular one can form Me_{i} which is again an R-submodule of Mℚ. Note that e_{i}g = ±e_{i} for any g ∈ G. For instance

e_{1}a = 1/4 · (1 + a + b + ab) · a = 1/4 · (a + 1 + ab + b) = e_{1},
e_{2}a = 1/4 · (1 − a + b − ab) · a = 1/4 · (a − 1 + ab − b) = −e_{2}

and similarly e_{1}b = e_{1}, e_{2}b = e_{2}. In other words a, b act identically on Me_{1}, while on Me_{2} the action of a is ‘skewed’ in the sense just explained. A skewed action of b or both a and b characterizes also Me_{3}, Me_{4}.

Thus the R-module structure on Me_{i} is always obtained from its abelian group structure by taking into account a trivial or skewed action of G. For instance the R-module Re_{2} can be identified with ℤ on which a acts as multiplication by −1 and b acts trivially (hence ab acts as multiplication by −1). Thus, for n and n_{i} integers, n·(n_{1} +n_{2}a+n_{3}b+ n_{4}ab) = n · n_{1} − n · n_{2} + n · n_{3} − n · n_{4}. Similarly Re_{1} is isomorphic to ℤ where a and b act trivially. Therefore every homomorphism between the underlying abelian groups of these modules can be regarded in a natural way as an R-module homomorphism.

Finally observe that, if M is a torsion-free R-module then Me_{i} (i = 1, …, 4) is isomorphic (as an R-module) to M · 4e_{i} ⊆ M, and in this way is pp-interpretable in M.

3. Pure injectivity

Recall that a module M over an arbitrary ring L is said to be pure injective if every finitely satisfiable (in M) system of linear equations over L has a solution in M. For various reformulations of this notion see [11, Chapter 4]. For instance (see [14, Theorem 4.3.6]) M is pure injective iff any summation map M^{(I)} → M can be lifted along the natural inclusion into the direct product module M^{(I)} ⊆ M^{I}. Every pp-definable submodule of a pure injective module M is pure injective. For instance, if L = R = ℤG and M is a pure injective ℤ-torsion-free R-module, then for every i = 1, …, 4 Me_{i} (as an R-module isomorphic to 4Me_{i}) is pure injective.

By [14, Theorem 4.4.8] every module with the descending chain condition on pp-definable subgroups is pure injective (in particular this is the case for ℤ̂_{p} considered as a module over R_{p} or over R with G acting trivially). It follows that every module of finite endolength is pure injective. Therefore (being of endolength 1) all the modules ℚGe_{i} are pure injective and indecomposable.

By [14, Lemma 4.2.8] every module linearly compact over its endomorphism ring is pure injective. This can be applied as follows.

Fact 3.1. (see [10, Theorem 2.1]) Every R_{p}-lattice M is a pure injective module over R_{p} and R.

Proof. Since M is finitely generated and torsion-free, it is isomorphic to ℤ̂_{p}^{(k)} for some k as a module over ℤ and ℤ̂_{p}. It is easily seen that ℤ̂_{p} is linearly compact over itself, therefore (since linear compactness is closed with respect to extensions) M is linearly compact over ℤ_{p}. It follows that M is linearly compact over the larger ring End_{R_{p}} M. Thus (by the above remark) M is pure injective over R_{p}, hence over R. □

Thus indecomposable R_{p}-lattices are an important source of indecomposable pure injective torsion-free R-modules. As we have already seen, the modules ℚGe_{i}, i = 1, …, 4 are also pure injective and indecomposable. Later we will construct more (infinitely generated) pure injective R-modules as a part of the Ziegler spectrum of R.

In fact it is time to start our analysis of Zg_{R}, more precisely of its torsion-free part Zg_{tf}. First note that for every positive integer n ((xn = 0)/(x = 0)) defines a basic open set of Zg_{R} consisting of points with n-torsion. It follows that the torsion-free part Zg_{tf} of Zg_{R}, as the intersection of complements of these open sets, is closed (but, by compactness of Zg_{R}, not open) in Zg_{R}. We will be interested in the topology of Zg_{tf} induced from Zg_{R}. Indeed in this paper we will relativize everything to the closed subset Zg_{tf} of the Ziegler spectrum of R. Being a closed subset of a compact space, Zg_{tf} is itself compact.

Important information on Zg_{tf} can be extracted from [10], where the more general case of L = DH, with D a commutative Dedekind domain of characteristic zero and H an arbitrary finite group, is considered. The authors of that paper call an L-module M reduced if ∩_{n}MP^{n} = 0 for every maximal ideal P of D. The following two facts correspond to [10, Theorems 2.1 and 2.2] as applied to our situation.

Fact 3.2. Every indecomposable pure injective torsion-free R-module M is either a simple ℚG-module (so is isomorphic to ℚGe_{i} for some i = 1, …, 4) or is an indecomposable pure injective torsion-free reduced module over R_{p} for a unique prime p.

For every prime p let Zg_{tfp} abbreviate Zg_{tf}(R_{p}), the torsion-free part of the Ziegler spectrum of R_{p}.

Fact 3.3. Every indecomposable pure injective torsion-free reduced R_{p}-module is pure injective and indecomposable also when viewed as a module over R, via restriction of scalars. Furthermore the set of torsion-free reduced points of Zg_{tfp} (as it is embedded in Zg_{tf}, with a divisible point excluded!) has the same topology whether it is considered as a subset of Zg_{tfp} or Zg_{tf}.

We describe now the closed points of Zg_{tf}.

Proposition 3.4. The modules ℚGe_{i}, i = 1, …, 4 are the only closed points of Zg_{tf}.

Proof. One implication is easy. In fact, by [14, Theorem 5.1.12], every finite endolength module is closed in Zg_{R}, therefore in Zg_{tf}. For a lazy proof of the converse we could use [14, Corollaries 5.3.21, 5.3.23]: since R is countable, every closed point M in Zg_{tf} is of finite endolength (relativized to the theory of torsion-free modules). But then for every prime p the descending chain of pp-subgroups Mp^{n} stabilizes, therefore (since M is torsion-free) M = pM. Thus M is divisible. It follows that M has a natural structure of a ℚG-module and it remains to apply the semisimplicity of ℚG.

However we could avoid in this proof the countability hypothesis (and extract more information) as follows.

Let M be an indecomposable pure injective torsion-free R-module, we will prove that one of the points ℚGe_{i} is in the closure of M. As M ≠ 0 and 1 = ∑_{i} e_{i}, there is at least one i = 1, …, 4 for which Me_{i} ≠ 0. Also, by Fact 3.2 we may assume that M is a reduced R_{p}-module, whence (by Fact 3.3) it suffices to show the following (for i = 1, …, 4):

(∗) if M is an R_{p}-module such that Me_{i} ≠ 0, then ℚGe_{i} is in the closure of M (as an R_{p}-module).

