INTERPRETATION FUNCTORS WHICH ARE FULL ON PURE-INJECTIVE
MODULES WITH APPLICATIONS TO R-TORSION-FREE MODULES OVER
R-ORDERS.
LORNA GREGORY
Abstract. Let R, S be rings, X ⊆ mod-R a covariantly finite subcategory, C the smallest definable subcategory of Mod-R containing X and D a definable subcategory of Mod-S. We show that if I : C → D is an interpretation functor such that IX ⊆ mod-S and whose restriction to X is full then I is full on pure-injective modules. We apply this theorem to an extension of a functor introduced by Ringel and Roggenkamp which, in particular, allows us to describe the torsion-free part of the Ziegler spectra of tame B¨ackstr¨om orders. We also introduce the notion of a pseudogeneric module over an order which is intended to play the same role for lattices over orders as generic modules do for finite-dimensional modules over finite-dimensional algebras.
1.		Introduction
Interpretation functors are a uniform additive version of the model theoretic notion of interpretation for (definable subcategories of) module categories. Algebraically, they are additive functors between definable subcategories of module categories which commute with direct limits and products. They are key tools for studying module categories from the perspective of model theory and for the algebraic study of purity.
When an interpretation functor I : C → D is full on pure-injective modules the connection between C and D is much closer than for other interpretation functors (more precisely, the connection between C “modulo” ker I and IC). Model theoretically being full on pure-injective modules corresponds, [Pre97, 3.17], to the interpretation given by I “preserving all induced structure”.
The main theorem of this paper (see 2.4) is the following. Here, mod-R denotes the category of finitely presented (right) R-modules.
Theorem. Let R, S be rings, X ⊆ mod-R be covariantly finite, C the smallest definable subcategory containing X and D a definable subcategory of Mod-S . If I : C → D is an interpretation functor and I is full on X and sends modules in mod-R to modules in mod-S then I is full on pure-injectives.
In general, interpretation functors do not send finitely presented modules to finitely presented modules. However, there are naturally occurring situations when they do. For instance, this is the case for k-linear interpretation functors between categories of modules over finite-dimensional k-algebras (see 2.7 for a generalisation of this observation). Moreover, many functors used in the literature to understand one category of finitely presented modules in terms of another extend to interpretation functors between categories of non-finitely presented modules. Thus, in these situations, our theorem can be applied.
One such example is Butler’s functor, [But74], between a large subcategory of lattices over the Z2-order Z2C2 × C2 and the category of finite-dimensional Z/2Z-vector spaces with 4 distinguished 2020 Mathematics Subject Classification. 03C60, 16G30 (primary), 03C98, 16H20.
Key words and phrases. Orders over a DVR, Pure-injective, Ziegler spectrum, B¨ackstr¨om orders, Interpretation functor.
The author was supported by PRIN 2017-Mathematical Logic: models, sets, computability.
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subspaces. This example fulfills the hypotheses of our theorem[1]. Using clever ad hoc methods, in [PT11], Puninski and Toffalori showed that its extension to an interpretation functor is full on pure-injective Z2-torsion free Z2C2 × C2 -mo dules.
The second half of this article, section 3, will give another such application using a functor defined by Ringel and Roggenkamp in [RR79]. Let R be a complete discrete valuation domain with maximal ideal generated by π and field of fractions Q. Let Λ be an R-order with A := QΛ a separable Q-algebra. Let Γ be a hereditary R-order such that Λ ⊆ Γ ⊆ A and let I ⊆ rad(Λ) be an ideal of Γ such that πn ∈ I for some n ∈ N. Let
Λ/I Γ/I
D := 0 Γ/I .
Ringel and Roggenkamp defined a full (but not faithful) additive functor from the category of Λ- lattices to the category of finitely generated D-modules. This functor extends to an interpretation functor F : TfΛ → Mod-D where TfΛ denotes the category of R-torsion-free Λ-modules. Since categories of Λ-lattices are covariantly finite and the smallest definable subcategory containing the Λ-lattices is TfΛ, Theorem 2.4 implies that F is full on pure-injectives (see 3.4). The kernel of F consists exactly of the R-divisible R-torsion-free Λ-module i.e. the restrictions of QΛ-modules.
A useful property of interpretation functors which are full on pure-injective modules is that they induce homeomorphisms between parts of Ziegler spectra. The Ziegler spectrum of a ring S, denoted ZgS, is the set of isomorphism types of indecomposable pure-injective S -modules with closed sets those of the form D ∩ ZgS where D is a definable subcategory. For C a definable subcategory of Mod-S, we write Zg(C) for the subspace ZgS ∩ C. Prest showed in [Pre97, 3.19] that if I : C → D is an interpretation functor which is full on pure-injective modules then the map N 7→ IN induces a homeomorphism from the open subset Zg(C)\kerI and its image in Zg(D), which is closed. The (R-)torsion-free part ZgtΛf := Zg(TfΛ) of the Ziegler spectrum of Λ was studied in [MPT02], [PT11], [GLT20] and [Gre23]. Under our hypotheses on Λ, every R-torsion-free Λ-module decomposes as D ⊕ N where D is R-divisible and N is R-reduced i.e. ∩i∈NN πi = {0}. Thus, 3.6, the functor of Ringel and Roggenkamp induces a homeomorphism from the open subset of R-reduced modules in ZgtΛf to a closed subset of ZgD . Note that since QΛ is semi-simple there are only finitely many points in ZgtΛf which are not R-reduced.
In subsection 3.2 we review the general features of ZgtΛf and explain how to use F to completely understand the topology on ZgtΛf in terms of ZgD .
Generic modules, i.e. indecomposable non-finitely presented modules which are finite length as modules over their endomorphism rings, play an important role in the representation theory of Artin algebras. However, if an R-torsion-free Λ-module is finite length over its endomorphism ring then it is R-divisible. So, although the indecomposable divisible modules are important they do not play the same role for lattices over orders as generic modules play for finitely presented modules over Artin algebras. We propose the notion of pseudogeneric R-torsion-free Λ-modules as a replacement for generic modules over Artin algebras. In subsection 3.3, after proving some routine results about pseudogenerics in analogy with generic modules, we show that Ringel and Roggenkamp’s functor F sends pseudogeneric modules to generic modules. Moreover, we show that, under a variety of conditions, F gives a bijection between the generic modules in D and the pseudogeneric Λ-modules. As a consequence, we show that tame B¨ackstr¨om orders of infinite representation type have finitely many pseudogeneric R-torsion-free Λ-modules.
When Λ is a tame B¨ackstr¨om order of infinite lattice type, D is a tame hereditary algebra. In subsection 3.4, we use the description of the Ziegler spectrum of tame hereditary algebras and the
[1]One needs to check that the category of “b-reduced” lattices is a covariantly finite subcategory of mod-Z2 C2 × C2 2
results in subsection 3.2, to give a description of ZgtΛf when Λ is a tame B¨ackstr¨om order of infinite lattice type.
Since the paper is relatively short, we present necessary background material when it is needed. For a general introduction to the algebraic study of Model Theory of Modules see [Pre88] or [Pre09].
2.		Interpretation functors which are full on pure-injectives
A (right) pp-n-formula is a formula in the language of (right) R-modules of the form
∃y (xy)A = 0
where x is an n-tuple of variables, y is a finite tuple of variables and A is an appropriately sized matrix with entries from R. For M ∈ Mod-R and φ a pp-n-formula we write φ(M) for the solution set of φ in M . Note that this is a subgroup of Mn. After identifying pp-n-formula whose solution sets are equal in all R-modules, the set of pp-n-formulae becomes a lattice under inclusion of solution sets, i.e. ψ ≤ φ if ψ(M) ⊆ φ(M) for all M ∈ Mod-R. We denote this lattice ppnR. For X a class of R-modules, we write ppnR (X) for the quotient of ppnR under the equivalence relation φ ∼X ψ if φ(M) = ψ(M) for all M ∈ X. For ψ ≤ φ ∈ ppnR, we will write [ψ, φ]X (and [ψ, φ]M when X = {M }) for the corresponding interval in ppnR (X).
A pp-pair, written φ/ψ, is a pair of pp formulae φ, ψ such that φ(M) ⊇ ψ(M) for all M ∈ Mod-R.
An embedding i : A → B of R-modules is pure if for all φ ∈ pp1R, φ(B) ∩ iA = iφ(A). An R-module M is pure-injective if every pure-embedding i : M → B splits. Equivalently, [Pre09, 4.3.11]	, N is pure-injective if and only if it is algebraically compact. That is, for all n ∈ N, if for each i ∈ I, ai ∈ N is an n-tuple and φi is a pp-n-formula then i∈I ai + φi(N) = ∅ implies there is some finite subset I′ of I with i∈I′ ai + φi (N) = ∅. We will use the formally stronger equivalent condition (see [Pre09, 4.2.3]) given in the following proposition.