Choose m ∈ M such that 0 ≠ me_{i} ∈ Me_{i}. Let T be the theory of M (as an R_{p}-module) and let p be the type consisting of the following formulae: x ≠ 0, e_{i} | x and n | x, n a positive integer. We prove that T′ = T ∪ p is consistent. By the compactness theorem it suffices to check that every finite subset of T′ is consistent. Clearly we can limit this check to finite subsets of the above form but with just one n. But such a set is realized in M by n′ = me_{i} · 4n = m · 4e_{i}n, in fact 0 ≠ n′ ∈ Mn ∩ Me_{i}. Thus p is realized in some model M′ of T by an element 0 ≠ m′ ∈ M′e_{i}, m′ ∈ M′n for all n. It easily follows that there is an R_{p}-module morphism f : ℚGe_{i} → M′ sending e_{i} to m′. Since ℚGe_{i} is a simple ℚGe_{i}-module, f is an isomorphism onto its image. Furthermore, since ℚGe_{i} is an injective R_{p}-module (see [10, p.1128, Claim 2]), it follows that M′ splits off a copy of ℚGe_{i}. □

Thus we have just 4 closed points in Zg_{tf}. In fact (∗) says more: each point ℚGe_{i} cannot be separated by an open set from any point M in Zg_{tf} with Me_{i} ≠ 0. Note that the last condition on M can be written as a basic open set O_{i} = ((e_{i} | x)/(x = 0)) or rather (to stay within the language of R-modules) as ((x(4e_{i} − 4) = 0)/(x = 0)). It is easily checked that for j ≠ i the module ℚGe_{j} does not belong to O_{i}. Thus Zg_{tf} = ∪_{i=1}^{4}O_{i} and for every i ℚGe_{i} is the unique closed point in O_{i}.

Having described the closed points of Zg_{tf} we next deal with the open ones. Clearly the complement of the 4 divisible points ℚGe_{i} is open. Let us denote it by Zg_{red} (‘red’ for reduced): Fact 3.2 explains this terminology. Note that [10, Theorem 2.4] claims that every indecomposable R_{p}-lattice is isolated in Zg_{red}. By Proposition 3.4 none of these points is closed in the whole space Zg_{tf}.

The following theorem also refines another aspect of the aforementioned result of [10].

Theorem 3.5. The only isolated points of Zg_{tf} are (indecomposable) R_{p}-lattices.

Proof. Let M be an indecomposable R_{p}-lattice. By [19] the category of R_{p}-lattices admits almost split sequences. Arguing as for modules over finite dimensional algebras (see [11, Proposition 13.11]) we conclude that M is isolated in Zg_{tfp}. By Fact 3.3 M is isolated in Zg_{red}. Since Zg_{red} is open in Zg_{tf}, it follows that M is isolated in Zg_{tf}.

For the converse let M be an isolated point of Zg_{tf}. By Fact 3.2 M is either a reduced isolated point of Zg_{tfp} or is isomorphic to one of the modules ℚGe_{i}, i = 1, …, 4. In the former case let (φ/ψ) be a pair of pp-formulae that isolates M. Representing M as a direct limit of R_{p}-lattices (as in [10, Theorem 2.3]) it is easily seen that (φ/ψ) already opens on one of these lattices N, therefore M is isomorphic to a direct summand of N.

It remains to notice that by Proposition 3.4 the point ℚGe_{i} cannot be separated by an open set from, say, R_{p}e_{i}, therefore is not isolated. □

At this point the topology of Zg_{tfp} for p an odd prime is easy to describe. In fact in this case each e_{i} is in R_{p}, whence R_{p} = ⊕_{i=1}^{4}R_{p}e_{i}, where R_{p}e_{i} ≅ ℤ̂_{2} and the action of G is trivial or skewed, as described in Section 2. In particular every torsion-free R_{p}-module is a direct sum of copies of the R_{p}e_{i} and ℚGe_{i}, i = 1, …, 4. It follows:

Lemma 3.6. If p is an odd prime, then the only points in Zg_{tfp} are R_{p}e_{i} and ℚGe_{i}, i = 1, 2, 3, 4. The former ones (as R_{p}-lattices) are exactly the isolated points of Zg_{tfp} and the latter points have CB rank 1.

So the only case to be examined is p = 2. We will deal with that in Section 5. But before that let us recall in the next Section 4 some basic facts about the 4-subspace quiver D̃_{4}. We only recall here that, as G is a 2-group, by [4, Corollary 5.25], the ring ℤ̂_{2}G is local with Jacobson radical consisting of elements n_{1} + n_{2}a + n_{3}b + n_{4}ab with n_{i} ∈ ℤ̂_{2} and ∑_{i} n_{i} ∈ 2ℤ̂_{2}.

4. Four subspace quiver

The representation theory of the 4-subspace quiver D̃_{4} (over an arbitrary field k) will play a crucial role in what follows. In this section we will recall for the reader some facts from this theory. More details on finite dimensional (f.d.) representations of D̃_{4} can be found in Simson and Skowroński [20, 13.3] (when k is algebraically closed) and in Brenner [1] (in the general case). Pure injective representations of D̃_{4} are treated in Prest [12] and Ringel [18] in the wider framework of modules over tame hereditary finite dimensional algebras.

Let A be the following matrix algebra

A = ( \begin{matrix}k & 0 & 0 & 0 & 0 \\ k & k & 0 & 0 & 0 \\ k & 0 & k & 0 & 0 \\ k & 0 & 0 & k & 0 \\ k & 0 & 0 & 0 & k \end{matrix} ) .

Then the representations of A are the ‘same’ as representations of the quiver D̃_{4} with the subspace orientation:

\xymatrix{1 \ar[drr] & 2 \ar[dr] &  & 3 \ar[dl] & 4 \ar[dll] \\  &  & 0}

Therefore every A-module corresponds to a quintet of k-vector spaces V^{∗} = (V, V_{1}, …, V_{4}) such that each arrow i → 0 corresponds to a kvector space morphism f_{i} : V_{i} → V.

For instance, A has four simple injective modules I(1), …, I(4), where I(1) is given by the following diagram

\xymatrix{k \ar[drr] & 0 \ar[dr] &  & 0 \ar[dl] & 0 \ar[dll] \\  &  & 0}

(with obvious values of morphisms) and the other modules I(2), I(3) and I(4) are described similarly.

It follows that every A-module is a direct sum of copies of these 4 simple injectives and a module built from a diagram where all morphisms are injective, that is, from a diagram where each V_{i} is a subspace of V. In particular let I(0) denote the (non-simple) injective module corresponding to the following diagram

\xymatrix{k \ar[drr] & k \ar[dr] &  & k \ar[dl] & k \ar[dll] \\  &  & k}

where all the maps are identities.

It follows from the general theory (see [20]) that most indecomposable f.d. A-modules are uniquely determined by their dimension vector \overline{x} = \binom{x_{1} x_{2} x_{3} x_{4}}{x_{0}} where x_{0} = dim V and, for i = 1, …, 4, x_{i} = dim V_{i}. For example the dimension vector of I(0) is \binom{1 1 1 1}{1} and that of I(1) is \binom{1 0 0 0}{0}.

Let us propose as further examples the simple projective module P (0) corresponding to \binom{0 0 0 0}{1} and the projective modules P (i) (i = 1, …, 4) where P (1) corresponds to \binom{1 0 0 0}{1} and P (2), P (3), P (4) are defined similarly.

Furthermore every indecomposable f.d. A-module is preinjective, preprojective or regular. An easy way to determine the type of a f.d. module M is to calculate its defect (see [20, p. 200]):

δ(M) = −2x_{0} + x_{1} + x_{2} + x_{3} + x_{4}.

Then

• M is preinjective iff δ(M) > 0 (for instance δ(I(0)) = −2 + 1 + 1 + 1 + 1 = 2).

• Similarly an indecomposable f.d. module M is preprojective iff δ(M) < 0 (and for those the P (i) (i = 0, …, 4) are examples).

• Finally an indecomposable f.d. module M is regular iff δ(M) = 0.

An example of a regular representation R(λ), λ ∈ k \ {0, 1} of A is given by the following diagram,

\xymatrix{k \ar[drr]_{f_{1}} & k \ar[dr]^{f_{2}} &  & k \ar[dl]_{f_{3}} & k \ar[dll]^{f_{4}} \\  &  & k^{2}}

(where for all r ∈ k, f_{1}(r) = (r, 0), f_{2}(r) = (0, r), f_{3}(r) = (r, r) and f_{4}(r) = (λr, r)), therefore of the dimension vector \binom{1 1 1 1}{2}.

In a standard way indecomposable f.d. A-modules are organized in the Auslander–Reiten (AR for short) quiver of A (that is, a locally finite graph whose vertices are indecomposable f.d. A-modules and morphisms correspond to a basis of irreducible maps). In this global picture the preinjective modules form a connected component of the following shape:

\xymatrix{ &  & τI(1) \ar[ddr] &  & I(1) \\  &  & τI(2) \ar@{-->}[dr] &  & I(2) \\ … & τI(0) \ar@{-->}[ur] \ar@{-->}[uur] \ar@{-->}[dr] \ar[ddr] &  & I(0) \ar@{-->}[ur] \ar@{-->}[uur] \ar@{-->}[dr] \ar[ddr] \\  &  & τI(3) \ar@{-->}[ur] &  & I(3) \\  &  & τI(4) \ar@{-->}[uur] &  & I(4)}

where τ stands for the AR-translate. In a similar way preprojective A-modules form a connected component in the AR-quiver which starts with the projective module P (0).

[figure: P (1); τ^{−1}P (1); P (2); τ^{−1}P (2); P (0); τ^{−1}P (0); …; P (3); τ^{−1}P (3); P (4); τ^{−1}P (4)]

The (indecomposable f.d.) regular A-modules are organized into infinitely many tubes (parameterized by irreducible polynomials over k), that is, they can be drawn on cylinders. Most of the tubes are homogeneous, that is they have just one simple regular module on the mouth and look like a line:

\xymatrix{∘ \ar[r] & ∘ \ar[r] \ar[l] & ∘ \ar[r] \ar[l] & ∘ \ar[l] & …}

However A has 3 exceptional tubes of period 2. A typical representative on the mouth of such a tube is the regular A-module R(∞) given by the following diagram

R(∞) = \xymatrix{0 \ar[drr] & k \ar[dr] & & 0 \ar[dl] & k \ar[dll] \\ & & k}

where the second module on this mouth is

τR(∞) = \xymatrix{k \ar[drr] & 0 \ar[dr] & & k \ar[dl] & 0 \ar[dll] \\ & & k}

In drawing the AR-quiver of A the preprojective modules are usually put on the left of regular modules, and preinjective modules are put on the right of regulars. Then all morphisms will go from the left to the right, for instance there are no nonzero morphisms from regular to preprojective modules.

To calculate the dimensions of Hom spaces, the bilinear form associated to A is very useful. If M and N are f.d. A-modules with the dimension vectors \overline{x} and \overline{y}, then let us define a bilinear form q(\overline{x}, \overline{y}) as dim Hom(M, N) − Ext^{1}(M, N) (where Hom, Ext^{1} abbreviate here for simplicity Hom_{A}, Ext_{A}^{1} respectively). An easy calculation shows that q(\overline{x}, \overline{y}) = ∑_{i=0}^{4} x_{i}y_{i} − y_{0} · ∑_{i=1}^{4} x_{i}.

As an example, if N is preinjective, then Ext^{1}(M, N) = 0, therefore q(\overline{x}, \overline{y}) = dim Hom(M, N). For instance, when N = I(0) = \binom{1 1 1 1}{1} and, say, M = R(∞) = \binom{0 1 0 1}{1}, then

dim Hom(R(∞), I(0)) = 3 − 1 · 2 = 1.

Similarly for N = I(1) = \binom{1 0 0 0}{0} we obtain

dim Hom(R(∞), I(1)) = 0 − 0 = 0

while, for N = I(2), dim Hom(R(∞), I(2)) = 1. The remaining Hom’s from R(∞) to preinjective modules could be calculated using the τperiodicity of R(∞) (see [18, after Proposition 6]).

Thus we know the structure of the category of the f.d. A-modules. Being of finite endolength every f.d. A-module M is pure-injective, hence (if indecomposable) it determines a closed point of Zg_{A}. Since the category of f.d. A-modules admits almost split sequences, by [11, Corollary 13.4], the f.d. points of Zg_{A} are just the isolated points . The following fact (which is again true for any f.d. algebra – see [11, Proposition 13.2 and Corollary 13.3]) will be useful later.

Fact 4.1. Every pure injective A-module is a direct summand of a direct product of f.d. A-modules, whence f.d. points are dense in Zg_{A}.

The description of the remaining (infinitely generated) points of Zg_{A} and the topology of this space we borrow from [18] (see also [12]). For each regular A-module S which is simple as an object in the category of regular modules there is a ray of irreducible monomorphisms S = S(1) ⟶ S(2) ⟶ S(3) ⟶ … winding around the tube. The direct limit along this ray is a pure injective indecomposable A-module S(∞) called the (S-) Prüfer module. This module is countably (infinitely) generated and has CB-rank 1. Note that S(∞) is in the closure of the regular modules S(i). Another way to obtain S(∞) is the following. By [18, Proposition 1] if V_{j}^{∗}, j ∈ J is an infinite set of (nonisomorphic) preinjective A-modules with Hom(S, V_{j}^{∗}) ≠ 0 for all j, then S(∞) is a direct summand of ∏_{j∈J} V_{j}^{∗} (thus S(∞) is in the closure of the V_{j}^{⋆}). For instance, if S is from a homogeneous tube, then Hom(S, V_{j}^{∗}) ≠ 0 for every preinjective module V_{j}^{∗}, therefore S(∞) is in the closure of any infinite set of preinjective points.

Thus for every homogeneous tube we have exactly one Prüfer module, and each nonhomogeneous tube produces two Prüfer modules.

Dually every simple regular A-module S is included in a coray of irreducible epimorphisms S = S(1) ⟵ S(2) ⟵ S(3) ⟵ . . .. The inverse limit along this coray, Ŝ, is said to be the S-adic module. This module is pure injective, indecomposable and has CB-rank 1. Furthermore, Ŝ is in the closure of the S(i), and in the closure of any infinite set of preprojective modules V_{j}^{∗}, j ∈ J with Hom(V_{j}^{∗}, S) ≠ 0 (see [18, Proposition 2].

Finally there is a unique point of Zg_{A} of CB-rank 2 — the generic module G with the following diagram

\xymatrix{k(x) \ar[drr]_{f_{1}} & k(x) \ar@{-->}[dr]^{f_{2}} &  & k(x) \ar@{-->}[dl]_{f_{3}} & k(x) \ar[dll]^{f_{4}} \\  &  & k(x)^{2}}

where k(x) is the field of rational functions in x over k and for every q = q(x) ∈ k(x), f_{1}(q) = (q, 0), f_{2}(q) = (0, q), f_{3}(q) = (q, q) and f_{4}(q) = (q, qx). For instance (see [18, Proposition 3]) G is a direct summand of any infinite power of an arbitrary Prüfer module S(∞). Being of finite endolength, G is a closed point of Zg_{A}.

The description of the topology on Zg_{A} is given by the following result.