Proposition 2.1. Let M be a pure-injective R-module. Let (xj)j∈J be a sequence of variables and for each finite subset E ⊆ J, write xE for the tuple of variables xj with j ∈ E. For i ∈ I, let φi be a pp formula in variables xEi and ai a tuple of elements of M indexed by Ei . If for each finite subset F ⊆ I there exists a sequence of elements (mj)j∈J in M such that mEi ∈ ai + φi(M) for all i ∈ F then there exists a sequence of elements (mj)j∈J in M such that mEi ∈ ai + φi (M) for all i ∈ I.
A full subcategory D ⊆ Mod-R is definable if it is closed under products, direct limits and pure-submodules. Equivalently, [Pre09, 3.4.7], a full subcategory D ⊆ Mod-R is definable if there exist pp-pairs φi/ψi for i ∈ I such that the modules M ∈ D are exactly those with φi (M) = ψi (M) for all i ∈ I. For any class of R-modules X, we write ⟨X⟩ for the smallest definable subcategory containing X .
A full subcategory X ⊆ mod-R which is closed under isomorphism, finite direct sums and taking direct summands is covariantly finite in mod-R if every M ∈ mod-R has a left X- approximation i.e. a homomorphism fM : M → MX with MX ∈ X such that all homomorphisms g : M → L with L ∈ X , factor through fM.
-→
For any X ⊆ Mod-R, let X denote the full subcategory of modules which are direct limits of modules in X.
Theorem 2.2. [Pre09, 3.4.37] An additive (full) subcategory X ⊆ mod-R is covariantly finite in -→
mod-R if and only if X is a definable subcategory.
Let C ⊆ Mod-R and D ⊆ Mod-S be definable subcategories. Let φ(x)/ψ(x) be a pp-m-pair over R and for each s ∈ S, let ρs(x, y) be a pp-2m-formula such that for each M ∈ C, the solution set ρs(M, M) ⊆ Mm × Mm defines an endomorphism ρsM of the abelian group φ(M)/ψ(M) and such that φ(M)/ψ (M) is a S -module in D when for all s ∈ S, the action of s on φ(M)/ψ(M) is given
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by ρsM . In this situation, (φ/ψ; (ρs)s∈S) defines an additive functor I : C → D. Following [Pre97], we call any functor equivalent to one defined in this way an interpretation functor. We will refer to φ/ψ as the underlying pp-pair of I . Of course this is not a well-defined notion as the same interpretation functor will have many different underlying pp-pairs.
The following theorem, due to Prest in full generality, and Krause in a special case which includes the cases in which we are interested in this paper, gives a completely algebraic characterisation of interpretation functors.
Theorem 2.3. [Pre11, 25.3][Kra98, 7.2] An additive functor I : C → D is an interpretation functor if and only if I commutes with direct limits and products.
Note that we will technically only use the forward direction of the previous theorem which follows easily from the definition of an interpretation functor.
Interpretation functors preserve pure-embeddings and pure-injectivity. Using the definition of an interpretation functor, solution sets of pp formulae in IN for N ∈ C can be translated into solution sets of pp formulae in N . From this it will follow that pure-embeddings are preserved. It can be seen that interpretation functors preserve algebraic compactness, and hence pure-injectivity, by translating systems of cosets of solution sets of pp formulae for IN into systems of cosets of solution sets of pp formulae for N via I .
Theorem 2.4. Let R, S be rings, X ⊆ mod-R be covariantly finite, C the smal lest definable subcategory containing X and D a definable subcategory of Mod-S . If I : C → D is an interpretation functor and I is full on X and sends modules in mod-R to modules in mod-S then I is full on pure-injectives.
Before starting the proof of the theorem, we briefly discuss its hypotheses. If X ⊆ mod-R is covariantly finite and C is the smallest definable subcategory containing it then C is locally finitely presented and the finitely presented objects of C are exactly those in X (see [CB94, 4.1]). So, we are in the situation of the more general theorem [Dea19, 3.12]. Hence we have the following.
Remark 2.5. If I in 2.4 is faithful when restricted to X , as well as full, then I is full (and faithful) on C.
As remarked in the introduction, the above statement does not hold when I restricted to mod-R is not faithful as the next example shows.
Example 2.6. Let R be a discrete valuation domain with maximal ideal generated by π which is not complete and let R be its completion. Let X be the category of torsion-free finitely presented R-modules. Then X is covariantly finite in mod-R and the category TfR of torsion-free R-modules is the smallest definable subcategory containing X . Let I : TfR → Mod-R/πR be defined by sending M ∈ TfR to M/Mπ i.e. I is the interpretation functor with underlying pp-pair x=x/∃yx=yπ. All non-zero modules in X are isomorphic to Rn for some n ∈ N, in particular they are all projective R-modules. From this it follows that I is full on X .
Since R is indecomposable as an R-module, R is not a direct summand of R. All submodules of R as a module over itself are isomorphic to R. Thus, every non-zero homomorphism f : R → R factors through a surjective homomorphism g : R → R. Since R is projective, g is split. This contradicts the fact that R is indecomposable. Therefore HomR(R, R) = 0. On the other hand, IR ∼= IR, so HomR/Rπ(IR, IR) ̸= 0. Therefore I is not full.
In general interpretation functors do not send finitely presented modules to finitely presented modules. There are however a few useful situations when they do.
Remark 2.7. Let k be a commutative noetherian ring and let R, S be k-algebras which are finitely generated as k-modules. If an interpretation functor I : C → D is k -linear then it sends finitely presented R-modules to finitely presented S -modules.
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Proof. Our hypothesis on R and S means that an R-module (respectively S-module) is finitely presented as an R-module (respectively S-module) if and only if it is finitely presented as a k- module. Let φ/ψ be the underlying pp-pair of I. Suppose M is a finitely presented R-module. Then φ(M) and ψ(M) are k-submodules of Mn with k-module structure inherited from M. Thus, since k is noetherian, they are finitely presented k-submodules of Mn. Hence φ(M)/ψ (M) is a finitely presented k-module with structure of a k-module inherited from M . Therefore, it is enough to note that since I is k-linear, the k-module structure on IM is the same as that on φ(M)/ψ (M) inherited from M.	□
Lemma 2.8. Let I, X and C be as in 2.4. For all L ∈ X and N ∈ C, the map
HomR(L, N ) → HomS (IL, IN ) defined by f 7→ If
is surjective.
Proof. Let η : HomR(L, -) → HomS (I L, I (-)) be the natural transformation in (C, Ab) with component maps ηM (f) = If for f ∈ HomR(L, M). Since IL is finitely presented and I commutes with direct limits, HomR (IL, I (-)) commutes with direct limits. Since I is full on X , ηM is surjective for all M ∈ X . By 2.2, every module in C is a direct limit of modules in X . Thus, for any N ∈ C, ηN is a direct limit of maps ηM for some directed system of M ∈ X. Therefore, since direct limits are exact, the direct limit of surjective maps is surjective and hence ηN is surjective. □
If m is an n-tuple of elements from a module M then the pp-type of m is the set of pp-n- formulas φ such that m ∈ φ(M). If M ∈ mod-R and m is an n-tuple of element from M then, [Pre09, 1.2.6], there exists φ ∈ ppnR which generates the pp-type of m as a filter in ppnR, i.e. ψ is in the pp-type of m if and only if ψ ≥ φ. In this situation, for any L ∈ Mod-R, l ∈ φ(L) if and only if there exists a homomorphism f : M → L with f(m) = l.
Lemma 2.9. Let A, B ∈ mod-R and δ : A → B . Let L ∈ mod-R, let c ∈ L be an n-tuple which generates L and let N ∈ Mod-R. For each α ∈ Hom(A, L), there exists χα(x) ∈ ppnR such that
χα(N) = {ϵ(c) ∈ N | ϵ ∈ Hom(L, N) and there exists β ∈ Hom(B, N) such that ϵ ◦ α = β ◦ δ}
Proof. Let a := (a1, . . . , am) generate A and let b generate B. Let σ generate the pp-type of c and τ generate the pp-type of b. For 1 ≤ j ≤ m, let rj and sj be tuples of elements from R, of appropriate lengths, such that δ(aj) = b · rj and α(aj) = c · sj. Let χα(x) be the pp formula
m
∃y τ(y) ∧σ(x) ∧	x· sj = y · rj.
j=1
Suppose m ∈ χα(N). Since m ∈ σ(N), there exists ϵ : L → N such that m = ϵ(c). Let m′ ∈ N be such that m′ ∈ τ(N) and m · sj = m′ · rj. Since m′ ∈ τ(N), there exists β : B → N such that β (b) = m′ . Then, for each 1 ≤ j ≤ m,
ϵ(α(aj)) = ϵ(c · sj) = ϵ(c) · sj = m · sj = m′ · rj = β(b) · rj = β(b · rj) = β(δ(aj)).
So, since a generates A, ϵ ◦ α = β ◦ δ.
Now suppose that ϵ ∈ Hom(L, N) and β ∈ Hom(B, N) are such that ϵ ◦ α = β ◦ δ. Then ϵ(c) ∈ σ(N) and β(b) ∈ τ (N). Then
ϵ(c) ·	sj = ϵ(c ·	sj)	= ϵ(α(aj))	= β(δ(aj)) = β(b · rj) = β(b) · rj.	□
Lemma 2.10. Let I,	X and C	be	as in 2.4.	There exist A, B ∈ X and δ : A →	B such that I composed with the forgetful functor to Ab is the cokernel of the natural transformation HomR(δ, -) in (C, Ab).

Proof. Abusing notation, we will write I for I composed with the forgetful functor from D to Ab. The functor I : D → Ab is defined by a pp-pair, the same pp-pair defines a functor Mod-R → Ab. Thus there exists C, D ∈ mod-R and ϵ : C → D such that I is the cokernel of HomR (ϵ, -) (see, for instance, [CB98, §2.1 Lemma 1]). Let πC : C → A be the left X -approximation of C. Let
D λ /P
OO
ϵ	λ′
C πC / A
be the pushout of πC : C → A and ϵ : C → D. Let πP : P → B be a left X -approximation of P and let δ = πP ◦ λ′ .
We will now show that the natural transformation η induced by the following diagram is an isomorphism.

Hom(ϵ,-) Hom(D, -)	,
Hom(λ◦πP ,-)

/ Hom(C, -)
Hom(πC ,-)

Hom(δ,-) Hom(B, -)	/ Hom(A, -)	/

/I
O✤
✤
✤η
✤
cokerHom(δ, -)

/0

/0

By definition of left approximation, for all M ∈ C, Hom(πC, M) is surjective. Therefore ηM is surjective for all M ∈ C . We now show that ηM is injective for all M ∈ C . Take M ∈ C and g : A → M. Suppose that g ◦ πC ∈ imHom(ϵ, M). By the defining property of pushouts, there exists h : P → M such that h ◦ λ′ = g. Since πP is a left X -approximation of P , there exists h′ : B → M such that h = h′ ◦ πP . Therefore h′ ◦ δ = h′ ◦ πP ◦ λ′ = g as required.	□
Proof of theorem 2.4. Suppose that N ∈ C is pure-injective, N′ ∈ C and g ∈ Hom(IN′, IN). We aim to show that there exists h ∈ HomR(N′, N) such that Ih = g.
Since X is covariantly finite and C is the smallest definable subcategory containing X , all modules in C are direct limits of modules in X . Fix J a directed partially ordered set, Li ∈ X and for each i ≤ j ∈ J, σij : Li → Lj such that N′ together with the canonical maps fi : Li → N′ is the direct limit of this directed system.
We will show below that there exists a set of homomorphisms hi : Li → N indexed by J such that for all i ≤ j ∈ J, hj σij = hi and Ihi = g ◦ Ifi . Since hj σij = hi for all i ≤ j, there exists a homomorphism h : N′ → N such that hfi = hi for all i ∈ I. Thus IhIfi = Ihfi = g ◦ Ifi for all i ∈ J. Since I commutes with direct limits, this implies that Ih = g. So h is the required homomorphism.
For each i ∈ J , let ci generate Li and let
Hi := {ϵ : Li → N | g◦Ifi = Iϵ} ⊆ Hom(Li,N).
By 2.8, each	Hi	is non-empty. For each i ∈ J, pick	ϵi	∈	Hi.	Note that	Hi	:=	ϵi	+ {ϵ ∈ HomR(Li,N) | Iϵ = 0}.
Since I is an interpretation functor, there exists A, B ∈ mod-R and δ : A → B such that, as a functor from C to Ab (i.e. after composing with the forgetful functor), I has a presentation
HomR(B, -)	omR( ,-) / HomR(A, -)	/ I / 0 .
By 2.10, we may assume that A, B ∈ X. Now Iϵ = 0 where ϵ : L → N if and only if for all α ∈ Hom(A, L) there exists β ∈ Hom(B, N) such that ϵ ◦ α = β ◦ δ. For each i ∈ J, let χiα be χα from 2.9, where L := Li and c := ci. Thus, for any ϵ : Li → N, we have Iϵ = 0 if and only if ϵ(ci) ∈ χα(N) for all α ∈ Hom(A, Li).
Now for all i ≤ j ∈ J, let tij ∈ Rlj be such that σij (ci) = cj · tij.
Consider the system of linear equations and cosets of pp-definable subsets