Fact 4.2. (see [18, Theorem]) A subset X of Zg_{A} is closed iff the following conditions are satisfied.

1) If S is a simple regular A-module and if there are infinitely many f.d. A-modules V^{∗} ∈ X with Hom(S, V^{∗}) ≠ 0, then S(∞) ∈ X.

2) If S is a simple regular A-module and if there are infinitely many f.d. A-modules V^{∗} ∈ X with Hom(V^{∗}, S) ≠ 0, then Ŝ ∈ X.

3) If there are infinitely many f.d. A-modules in X or if there exists at least one infinite dimensional module in X, then G ∈ X.

For instance, the basis of open sets for G is given by Zg_{A} \F, F a finite set of f.d. points in Zg_{A}.

As has been noticed in [18], the functors Hom(S, −) with S simple regular (from the category of f.d. A-modules to the category of k-vector spaces) play a crucial role in the classification of indecomposable pure injective A-modules. These functors can be described by the so-called patterns (see [18] or rather [17, 3] for a definition and numerous examples). For instance (see [17, p. 254]) if S = R(∞) (or any simple regular A-module from a nonhomogeneous tube) then its pattern has the following shape

[figure: ∘; ∘; ∘; ∘; ∘; …; …; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘]

(its vertices correspond to nonzero morphisms f from S to indecomposable f.d. A-modules V^{∗} and the arrows correspond to irreducible maps g : V^{∗} → W^{∗} such that gf ≠ 0). Notice that in this case all nonzero spaces Hom(S, V^{∗}) are 1-dimensional. We will give a model-theoretic interpretation of the above diagram.

Choose a set of generators \overline{x} for S and let the pp-formula φ generate the pp-type of \overline{x} in S. Thus φ is equivalent to the annihilator condition \overline{x}B = 0 describing the relations on \overline{x}. Since Hom(S, V^{∗}) is (at most) 1-dimensional for every indecomposable f.d. module V^{∗}, it follows from general theory (see [15, Chapter 11]) that the interval [0, φ] is a distributive lattice which is obtained from the above pattern by making ‘free sums’ of points (but taking into account all order relations on the original poset). Thus this interval of pp-formulae has the following shape (where the order relation goes from the left to the right)

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

(the new ‘free sums’ formulae are marked by bullets).

Let p be the pp-type of the tuple \overline{x} ∈ S(∞) (via the natural inclusion S ⊆ S(∞) along the ray of irreducible monomorphisms). Clearly p defines a cut between two dots in the above diagram. Furthermore p is the unique non-finitely generated pp-type containing φ.

The pattern of a simple regular A-module S from a homogeneous tube is more involved. For instance, (see [17, p. 149]) some spaces Hom(S, V^{∗}) are 2-dimensional (e.g., dim Hom(R(λ), I(0)) = 2), hence the interval below φ is not a distributive lattice, but the conclusion stays the same:

Lemma 4.3. For every simple regular A-module S there is a unique non-finitely generated indecomposable pp-type p containing φ. (Recall that φ is isomorphic to Hom(S, −)).

Proof. If p is the pp-type of the generating set \overline{x} of S = S(1) in S(∞), then p is not finitely generated and contains φ. Furthermore, if φ_{n} generates the pp-type of \overline{x} in S(n) (via the standard embedding S(1) ⊆ S(n)), then p is generated by the φ_{n}.

Suppose that q is another indecomposable non-finitely generated pptype over A containing φ. Since q is not finitely generated, using almost split sequences (see [16, Proof of Proposition 7.3] for similar arguments) we conclude that φ_{n} ∈ q for every n, therefore p ⊆ q. Let (V^{∗}, \overline{y}) be a realization of q in an indecomposable pure injective module V^{∗}. From φ ∈ q it follows that Hom(S, V^{∗}) ≠ 0. By the classification of indecomposable pure injective A-modules we obtain that V^{∗} ≅ S(∞), therefore there is a morphism f : S(∞) → S(∞) sending \overline{x} to \overline{y}. If p ⊂ q (strict inclusion) then f ∈ Jac(S(∞)), therefore f annihilates the simple regular socle S(1) of S(∞), in particular \overline{y} = f(\overline{x}) = 0, a contradiction. □

5. The Butler functor

In this section we will recall Butler’s functor and some of its properties (see [2]) and extend this functor to a certain category of pure injective modules over R_{2} = ℤ̂_{2}C(2)^{2} (indeed Butler’s original theory applies to the category of R_{2}-lattices, as ℤ̂_{2} is a complete noetherian commutative valuation domain).

We say that a (ℤ)-torsion-free R_{2}-module is b-reduced (reduced in the terminology of [2] in case of lattices) if M ∩ Me_{i} = 2Me_{i} for every i = 1, …, 4.

Warning: The reader should take care of not confusing this notion of b-reduced with the notion of reduced which was defined in Section 3 before Fact 3.2.

Let C denote the category of b-reduced torsion-free R_{2}-modules. Observe that C is definable (in the sense of [14, 3.4.1]). Indeed the breduction condition defines a closed set of Zg_{R}, that is, the intersection (for i = 1, …, 4) of ((e_{i} | x)/(2e_{i} | x))^{c} –or rather ((4e_{i} | 4x ∧ 4 | x)/(8e_{i} | 4x))^{c}– where ‘c’ stands for the complement. Let us denote by Zg_{C} this closed subset.

Note also that R_{2} ∩ R_{2}e_{i} = 4R_{2}e_{i} ≠ 2R_{2}e_{i} and R_{2}e_{i} ∩ (R_{2}e_{i})e_{i} = R_{2}e_{i} ≠ 2R_{2}e_{i}, therefore R_{2} and R_{2}e_{i}, i = 1, …, 4 are not b-reduced.

On the other hand the four modules ℚGe_{i} (i = 1, …, 4) are in C. Thus the set of reduced b-reduced indecomposable pure injective torsion-free R_{2}-modules is the intersection of Zg_{C} with the open set ((x = x)/(2 | x)). In particular it is locally closed, but neither open nor closed in Zg_{tf}. In the following we will be mainly concerned with this locally closed set, rather than with the whole Zg_{C}.

The following proposition extends [2, Proposition 1.5] from lattices to torsion-free pure injective modules and has a similar proof.

Proposition 5.1. Let M be a ℤ-torsion-free R_{2}-module.

1) If 2Me_{i} ⊈ M for some i = 1, …, 4, then M contains a copy of R_{2} as a direct summand.

2) If i = 1, …, 4 and 2Me_{i} ⊂ M ∩ Me_{i} (a strict inclusion) then M contains a copy of the rank 1 lattice R_{2}e_{i} as a direct summand.

3) Otherwise M is b-reduced.

Proof. 1) Choose m ∈ M such that 2me_{i} ∉ M, therefore (since M is torsion-free) 4me_{i} ∉ 2M. From 4e_{1}−4e_{i} ∈ 2R_{2} it follows that n = 4me_{1} is in M but not in 2M (otherwise 4me_{i} = 4me_{1} + m(4e_{i} − 4e_{1}) ∈ 2M + M · 2R_{2} ⊆ 2M). We claim that N = nℤ̂_{2} ≅ ℤ̂_{2} is a direct summand of M (as a ℤ̂_{2}-module). Since ℤ̂_{2} is pure injective (over itself), it suffices to prove that the pp-type p (over ℤ̂_{2}) of n in N is equal to q = pp_{M}(n). Clearly p ⊆ q. Let φ ∈ q be a pp-formula. By an elimination procedure for commutative valuation domains (see [11, Theorem 2.ℤ.1]) and the fact that M is torsion-free we may assume that φ ≐ 2^{l} | x for some l ∈ ℕ. Since n ∈ M \ 2M it follows that l = 0, therefore φ becomes trivial. Thus N = nℤ̂_{2} ≅ ℤ̂_{2} is a direct summand of M as a ℤ̂_{2}-module.

Therefore there exists a ℤ̂_{2}-morphism π : M → ℤ̂_{2} such that π(n) = 1. Let us define a map τ : M → R_{2} by putting, for every s ∈ M, τ(s) = ∑_{g∈G} π(sg^{−1})g. It is easily checked that τ is an R_{2}-morphism.

Recall that n = 4me_{1}. Since e_{1}g = e_{1} for every g ∈ G we conclude that ng = n, therefore

τ(n) = ∑_{g∈G} π(ng^{−1})g = ∑_{g∈G} π(n)g = ∑_{g∈G} g = 4e_{1}.

It follows that τ(m) · 4e_{1} = τ(m · 4e_{1}) = τ(n) = 4e_{1}. Since R_{2} is a local ring we conclude that τ(m) is invertible in R_{2}, therefore τ is onto. Since R_{2} is projective, τ splits, therefore M contains a direct summand isomorphic to R_{2}.

2) Suppose that 2Me_{i} ⊂ M ∩ Me_{i} (a strict inclusion) and choose m ∈ (M ∩ Me_{i}) \ 2Me_{i}. Since m ∈ Me_{i} \ 2Me_{i} then (as above) we conclude that N = mℤ̂_{2} = mR_{2} ≅ R_{2}e_{i} is a pure submodule of Me_{i} as a module over ℤ̂_{2} or R_{2}. Because R_{2}e_{i} is pure injective, it follows that N is a direct summand of Me_{i}, therefore of M^{∗} = ⊕_{i=1}^{4}Me_{i}. Because N ⊆ M ⊆ M^{∗} it follows that N ≅ R_{2}e_{i} is a direct summand of M. □

Note that in the above proof we used only the fact that M is a ℤ-torsion-free R_{2}-module (although the result obviously applies to ℤtorsion-free pure-injective modules). Another elegant way to formulate the previous proposition is the following (see [14, 18.2.4]): any torsionfree R_{2}-module not in C contains one of the lattices R_{2}, R_{2}e_{i}, i = 1, …, 4 as a pure submodule (and indeed as a direct summand).

In fact (again using Prest’s remark [14, 18.2.4, p. 691]) we can single out minimal pairs associated with these modules.

Lemma 5.2. The following are minimal pairs in the theory T_{tf} of torsion-free modules over R_{2} (or R):

1) ((8e_{1} | 4x)/(4 | x ∧ 4e_{1} | x)) isolates R_{2};

2) ((4 | x ∧ 4e_{i} | x)/(8e_{i} | 4x)) isolates R_{2}e_{i}, i = 1, …, 4.

Proof. 1) The set of indecomposable torsion-free R_{p}-lattices (with p ranging over primes) is dense in Zg_{tf}. Evaluating this pair (φ/ψ) on these lattices we see that it takes a nontrivial value just on R_{2}, and this value (φ/ψ)(R_{2}) is a one-dimensional vector space over k = GF (2). The conclusion follows easily (were φ > θ > ψ for some pp-formula θ, then we would obtain (φ/θ)(R_{2}) ≠ 0 and (θ/ψ)(R_{2}) ≠ 0, a contradiction).

Similar arguments prove 2). □

Now, following Butler [2], we consider b-reduced R_{2}-modules M in the definable category C and we extend Butler’s functor Δ in their setting. Let us preliminarily recall the notation M^{∗} = ⊕_{i=1}^{4}Me_{i}. Note that M b-reduced implies that 2M^{∗} ⊂ M ⊂ M^{∗} (strict inclusions).

We refer to the field k = GF (2) and we associate to M a module over kD̃_{4} (D̃_{4}-module for short) Δ(M) = V^{∗} where V = M^{∗}/M and, for every i = 1, …, 4, V_{i} = (Me_{i} + M)/M ≅ Me_{i}/M ∩ Me_{i} = Me_{i}/2Me_{i}.

Note also that, since m = ∑_{1≤i≤4} me_{i} for every m ∈ M, it easily follows that for every i, V = ∑_{j≠i} V_{j}. This leads us to consider the definable category D consisting of the D̃_{4}-modules V^{∗} with V = ∑_{j≠i} V_{j} for every i = 1, …, 4. Observe that this condition determines a closed subset Zg_{D} of Zg_{kD̃_{4}}.

The map Δ clearly extends also to morphisms, therefore gives rise to a functor from C into the category of D̃_{4}-modules, and indeed into D. Butler [2] noticed that Δ is full when restricted to the category of R_{2}-lattices and gives a representation equivalence from this category to the category of f.d. representations of D̃_{4} in D. It is easily seen that a f.d. D̃_{4}-module is in the image of Δ iff it contains no modules I(1), …, I(4) and P (0), P (1), …, P (4) as direct summands.

It follows from general theory that the image of C with respect to Δ is a definable category of D̃_{4}-modules (and defines a clopen cofinite subset if we think in terms of the Ziegler spectrum). Since Δ commutes with direct products and direct limits, it provides an example of a definable functor between definable categories (see [14, 18.2] or just notice that Δ(M) is clearly definable in M, hence Δ is an interpretation).

We will prove later that the image of Δ is just D and we will also describe in more detail the preimage of any f.d. point in D. But to illustrate here Butler’s ideas let us construct an R_{2}-lattice whose image will be a simple regular module R(1) with the following diagram

\xymatrix{k \ar[drr] & k \ar[dr] &  & 0 \ar[dl] & 0 \ar[dll] \\  &  & k}

(living on a non-homogeneous tube). What we have to do is to construct a module R_{2}e_{1} ⊕ R_{2}e_{2} (so ℤ̂_{2} ⊕ ℤ̂_{2} where G acts on the two copies of ℤ̂_{2} as suggested by e_{1}, e_{2} respectively) and to choose the elements in this direct sum which are identified via embeddings ℤ̂_{2}/2ℤ̂_{2} ≅ k ⟶ k^{2} ⟵ k ≅ ℤ̂_{2}/2ℤ̂_{2} as D̃_{4}-modules. Hence we are led to define M = {(n, m) ∈ ℤ̂_{2} ⊕ ℤ̂_{2} | n − m ∈ 2ℤ̂_{2}}. In particular, Me_{1} = ℤ̂_{2} ⊕ 0, Me_{2} = 0 ⊕ ℤ̂_{2}, therefore Δ(M) ≅ R(1).

Note also that the kernel of the induced map from Hom(M, N) to Hom(Δ(M), Δ(N)) consists of morphisms f such that f(Me_{i}) ⊆ N for every i, and that it equals ∑_{i} 2 Hom(Me_{i}, Ne_{i}).