(1)iα

xi ∈ ϵi(ci) + χiα(N)
for α ∈ HomR(M, Li) and

(2)ij

xi = xj tij .
Suppose (mi)i∈J is a solution to this system in N. Since mi - ϵi(ci) ∈ χiα (N), there is a (unique) homomorphism h′i : Li → N defined by h′i (ci) = mi - ϵi (ci). By the argument above Ih′i = 0. Therefore hi := h′i + ϵi ∈ Hi. The second condition (2)ij implies that for all i ≤ j, hj ◦ σij = hi since
hi(ci) = mi = mj tij = hj (cj) · tij = hj (cj · tij) = hj ◦ σij (ci).
Since N is pure-injective, in order to show that the system of equations has a solution, we just need to show that it is finitely solvable. Let J0 ⊆ J be a finite subset of J . Since J is directed, by adding an element to J0 , we may assume that there is a k ∈ J0 such that i ≤ k for all i ∈ J0 .
Let mk := ϵk(ck) and for each i ∈ J0, let mi = mk · tik. By definition mk is in ϵk(ck) +χkα(N) for all α ∈ HomR(M,Lk). Now I(ϵk◦σik) = I(ϵk) ◦I(σik) = g◦I(fk)◦I(σik) = g◦I(fkσik) = g◦I(fi). So ϵk ◦ σik ∈ Hi. Thus ϵk ◦ σik(ci) = mi satisfies ϵi(ci) + χkα (N) for all α ∈ HomR(M, Lk). Thus we have shown that the system is finitely satisfiable. □
Now that we have proved the main theorem of this section we recall some consequences of an interpretation functor being full on pure-injectives which we will use in the next section to compute dimensions on pp-lattices of B¨ackstr¨om orders and to describe the Ziegler spectra of tame B¨ackstr¨om orders.
The (right) Ziegler spectrum, ZgR, of R is a topological space whose set of points is the set pinjR of isomorphism classes of indecomposable pure-injective modules and which has basis of open sets given by
(φ/ψ) := {M ∈ pinjR | φ(M) ⊋ ψ(M) ∧ φ(M)} where φ, ψ ∈ pp1R. The open sets (φ/ψ), and ZgR itself, are compact.
There is a bijective correspondence between the definable subcategories of Mod-R and the closed subsets of closed subsets of ZgR given by sending a definable subcategory D ⊆ Mod-R to pinjR ∩ D and sending a closed subset C ⊆ ZgR to ⟨C⟩ ⊆ Mod-R. For C a definable subcategory of Mod-R, we write Zg(C) for the set ZgR ∩ C equipped with the subspace topology.
Note that if I : C → D is an interpretation functor then ker I ∩ Zg(C) is the closed subset Zg(C)\ (φ/ψ) where φ/ψ is the underlying pp-pair of I. The image of a definable subcategory C′ ⊆ C under an interpretation functor I is not in general a definable subcategory. However, it follows from [Pre09, 3.4.7 (iii)] that the closure of IC′ under pure-submodules is a definable subcategory of D.
Theorem 2.11. [Pre97, 3.19][Pre09, 18.2.27] Let C, D be definable subcategories and let I : C → D be an interpretation functor. If I is full on pure-injectives then the assignment N 7→ IN induces a homeomorphism from the open subset Zg(C)\ ker I of Zg(C) to the closed subset IC ∩ Zg(D) of Zg(D).
Let L be a non-empty class of lattices closed under sublattices and quotient lattices. The L- dimension, denoted L- dim L, of a bounded modular lattice L measures the ordinal number of times it takes to reach the one element lattice by iteratively collapsing intervals in L. The one element lattice is formally given L-dimension -1 and a lattice L has L-dimension 0 if and only if there exist ⊤ = an+1 > an > . . . > a0 = ⊥ such that [ai, ai+1] ∈ L for all 0 ≤ i ≤ n where ⊤ (respectively ⊥) is the greatest (respectively least) element of L.
The two most important dimensions are m-dimension and breadth. The m-dimension, respectively breadth, of a modular lattice L is L- dim L where L is the class of lattices of size 2, respectively 7
the class of total orders. If pp1R(C) has m-dimension then it is equal to the Cantor-Bendixson rank of Zg(C) and, if R, is countable then the two values are always equal. The m-dimension of pp1R is equal to the Krull-Gabriel dimension, in the sense of [Gei85], of the category (mod-R, Ab)f p of finitely presented functors from mod-R, the category of finitely presented right R-modules, to Ab, the category of abelian group. This dimension measures the complexity of factorisation of morphisms in mod-R. Recall that a superdecomposable module is a non-zero module without indecomposable direct summands. If ppR (C) has breadth then there are no superdecomposable pure-injective modules in C and, if R is countable, the converse holds.
To formally define the L-dimension (see [Pre88, Section 10.2] and [Pre09, Section 7.1]) of a bounded modular lattice L one defines a sequence of congruence relations ∼α on L for each ordinal α which is the congruence relation induced on L by iteratively collapsing intervals in L α times. The L-dimension of L is then the least ordinal α (or -1, to deal with the case when L is already the one element lattice) such that L/ ∼α+1 is the one element lattice.
Remark 2.12. Let L be a non-empty class of modular lattices closed under sublattices and quotients.
(1)		Let L be a bounded modular lattice and let a ≤ b ≤ c. Then
(a)		L- dim[a, b] ≤ L- dim L; and
(b)		L- dim[a, c] = max{L- dim[a, b], L- dim[b, c]}.
(2)		Let n ∈ N and let 1 ≤ l ≤ n + 1. Define
l-1	n
yl :=	xi = xi ∧ xi = 0 ∈ ppnR.
i=1	i=l
Then, for 1 ≤ l ≤ n, the map
pp1R → [yl,yl+1]
l-1	n
φ 7→	xi = xi ∧ φ(xl ) ∧	xi = 0.
i=1	i=l+1
is a lattice isomorphism. Hence, by (1), L-dim ppnR = L-dim pp1R.
Proposition 2.13. Let L be a non-empty class of lattices closed under sublattices and quotient lattices. Let C ⊆ Mod-R and D ⊆ Mod-S be definable subcategories and let I : C → D be an interpretation functor with ⟨IC⟩ = D. If I is full on pure-injectives then
sup{L- dim pp1S (D), L- dim pp1R(ker I)} ≤ L-dimpp1R(C) ≤ L- dim pp1S(D) + 1 + L-dimpp1R(ker I).
So, in particular, L- dimpp1R(C) exists if and only if L- dim pp1R(ker I) and L- dim pp1S(D) exist.
Lemma 2.14. Let C ⊆ Mod-R and D ⊆ Mod-S be definable subcategories and let I : C → D be an interpretation functor with underlying pp-pair φ/ψ. For each σ ∈ ppnS, there exists a unique σ′ ∈ [ψ, φ]C such that for all M ∈ D, σ(IM) = σ′(M)/ψ(M). The assignment σ 7→ σ′ gives a lattice homomorphism pp1S (D) → [ψ, φ]C . If ⟨IC⟩ = D then this lattice homomorphism is injective and if I is full on pure-injectives then it is surjective.
Proof. That such a σ′ exists follow easily from the definition of an interpretation functor. Uniqueness and that the assignment σ 7→ σ′ gives a lattice homomorphism follows directly from the defining property of σ′. Suppose ⟨IC⟩ = D and σ > τ in pp1S(D). Since ⟨IC⟩ = D, there exists M ∈ C such that σ(IM) ⊋ τ(IM). Thus σ′(M) ⊋ τ′(M). So the lattice homomorphism is injective. The final claim is a consequence of [Pre97, 3.17].	□

In order to prove 2.13 we need a folklorish result which we were unable to find an explicit reference for. It can be proved using model theoretic methods or using functor categories methods, it follows from [Pre09, 12.3.18 & 13.1.4].
Lemma 2.15. Let L be a non-empty class of lattices closed under sublattices and quotient lattices. Let E be a definable subcategory of Mod-R. For α an ordinal, let ∼α be the congruence relation on pp1R(E) induced by iteratively collapsing intervals in L α times. Let
Eα := {M ∈ E | σ(M ) = τ (M ) for all σ, τ with σ ∼α τ }.
Then, for all σ, τ ∈ pp1R, σ ∼α τ if and only if σ(M ) = τ (M ) for all M ∈ Eα .
Proof of 2.13. Let φ/ψ be the underlying pp-pair of I. By 2.14, the interpretation functor I induces a lattice isomorphism from pp1S (D) to [ψ, φ]C .
The lattice pp1R(ker I) is a quotient of pp1R(C) and the lattice [ψ, φ]C ∼= pp1S (D) is an interval of ppnR(C). Hence if L- dimpp1R(C) = L- dim ppnR(C) exists then so does L- dim pp1R(ker I) and L- dim pp1S (D). Moreover, the left hand inequality holds.
Let α := L- dim pp1S (D). Using the notation of 2.15, since L- dim[ψ, φ]C = α, we have that ψ ∼α+1 φ. Hence Cα+1 ⊆ ker I. Therefore the quotient map pp1R(C) → pp1R(C)/ ∼α+1 factors
3.		
through the quotient map pp1R(C) → pp1R(ker I). So the right hand inequality holds.

Applications to R-torsion-free modules over R-orders
Throughout this section, we fix the following assumptions and notation. Let R be a complete discrete valuation domain with maximal ideal generated by π and field of fractions Q. Let Λ be an R-order with A := QΛ a separable Q-algebra. Recall that an R-order is an R-algebra which is torsion-free and finitely generated as an R-module. Let LattΛ denote the category of right Λ- lattices, i.e. the category of finitely generated Λ-modules which are torsion-free as R-modules, and let TfΛ denote the category of R-torsion-free Λ-modules.
3.1.		The functor of Ringel and Roggenkamp. Under the hypotheses of this section, by [CR81, 26.5], there always exists a hereditary R-order Λ ⊆ Γ ⊆ A. Let I ⊆ rad(Λ) be an ideal of Γ such that πn ∈ I for some n ∈ N. Let
Λ/I Γ/I
D :=	0 Γ/I .
Note that D is an RI -Artin algebra where RI := R/R ∩ I.
We will be particularly interested in the case of B¨ackstr¨om orders, that is, when Γ ⊋ Λ and I = rad(Λ) = rad(Γ). In this case, D is a hereditary R/πR-algebra since Λ/rad(Λ) and Γ/rad(Γ) are both semisimple.
The category of right D-modules is equivalent to the category of triples (U, V, f) where U ∈ Mod-Λ/I , V ∈ Mod-Γ/I and f : U → V is a homomorphism of Λ/I -modules. Morphisms are given by (α, β) : (U1, V1, f1) → (U2, V2, f2) where α : U1 → U2 is a Λ/I homomorphism, β : V1 → V2 is a Γ/I homomorphism and the diagram
U1 f1 / V1
commutes.