Dieterich [5] showed that Butler’s functor Δ (in its original framework of lattices) induces an isomorphism from the AR-quiver of C to the AR-quiver of its image. By adjoining the points out of the domain of Δ one gets the complete AR-quiver of R_{2}-lattices (see [5, p. 54]). Similar to D̃_{4} this quiver contains infinitely many regular tubes (and 3 of them are nonhomogeneous), but the end of the preinjective component is sewn with the beginning of the preprojective component as follows

[figure: R_{2}; τ^{−1}P (1)′; R_{2}e_{1}; τI(1)′; τ^{−1}P (2)′; R_{2}e_{2}; τI(2)′; …; τ^{−1}P (0)′; I(0)′; …; τ^{−1}P (3)′; R_{2}e_{3}; τI(3)′; τ^{−1}P (4)′; R_{2}e_{4}; τI(4)′]

(where to the right and to the left we denote by N′ the R_{2}-lattice corresponding to a given D̃_{4}-module N). Thus the category of R_{2}-lattices is essentially richer than the category of D̃_{4}-modules, in particular there are some nonzero morphisms from “preinjectives” to “preprojective” lattices.

Furthermore to get an irreducible morphism, say from I(0)′ to R_{2}, we represent I(0)′ as the submodule of R_{2}^{∗} = ⊕_{i=1}^{4}R_{2}e_{i} consisting of tuples (m_{1}, …, m_{4}) such that 2 | m_{i} − m_{j} for all i ≠ j, and then multiply it by 2 to get inside R_{2}.

Now we are going to prove the main result of this section: that Δ is full when restricted to the category of b-reduced (and reduced) pure injective torsion-free R_{2}-modules in C. The following remark will be helpful.

Remark 5.3. Every indecomposable pure injective torsion-free module over ℤ̂_{2} is isomorphic to ℤ̂_{2} or ℚ. Thus every reduced pure injective torsion-free R_{2}-module M, if viewed as a ℤ̂_{2}-module, is the pure injective envelope of a module K = ℤ̂_{2}^{(α)} for some α, in particular M is pp-essential over K (as a ℤ̂_{2}-module).

Theorem 5.4. The functor Δ is full when restricted to the category of reduced b-reduced pure injective torsion-free R_{2}-modules.

Proof. Let M, N be reduced (and b-reduced) pure injective torsion-free R_{2}-modules, we wish to prove that the induced map from Hom(M, N) to Hom(Δ(M), Δ(N)) is onto. Put Δ(M) = V^{∗}, Δ(N) = W^{∗}. From the construction of Δ we have that V = M^{∗}/M and V_{i} = Me_{i}/M ∩ Me_{i} = Me_{i}/2Me_{i} ⊆ V (i = 1, …, 4) are k-vector spaces, similarly W = N^{∗}/N and W_{i} = Ne_{i}/2Ne_{i}. Thus a morphism f : Δ(M) → Δ(N) is given by a morphism of vector spaces V → W such that f(V_{i}) ⊆ W_{i} for every i (thus f_{i} : V_{i} → W_{i} will denote the induced map). Let π_{M} denote the projection Me_{i} → Me_{i}/2Me_{i} = V_{i} and similarly for π_{N} : Ne_{i} → W_{i}.

By Remark 5.3 each Me_{i} (if viewed as a ℤ̂_{2}-module) is isomorphic to the pure injective hull of a submodule K = ℤ̂_{2}^{(α)} (as a ℤ̂_{2}-module) for some α. Since ℤ̂_{2}^{(α)} is a projective ℤ̂_{2}-module, f_{i} can be lifted to a morphism g′_{i} : ℤ̂_{2}^{(α)} → Ne_{i} of ℤ̂_{2}-modules such that f_{i}π_{M}(m) = π_{N}g′_{i}(m) for every m ∈ K = ℤ̂_{2}^{(α)} (see the following diagram).

[figure: ℤ̂_{2}^{(α)} ⊆; g′_{i}; Me_{i}; g_{i}; π_{M}; V_{i}; f_{i}; Ne_{i}; π_{N}; W_{i}]

Since K is a pure submodule of Me_{i}, and Ne_{i} is pure injective, there is a ℤ̂_{2}- module morphism g_{i} : Me_{i} → Ne_{i} extending g′_{i}, that is, g_{i} |_{K}= g′_{i}. This morphism g_{i} can be regarded in a natural way as an R_{2}- module morphism with respect to the action of a and b corresponding to e_{i}. We claim that g_{i} makes the above diagram commutative, that is, π_{N}g_{i}(m) = f_{i}π_{M}(m) for every m ∈ Me_{i}.

Indeed, since K is pp-essential in Me_{i} as a ℤ̂_{2}-module, there exist n ∈ K and a pp-formula r | x + ys (to be read “r divides x + ys”) over ℤ̂_{2} (in particular r, s ∈ ℤ̂_{2}) that connects n and m in Me_{i}, meaning that n + ms ∈ Me_{i}r but n, mb ∉ Me_{i}r (use [11, Theorem 4.10.(d)] and observe that, due to the torsion-free assumption, the only interesting pp-formulae over ℤ̂_{2} regard divisibility conditions). Choose m′ ∈ Me_{i} such that m′r = n + ms and apply π_{M}. Clearly 2 | r (otherwise r is invertible), whence m′r ∈ 2Me_{i} and we can conclude that π_{M}(n) = π_{M}(ms). Therefore

(∗) f_{i}π_{M}(ms) = f_{i}π_{M}(n) = π_{N}g′_{i}(n).

From m′r ∈ 2Me_{i} it also follows that g_{i}(m′r) = g_{i}(m′)r ∈ 2Ne_{i}. Thus applying π_{N} to the equality g_{i}(m′r) = g′_{i}(n) + g_{i}(ms) we obtain π_{N}g′_{i}(n) = π_{N}g_{i}(ms). Taking into account (∗) we conclude f_{i}π_{M}(ms) = π_{N}g_{i}(ms), therefore (since M is torsion-free) f_{i}π_{M}(m) = π_{N}g_{i}(m), as desired.

Let g be the R_{2}-module morphism from M^{∗} to N^{∗} whose restriction on Me_{i}, i = 1, …, 4 is g_{i}. Then fπ_{M} = π_{N}g. Finally for every m ∈ M we have π_{M}(m) = 0, hence π_{N}g(m) = 0 and therefore g(m) ∈ N. Thus the restriction of g to M defines a morphism from M to N that lifts f. □

Note that in the proof of the above theorem we only used that M is reduced (b-reduced pure injective torsion-free), but N could be an arbitrary (not necessarily reduced) b-reduced pure injective torsion-free R_{2}-module.

A small adjustment of the above theorem gives its full strength.

Corollary 5.5. The functor Δ is full on b-reduced pure injective torsionfree R_{2}-modules.

Proof. Note that every pure injective torsion-free R_{2}-module M is a direct sum of copies of the ℚGe_{i} (i = 1, …, 4) and a reduced module. Indeed if M is not reduced, take a nonzero element m in its divisible part. Then for some i n_{i} = 4me_{i} is a non-zero element in Me_{i} ∩ M, moreover n_{i} is still divisible. Then standard arguments show that n_{i} is included into a direct summand of M isomorphic to ℚGe_{i}. Now let N be a maximal direct sum of copies of the ℚGe_{i}. Since each of these modules is of finite endolength, N is pure injective, therefore M = N ⊕ M/N and M/N is reduced.

It remains to notice that the divisible part of M is annihilated by Δ, therefore can be ignored when extending morphisms. □

Since Δ preserves direct products and direct limits it follows that Δ preserves pure injectivity, that is, if M is a pure injective R_{2}-module, then Δ(M) is a pure injective D̃_{4}-module. In fact (as we will show later) Δ preserves indecomposability, therefore induces a map Zg_{C} → Zg_{D}. Again we will show later that this map is one-to-one (after removing the ℚGe_{i}). But one important theorem can be already formulated.