Under this equivalence a triple (U, V, f) is mapped to the D-module with underlying abelian group U × V and D-action given by
(u, v) · ^0a bc = (ua, f (u)b + vc).
We will generally identify Mod-D with the category of triples.
For any M ∈ TfΛ , define MΓ to be the Γ-module generated by M inside M Q. Then the inclusion of M in MΓ induces an embedding σM : M/MI → MΓ/MI . Let
F : TfΛ → Mod-D, M 7→ (M/MI,MΓ/MI,σM).
This is the natural extension to TfΛ of the functor of Ringel and Roggenkamp defined in [RR79] as a functor from LattΛ to mod-D.
We now give a description of F in terms of pp formulae. So, in particular, we show that F is an interpretation functor. Every ideal K ◁ Λ is finitely generated as an R-submodule. If K = im=1 αiR then let ΘK (x) be the pp formula ∃y x = y1α1 + . . . + ymαn. For any M ∈ Mod-Λ, ΘK(M) is MK. Since Λ ⊆ Γ ⊆ QΛ and Γ is an R-order, there exists n ∈ N such that πnΓ ⊆ Λ. Then πnΓ is a finitely generated ideal of Λ. For any M ∈ TfΛ , M πnΓ/M πnI is isomorphic to MΓ/MI and M πn /M πn I is isomorphic to M/M I . Therefore F is isomorphic to the functor
TfΛ → Mod-D, M 7→ (Mπn/MπnI, MπnΓ/MπnI, σM′ )
where σM′ : Mπn/MπnI → M πnΓ/M πnI is the embedding induced by the inclusion of Mπn in M πnΓ. Therefore F is isomorphic to the functor given by (φ/ψ; (ρδ)δ∈D) where
•		φ is ΘπnΛ (x1) ∧ ΘπnΓ (x2);
•		ψ is ΘπnI (x1) ∧ ΘπnI (x2); and
•		for each δ := ^ a+0 I cb++II where a ∈ Λ and b, c ∈ Γ,
ρδ(x,y) is y1 = x1a ∧ y2 = x1b + x2c.
Let D denote the full subcategory of D-modules (U, V, f) such that f is a monomorphism and imf generates V as a Γ/I -module.
Remark 3.1. The category D is a definable subcategory of Mod-D.
Proof. This is particularly easy to see in terms of pp formulae. Note f is a monomorphism if and only if the subset defined by x · ( 00 10 ) = 0 is {0}. Let b1, . . . , bn generate Γ/I as an RI -module. Then imf generates V if and only if the pp formulae ∃y y · ( 00 01 ) = x and ∃y1, . . . , yn x = Pin=1 yi · 00 b0i define the same subset in the D-modules corresponding to (U, V, f).	□
Theorem 3.2. [RR79, 1.3 & 1.4] The functor F|LattΛ is full and for all M ∈ D ∩ mod-D there exists M′ ∈ LattΛ such that FM′ ∼= M .
The next remark follows from [CR81, 5.22 (iv)].
Remark 3.3. There exists m ∈ N such that rad(Λ)m ⊆ πΛ. Hence, Im ⊆ πΛ.
For any L ∈ mod-Λ, the canonical surjection πL : L → L/tor(L) is a left LattΛ-approximation of L, where tor(L) denotes the Λ-submodule of R-torsion elements of L. Therefore, the category LattΛ is covariantly finite in mod-Λ.
Theorem 3.4. The functor F : TfΛ → Mod-D is full on pure-injectives, kerF is the class of R-divisible R-torsion-free Λ-modules and ⟨FTfΛ⟩ = D.
Proof. The first statement follows from 2.4 and 3.2. Let M ∈ TfΛ. Then FM = 0 if and only if M/MI = 0. Since Im ⊆ πΛ and πl ∈ I for some l, m ∈ N, M = MI if and only if Mπ = M . Therefore M ∈ ker F if and only if M is R-divisible. For the final claim, it is enough to show that every M ∈ D is a directed union of finitely presented modules in D. It follows from the definition of F that FM ∈ D for all M ∈ TfΛ. Suppose M := (U, V, f) ∈ D and M0 := (U0, V0, f0) is finitely presented (equivalently finitely generated) submodule of M. Then f0 is injective. Take u1, . . . , un ∈ U such that the Γ/I -submodule generated by f (u1), . . . f(un) contains V0. Let U1 be the Λ/I -submodule generated by U0 and u1, . . . , un and let V1 be the Γ/I-submodule generated by f(u1), . . . f (un). Then (U1, V1, f|U1) is finitely presented, in D and contains M0.	□
3.2.		Pure-injective R-torsion-free Λ-modules and the torsion free part of the Ziegler spectrum. The torsion free part of ZgΛ, denoted ZgtΛf , is the closed set Zg(TfΛ) of ZgΛ. This space was studied in [MPT02], [PT11], [GLT20] and [Gre23]. In this section, we first briefly review the general features of this space and then show how 3.4 can be used to understand the topology on ZgtΛf in terms of the topology on ZgD .
The R-divisible R-torsion-free Λ-modules are a definable subcategory equivalent to Mod-QΛ. Since QΛ is separable, and hence semisimple, all R-divisible modules are injective (see [MPT02, Claim 2 p. 1128]) and hence pure-injective as Λ-modules. An R-torsion-free Λ-module M is R- reduced if ∩i∈NMπi = {0}. For any M ∈ TfΛ,
DM := {m ∈ M | πn|m for all n ∈ N}
is a divisible submodule of M and M/DM is R-reduced. Since DM is injective, M ∼= DM ⊕ M/DM . Since we assume that R is complete, Λ-lattices are pure-injective, see for example [GLT20, 2.2].
The indecomposable Λ-lattices are isolated in ZgtΛf [GLT20, 2.4]. The set of indecomposable Λ- lattices are dense in ZgtΛf [GLT20, 2.2] and hence, the previous sentence implies that they are exactly the isolated points. By the discussion above, all N ∈ ZgtΛf are either R-divisible or R- reduced. There are finitely many indecomposable R-divisible R-torsion-free Λ-modules; they are the (restrictions to Λ of the) simple QΛ-modules. These are exactly the closed points of ZgtΛf (this follows from [GLT20, 2.8]). If Λ is of finite lattice type then, [GLT20, 5.4], all points in ZgtΛf are either Λ-lattices or R-divisible. The R-reduced modules in ZgtΛf are those modules in the open set (x=x/π|x) and the complement of this set may be identified with ZgQΛ .
For each simple QΛ-module S there exists an open subset V(S) such that for all N ∈ ZgtΛf , S is in the closure of N if and only if N ∈ V(S). Moreover N ∈ V(S) if and only if S is a direct summand of NQ. Based on this fact we get the following description of open subsets of ZgtΛf .
Lemma 3.5. [Gre23, 5.11] Let U be an open subset of ZgtΛf . Then
U = (U ∩ (x = x/π|x)) ∪ [ V(S)
S∈λ(U)
where λ(U) := {S ∈ ZgQΛ | S ∈ U}.
Using 2.11, we get the following corollary to 3.4.
Corollary 3.6. The functor F : TfΛ → Mod-D induces a homeomorphism between the open set (x=x/π|x) of R-reduced points in ZgtΛf and the closed subset Zg(D) := ZgD ∩ D.
This corollary allows us to understand the topology on ZgtΛf in terms of the topology on ZgD .
Definition 3.7. For V a closed subset of ZgD, define
V0 := {N ∈ (x=x/π|x) | FN ∈ V}
11
and define V to be the closure of V0 in ZgtΛf .