Theorem 5.6. Δ induces a homeomorphism from Zg_{C} \{ℚGe_{i}}_{i=1}^{4} onto its image in Zg_{D}.

Proof. Apply Corollary 5.5 and Prest [14, 18.2.27] □

Indeed we are going to see that the image of Δ is just the whole space Zg_{D}. But for this we need one more property of Δ.

Proposition 5.7. Let M be a pure injective b-reduced reduced torsionfree R_{2}-module. Then the kernel of the induced map from End(M) to End(Δ(M)) is contained in Jac(End(M)).

Proof. Let f ∈ ker(Δ_{M}). Extend f to f^{∗} : M^{∗} → M^{∗} in the obvious way. Since f ∈ ker(Δ_{M}), therefore f^{∗}(e_{i}M) ⊆ 2e_{i}M, and then f^{∗}^{k}(e_{i}M) ⊆ 2^{k}e_{i}M for every positive integer k and every i = 1, …, 4. Clearly M^{∗} = ⊕_{i=1}^{4}Me_{i} is pure injective reduced and torsion-free. It follows (see [10, p. 1128] for similar arguments) that a formal inverse (1 − f^{∗})^{−1} = 1 + f^{∗} + f^{∗}^{2} + … can be given the unique R_{2}-module action on M^{∗} that inverts 1 − f^{∗}. Since f^{∗}(M) ⊆ M it follows that the above map sends M to M, therefore 1 − f is invertible in End(M). □

The following corollary is standard.

Corollary 5.8. Let M be a b-reduced pure injective reduced torsionfree R_{2}-module. Then there is a natural 1-1 correspondence between direct sum decompositions of M and Δ(M). In particular Δ preserves and reflects indecomposability (within this class of modules).

Proof. Since (see [8, Corollary 7.5]) the endomorphism ring of every pure injective module is F -semiperfect (in particular, idempotents lift modulo the Jacobson radical) the statement of the corollary is a consequence of general theory (see [8, p. 212] for this kind of argument).

For instance, if M is indecomposable, then End(M) is a local ring. Since ker(Δ_{M}) ⊆ Jac(End(M)), it follows that End(Δ(M)) is local, therefore Δ(M) is indecomposable. For the converse, if Δ(M) is indecomposable, then End(M) cannot have non-trivial idempotents, otherwise they will survive when factoring out ker(Δ_{M}). □

Theorem 5.9. Δ is a bijection between Zg_{C} \{ℚGe_{i}}_{i=1}^{4} and Zg_{D}.

Proof. At the level of lattices this was shown by Butler. Let us briefly recall his proof. We are given a 5-uple of vector spaces V^{∗} = (V, (V_{i})_{i}) in D with V finite dimensional. For every i = 1, …, 4 let V_{i} ≅ GF (2)^{d_{i}} for some non-negative integer d_{i}. Lift V_{i} to ℤ̂_{2}^{d_{i}} viewed as an R_{2}-module with respect to the action of a and b determined by e_{i}. Let σ_{i} denote the canonical ℤ̂_{2}- (indeed R_{2}-) module morphism of ℤ̂_{2}^{d_{i}} onto V_{i}. Thus σ = ∑_{i} σ_{i} is an R_{2}-module morphism of ⊕_{i}ℤ̂_{2}^{d_{i}} to V. Then the kernel M of σ is a b-reduced R_{2}-lattice and Δ(M) ≅ V^{∗} (see [2, (1.3)] for more details).

In particular for each simple regular D̃_{4}-module S there exists an indecomposable R_{2}-lattice S′ such that Δ(S′) ≅ S. We will also call this lattice simple regular, because it is on the mouth of a regular tube of R_{2}-lattices.

Now let us deal with infinite dimensional indecomposable pure injective D̃_{4}-modules V^{∗} in D.

Use Fact 4.1 and represent V^{∗} as a direct summand of a direct product ∏_{j∈J} V_{j}^{∗} of f.d. indecomposable pure injective D̃_{4}-modules V_{j}^{∗} (or of indecomposable pure injective D̃_{4}-modules V_{j}^{∗} already in the image of Δ) such that V_{j}^{∗} is in D for every j ∈ J. Lifting the corresponding idempotent we obtain an indecomposable pure-injective torsion-free breduced reduced R_{2}-module M as a direct summand of ∏_{j∈J} M_{j}, where for every j M_{j} is an indecomposable pure injective R_{2}-module such that Δ(M_{j}) ≅ V_{j}^{∗}. Since Δ reflects isomorphisms this module is even unique (up to isomorphism).

This is the general strategy. But let us treat separately the cases when V^{∗} is S-Prüfer or S-adic for some simple regular S, or G, to obtain more information on the topology Zg_{C}.

First take V^{∗} = S(∞), the S-Prüfer module. Choose an infinite set of preinjective D̃_{4}-modules V_{j}^{∗}, j ∈ J such that Hom(S, V_{j}^{∗}) ≠ 0. Then (see [18, Proposition 1]) S(∞) is a direct summand of the (pure injective) direct product ∏_{j∈J} V_{j}^{∗}. This provides a preimage M of S(∞) as illustrated above. Let us denote it S′(∞) and call it the S′-Prüfer module.

Now the generic module G is a direct summand of any infinite product of copies of S(∞). Lifting this decomposition we obtain an indecomposable pure injective torsion-free reduced module G′ which we call pseudo-generic (since G′ is reduced, the strictly descending chain G′ ⊃ 2G′ ⊃ 4G′ ⊃ … shows that it is not of finite endolength). Note that G′ can be written in terms of generators and relations using the diagram for G (see above).

Finally, in the case of the S-adic D̃_{4}-module Ŝ we simply recall that it is (as any pure injective D̃_{4}-module) a direct summand of a direct product of indecomposable f.d. D̃_{4}-modules. Also, we can assume that these D̃_{4}-modules are in D, in other words exclude P (0), …, P (4) and I(1), …, I(4). Lifting this decomposition we obtain its R_{2} analogue, the S′-adic module Ŝ′.

Thus every module in Zg_{D} has gotten its preimage in Zg_{C}, therefore the induced map Δ : Zg_{C} \{ℚGe_{i}}_{i=1}^{4} → Zg_{D} is a homeomorphism. □

Note that on the way we proved the following remark.

Remark 5.10. Every pure injective reduced torsion-free R_{2}-module is a direct summand of a direct product of R_{2}-lattices.

Although the topology on Zg_{D} and consequently that on Zg_{C} is known, it may be worth recalling its concise description à la Ringel [18]. Let S′ be a simple regular R_{2}-lattice corresponding to a simple regular D̃_{4}-module S = Δ(S′). Let us define a functor Hom_{R_{2}}^{∗}(S′, −) (from the category of R_{2}-lattices to the category of k-vector spaces) as follows. Let \overline{x} be a set of generators for S′ and let \overline{x}B = 0 be the set of relations defining S′ (this gives a subfunctor of the forgetful functor Hom(R^{k}, −), where k is the length of x̄). Then Hom_{R_{2}}^{∗}(S′, −) is a quotient of this functor given by the pp-pair ((\overline{x}B = 0)/(8 | 4x̄e_{i})), therefore Δ provides a natural isomorphism Hom_{R_{2}}^{∗}(S′, −) ≅ Hom_{kD̃_{4}}(S, −) in particular these functors have the same lattice of finitely generated subfunctors.

Taking into account Fact 4.2 and Theorem 5.6, we obtain (here, again Zg_{C} is considered within the category of R_{2}-modules, that is, as a subset of Zg_{tf2} regarded as embedded in Zg_{tf}).

Theorem 5.11. A subset X of Zg_{C} is closed iff the following holds true.

1) If S′ is a simple regular R_{2}-lattice and there are infinitely many lattices M_{j} ∈ X, j ∈ J with Hom^{∗}(S′, M_{j}) ≠ 0, then S′(∞) ∈ X.

2) If S′ is a simple regular R_{2}-lattice and there are infinitely many lattices M_{j} ∈ X, j ∈ J with Hom^{∗}(M_{j}, S′) ≠ 0, then Ŝ′ ∈ X.

3) If there are infinitely many R_{2}-lattices in X or X contains at least one non-finitely generated reduced module, then the pseudo-generic module G′ ∈ X.

4) If Me_{i} ≠ 0 for some M ∈ X and i = 1, …, 4, then ℚGe_{i} ∈ X.

6. Topology

In this section we will describe the topology on Zg_{tf}, the torsion-free part of the Ziegler spectrum of R = ℤC(2)^{2}. Let X be a subset of Zg_{tf}; we have to decide whether X is open or not.

Recall that in Section 3 we defined open sets O_{i}, i = 1, …, 4 such that Zg_{tf} = ∪_{i=1}^{4}O_{i} and ℚGe_{i} is the only closed point within O_{i}. Thus X is open iff X ∩ O_{i} is open for every i, therefore we may assume that X ⊆ O_{i} for some i. By the remark after Proposition 3.4, ℚGe_{i} cannot be separated by an open set from any point of O_{i}, therefore, if ℚGe_{i} ∈ X then X is open iff X = O_{i}.

Otherwise X ⊆ O_{i} \ {ℚGe_{i}}. Since ℚGe_{i} is a closed point, the last set is open in Zg_{tf}. Thus we may assume that X consists of reduced points.

For each prime p let O_{p} denote the open set ((x = x)/(p | x)). Note that O_{p} is the torsion-free non-divisible part of the Ziegler spectrum of R_{p}, Zg_{tfp} regarded as embedded in Zg_{tf} (the divisible part is {ℚGe_{i}}_{i=1}^{4}).

If p ≠ 2, then, since X contains no divisible points, X ∩ O_{p} consists of (at most four) lattices R_{p}e_{i}, therefore is open. Thus X is open iff X ∩ O_{2} is open, and therefore (by [10, Theorem 2.2]) iff X ∩ O_{2} is open considered as a subset of Zg_{tf2}. But Theorem 5.11 provides a complete answer to this question.

For instance, if X contains a pseudo-generic point G′ then to be open X must contain all Prüfer and adic points and almost all R_{2}- lattices (and there are no further restrictions but, by the assumption, ℚGe_{i} ∉ X).

Now it is not difficult to execute the Cantor–Bendixson analysis for Zg_{tf}.

Theorem 6.1. Let R = ℤC(2)^{2} and let Zg_{tf} denote the torsion-free part of the Ziegler spectrum of R.

1) The only isolated points of Zg_{tf} are R_{p} = ℤ̂_{p}C(2)^{2}-lattices (again, there are infinitely many of them if p = 2, and just finitely many otherwise).

2) The only points of CB-rank 1 of Zg_{tf} are the (S′-) Prüfer and adic points for every simple regular R_{2}-lattice S′.

3) The only point of CB-rank 2 is the pseudo-generic point G′.

4) The only points of CB-rank 3 are the divisible modules ℚGe_{i}, i = 1, …, 4.

Proof. 1) follows from Theorem 3.5. Furthermore by Proposition 3.4 and the discussion afterwards, the ℚGe_{i} are points of maximal CBrank and they are the only closed points of Zg_{tf}. In particular 4) will follow from 2) and 3).

2) By 1) the CB-rank of any Prüfer or adic point of Zg_{tf} is at least 1. Let M′ = S′(∞) be an S′-Prüfer point over R_{2}. We claim that M′ is isolated in the first derivative (that is, the subspace of non-isolated points) of Zg_{tf} by the intersection of ((x = x)/(2 | x)) with the pp-pair defined by the functor Hom^{∗}(S′, −). Indeed, if M is an indecomposable pure injective torsion-free R-module in this intersection, then M has a natural structure of an R_{2}-module and Hom^{∗}(S′, M) ≠ 0. If S = Δ(S′) and V^{∗} = Δ(M) then from the functorial isomorphism we obtain Hom_{kD̃_{4}}(S, V^{∗}) ≠ 0. Since V^{∗} is not finitely generated, it follows from the description of Zg_{kD̃_{4}} that V^{∗} ≅ S(∞), therefore M ≅ S′(∞) by the definition of that.

Note that there exists a homeomorphism of ZgD̃_{4} (given by elementary duality composed with standard duality Hom(−, k)) that interchanges Prüfer and adic points, therefore there exists such a homeomorphism on Zg_{C}. It follows that each adic point has CB-rank 1.

Now the pseudo-generic point G′ is a direct summand of any infinite product of copies of any Prüfer point S′(∞). It follows that G′ is in the closure of any such point, therefore cannot be separated from S′(∞) at level one. Thus CB(G′) ≥ 2, and it is exactly 2 because the open set ((x = x)/(2 | x)) separates G′ from the remaining points ℚGe_{i}.

At the final step we can use the open sets O_{i} to separate the points ℚGe_{i}, i = 1, …, 4 from each other. □

Thus we have completed a description of the points and topology of Zg_{tf}. However, because the construction of points has not been direct, their algebraic structure is still enigmatic. To give an example, let us consider a simple regular R_{2}-module S′. Then there is a sequence of irreducible maps S′ = S′(1) → S′(2) → … in the category of R_{2}-lattices that goes (applying Δ) to the sequence of irreducible monomorphisms S = S(1) → S(2) → … in the category of f.d. D̃_{4}- modules. Let T = \underrightarrow{lim} S′_{i}. Since Δ commutes with direct limits, it follows that Δ(T) ≅ S(∞).

Question 6.2. Is it true that T is an indecomposable pure injective R_{2}-module?

Recall that there exists an indecomposable pure injective torsion-free reduced R_{2}-module S′(∞) such that Δ(S′(∞)) ≅ S(∞) ≅ Δ(T). The problem is that, without knowing that T is pure injective, it is not clear how to lift this isomorphism. The best we can do is the following.

Lemma 6.3. S′(∞) is a direct summand of the pure injective envelope of T.

Proof. As we have already mentioned Δ provides a natural isomorphism of functors Hom_{R_{2}}^{∗}(S′, −) and Hom_{kD̃_{4}}(S, −). Suppose that the former functor is given by a pp-pair (φ/ψ) over R_{2}. By Lemma 4.3 there is a unique non-finitely generated pp-type over kD̃_{4} containing Hom(S, −), therefore there exists a unique non-finitely generated pptype p over R_{2} in the sort (φ/ψ). It follows easily that both T and S′(∞) realize p, therefore S′(∞) = PE(p) is a direct summand in PE(T). □

Is it true at least that PE(T) has no divisible part?

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Department of Mathematics, Plekhanov Academy of Economics, Stremyannyi 36, Moscow, 117997 Russia

  E-mail address: punins@mail.ru

School of Sciences and Technologies, University of Camerino, Via Madonna delle Carceri 9, 62032 Camerino, Italy

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