Proposition 3.8. Let V be a closed subset of ZgD . Then
V = V0 ∪ {S ∈ ZgQΛ | there exists N ∈ (x = x/π|x) with S|NQ and FN ∈ V }.
Moreover, all closed subsets of ZgtΛf are of the form V ∪ W where V ⊆ ZgD is a closed subset and W ⊆ ZgQΛ .
Proof. Let X := {S ∈ ZgQΛ | there exists N ∈ (x = x/π|x) with S|NQ and FN ∈ V }. We first show that V0 ∪X is closed. Since V is closed, so is V1 := {N ∈ ZgtΛf | FN ∈ V ∪ {0}}. By definition, V1 ∩ ZgQΛ = ZgQΛ. For all S, S′ ∈ ZgQΛ, S′ ∈ V(S) if and only if S = S′. Therefore
ZgQΛ∩V1∩ \ ZgtΛf\V(S)=X.
S∈/X
By definition of X and V(S), if N ∈ V0 and S ∈/ X then N ∈/ V(S). Hence
V0 ⊆ (x=x/π|x) ∩ \ ZgtΛf\V(S).
S∈/X
By definition, V1 ∩ (x=x/π|x) = V0. Therefore
(x=x/π|x) ∩ V1 ∩ \ ZgtΛf\V(S) =V0.
S∈/X
Thus
V0 ∪ X = V1 ∩ \ ZgtΛf\V(S)
S∈/X
and hence V0 ∪ X is closed.
By definition V0 ⊆ V . For each S ∈ X ∩ZgQΛ , there exists N ∈ V0 such that S |N Q or equivalently S is in the closure of N . Therefore X ⊆ V . Thus V0 ∪ X ⊆ V and hence V0 ∪ X = V .
Finally, for any closed set C ⊆ ZgtΛf, C ∩ (x=x/π|x) is equal to V for some V ⊆ ZgD and C\C ∩ (x=x/π|x) ⊆ ZgQΛ.	□
3.3.		Pseudogeneric modules. A module over an arbitrary ring S is generic if it is indecomposable, of finite length over its endomorphism ring and not finitely presented. A module M is finite length over its endomorphism ring if and only if ppnS (M) is finite length for some, equivalently all, n ∈ N (see [Pre09, 4.4.25]). This implies that M is pure-injective. An R-torsion-free Λ-module is finite length over its endomorphism ring if and only if it is R-divisible. This is because for all n ∈ N, Mπn is an End(M)-submodule of M and since M is R-torsion-free Mπn = Mπn+1 implies Mπ = M . Although the indecomposable divisible modules are important, they do not play the same role for lattices over orders as generic modules play for finitely presented modules over Artin algebras. For instance, [CB92, Thm 9.6], if S is an Artin algebra then there exists a generic S-module if and only if for some n ∈ N, there are infinitely many non-isomorphic indecomposable S-modules of endolength n. However, there are always finitely many indecomposable R-divisible R-torsion-free Λ-modules.
We propose the notion of a pseudogeneric R-torsion-free Λ-module as a replacement for the generic modules over Artin algebras. To motivate this definition we first need to summarise some results around Maranda’s theorem. Since QΛ is separable, by [CR81, 29.5] and the discussion just after [CR81, 30.12], there exists k ∈ N such that for all L, M ∈ LattΛ, πk Ext(L, M) = 0. Let k0 be the least such k. Under the hypothesis of this section, Maranda’s theorem, [CR81, 30.14 & 30.19], states that for any k ≥ k0 + 1, a pair of Λ-lattices M, N are isomorphic if and only if M/Mπk ∼= L/Lπk . Moreover, if M is an indecomposable Λ-lattice then M/M πk is also indecomposable. A version of this result holds for R-reduced R-torsion-free pure-injective Λ-modules.
Theorem 3.9. [Gre23, 3.4, 3.5 & 5.1] Let N, M ∈ TfΛ be R-reduced and pure-injective and let k ≥ k0 + 1. If N is indecomposable then N/N πk is indecomposable and if N/N πk ∼= M/Mπk then N ∼= M. Moreover, the map which sends N ∈ (x = x/π|x) ⊆ ZgtΛf to N/N πk ∈ ZgΛ/πkΛ induces a homeomorphism onto its image which is closed.
For k ≥ k0 + 1, we will refer to the functor taking M ∈ TfΛ to M/Mπk as Maranda’s functor. When the exact value of k ≥ k0 + 1 is not important we will write J for this functor. Although Maranda’s functor is not full (even on pure-injectives) it preserves the structure on the interval [π|x, x = x].
The next proposition is part of [GLT20, 4.6]. Recall that for ψ ≤ φ ∈ ppnS and X some collection of S-modules, we write [ψ, φ]X (and [ψ, φ]M when X = {M }) for corresponding interval in the lattice ppnS (X).
Proposition 3.10. The map from [π +Λπk|x, x = x]⟨ITfΛ⟩ to [π|x, x = x]TfΛ induced by Maranda’s functor TfΛ → Mo d-Λ/π k Λ is an isomorphism. In particular, for any M ∈ TfΛ, the induced map from [π + Λπk|x, x = x]M/Mπk to [π|x, x = x]M is surjective.
We say M ∈ TfΛ is pseudogeneric if M is indecomposable, pure-injective, R-reduced, not finitely generated and M/Mπk0+1 is finite-length over its endomorphism ring. For M ∈ TfΛ , we define the pseudoendolength of M to be the length of [π|x, x = x]M . Note that this means that R-divisible Λ-modules have pseudoendolength 0.
Lemma 3.11. Let M ∈ TfΛ and let k ≥ k0 + 1.
(1)		The endolength of M/Mπk is equal to k times the pseudoendolength.
(2)		The module M is pseudogeneric if and only if it is indecomposable, pure-injective, R-reduced and has finite pseudoendolength.
Proof. The second statement follows directly from the first. To prove (1), take M ∈ TfΛ. Let p := π + Λπk. For each 1 ≤ i ≤ k - 1, [pi+1 |x,pi|x]M/Mπk is isomorphic to [p|x, x = x]M/Mπk . Hence, since pk |x is equivalent to x = 0 in M/Mπk, the length of ppΛ/Λπk (M/Mπk) is equal to k times the length of [p|x, x = x]M/Mπk . By 3.10, [p|x, x = x]M/Mπk is isomorphic to [π|x, x = x]M. Therefore k times the pseudoendolength is equal to the length of ppΛ/Λπk (M/Mπk). By [Pre09, 4.4.25], the length of ppΛ/Λπk (M/M πk) is equal to the endolength of M/Mπk.	□
Indecomposable modules of finite length over their endomorphism ring are closed points of the Ziegler spectrum, [Pre09, 5.1.12]. Thus by 3.9 and [GLT20, 2.2 & 2.4] we get the following remark.
Remark 3.12. If M ∈ TfΛ is pseudogeneric then it is closed and not isolated in the open set (x = x/π|x) equipped with the subspace topology.
Lemma 3.13. Suppose that one of the fol lowing conditions hold.
(1)		R/Rπ is countable.
(2)		The isolation condition holds for Λ/Λπk0+1 .
(3)		pp1Λ(TfΛ) has m-dimension.
An indecomposable pure-injective module M is pseudogeneric if and only if it is closed and not isolated in the open set (x = x/π|x) equipped with the subspace topology.
Proof. The forwards direction is always true. Recall that the isolated points of ZgtΛf are Λ-lattices and that N ∈ (x = x/π|x) if and only if N is R-reduced. Thus, under the hypothesis that (1), (2) or (3) holds, we just need to show that if N is closed in (x = x/π|x) then N has finite pseudoendolength. If R/Rπ is countable then so is Λ/Λπ k0 +1. Thus (1) implies (2). Suppose that (2) holds and N ∈ (x = x/π|x) is closed in (x = x/π|x). Then N/Nπk0+1 is closed in ZgΛ/Λπk0+1 . Since 13
the isolation condition holds, by [Pre09, 5.3.23], N/N πk0+1 is of finite endolength. Hence N is of finite pseudoendolength. If (3) holds then, by [Pre09, 5.3.17], the isolation condition holds for ZgtΛf. Hence [Pre09, 4.3.22] implies that if N is closed in (x = x/π|x) then [π|x, x = x]N is finite length as required. □
The proof of the next proposition copies Herzog’s proof, [Her97, 9.6], of the analogous result for Artin algebras.
Proposition 3.14. If there are infinitely many non-isomorphic indecomposable Λ-lattices of pseudoendolength l then there exists a pseudogeneric of pseudoendolength ≤ l.
Proof. By [Her97, 9.3], for any ring S, pp-pair φ/ψ and n ∈ N0, the set of N ∈ ZgS such that [ψ, φ]N has length ≤ n is a closed subset of ZgS . Therefore the set Cl of N ∈ ZgtΛf with pseudoendolength ≤ l is a closed subset (note that this set also contains the divisible points). Since (x = x/π|x) is compact, it follows that (x = x/π|x) ∩Cl is compact. The indecomposable Λ-lattices are isolated in ZgtΛf . Thus if there are infinitely many non-isomorphic indecomposable Λ-lattices of pseudoendolength l then Cl ∩ (x = x/π|x) contains infinitely many isolated points and hence must contain a non-isolated point N . □
Let e1 := ( 10 00 ) , e2 := ( 00 01 ) ∈ D. For any N ∈ Mod-D, pp1D (N) is finite length if and only if [x = 0, e1|x]N and [x = 0, e2|x]N are finite length because
[e1 |x, x = x]N = [xe2 = 0, x = x]N ∼= [x = 0, e2|x]N.
Proposition 3.15. If N is pseudogeneric then FN is a generic module. If any of the conditions (1), (2) or (3) from 3.13 hold then if N ∈ TfΛ is R-reduced, pure-injective and FN is generic then N is pseudogeneric.
Proof. Suppose N is pseudogeneric. Let n ∈ N be such that πn ∈ I . Since N is pseudogeneric the interval [π|x, x = x]N is finite length. Therefore, the interval [πl |x, x = x]N is finite length for any l ∈ N. Since πn ∈ I, πn|x ≤ I|x. Hence [I |x, x = x]N is finite length. Let m ∈ N be such that πmΓ ⊆ Λ. Then [πmI|x, x = x]N and hence [πmI|x, πmΓ]N are finite length. Since F is full on pure-injectives, by [Pre09, 3.17], [I |x, x = x]N is isomorphic to [x = 0, e1 |x]FN and [πmI |x, πmΓ]N ∼= [x = 0, e2 |x]FN. Thus ppD (FN) is finite length. By 3.4, FN is indecomposable. Every finitely presented module in D is of the form FM for some M ∈ LattΛ . Since R is complete, Λ-lattices are pure-injective. So, if FN were finitely presented then N ∼= M for some Λ-lattice M . Thus we have proved the first statement.
Suppose that one of (1), (2) or (3) from 3.13 holds. Since FN is generic, it is closed in Zg(D). So, by 3.6, N is closed in (x=x/π|x). Since FN is generic, N is not a Λ-lattice. Thus N is not isolated in (x=x/π|x). So, by 3.13, N is pseudogeneric. □
We now specialise to the case of B¨ackstr¨om orders i.e. an order Λ where there exists a hereditary over order Γ ̸= Λ such that rad(Λ) = rad(Γ). In this situation we set I = rad(Λ). Now Λ/I and Γ/I are both semi-simple R/πR-algebras. Hence D is a hereditary R/πR-algebra.
Remark 3.16. For Λ a B¨ackstr¨om order, the only indecomposable pure-injective D-modules which are not in D are the simple D-modules.
Proof. Since Λ is a B¨ackstr¨om order, I := radΛ = radΓ and both Λ/radΛ and Γ/radΓ are semisimple R/πR-algebras. Let (U, V, f) be a D-module and let V1 be the Γ-module generated by the image of f. Since Γ/radΓ is semi-simple there exists V′ such that V = V1 ⊕ V′. Let f : U/ker f → V1 be the map induced by f. Then
(U,V,f) = (kerf,0,0)⊕(U/kerf,V1,f)⊕(0,V′,0).
14
Since (U/ ker f, V1, f) is in D, the only indecomposable members of Mod-D which are not in D are simple D-modules.	□
Following Ringel and Roggenkamp, we say that a B¨ackstr¨om order Λ is tame (respectively wild) if D is tame (respectively wild). It is shown in [GLT20, 5.1] that when an arbitrary order Λ over a Dedekind domain is of finite lattice type, i.e. there are only finitely many indecomposable Λ-lattices up to isomorphism, then m-dim pp1Λ (TfΛ) = 1.
Proposition 3.17. If Λ is a wild B¨ackstr¨om order then m-dim pp1Λ TfΛ is undefined. If Λ is a tame B¨ackstr¨om order of infinite lattice-type then m-dim pp1Λ TfΛ is 3.
Proof. By 2.13 and since the m-dimension of pp1Λ kerF is 0, the m-dimension of pp1Λ TfΛ is undefined if and only if the m-dimension of pp1D (D) is undefined. If Λ is a wild B¨ackstr¨om order then the m-dimension of pp1D is undefined. Since there are only finitely many modules in ZgD which are not in Zg(D) and all of these modules are finite (endo)length, for any pp-pair φ/ψ, if φ(M) = ψ(M) for all M ∈ D then [ψ, φ] is finite length in ppnD. This implies that pp1D (D) also has undefined m-dimension.
Suppose that Λ is a tame B¨ackstr¨om order of infinite lattice-type. Then, [Gei85, 4.3], the m- dimension of pp1D is 2 and the m-dimension of pp1Λ ker F is 0. Thus, by 2.13, 2 ≤ m-dimpp1ΛTfΛ ≤ 3. Thus, [Pre09, 5.3.60], the Cantor-Bendixson (CB) rank of ZgtΛf is equal to the m-dimension of pp1Λ(TfΛ). Let N ∈ ZgD be a point of CB rank 2. The finite-dimensional D-modules are isolated in ZgD . So N also has CB rank 2 in the subspace of ZgD with the (finitely many) simple modules removed. Let M ∈ ZgtΛf be such that FM = N. By 3.6, M has CB rank 2 in (x = x/π|x) and hence also in ZgtΛf . Let S be a simple QΛ-module which is a direct summand of QM . Then, by [GLT20, 2.7], S is in the closure of N. Thus S has Cantor-Bendixson rank 3 as required. □
The previous lemma plus 3.15 gives that when Λ is a tame B¨ackstr¨om order F gives a bijective correspondence between the generic modules of D and the pseudogeneric modules of Λ. When D is tame hereditary, it only has finitely many generic modules (for a proof, make minor modifications to [Rin79b, 6.8]).
Corollary 3.18. If Λ is a tame B¨ackstr¨om order of infinite representation type then Λ has finitely many pseudogeneric R-torsion-free modules.
3.4.		Tame B¨ackstr¨om orders. Throughout this section we assume that Λ is a tame B¨ackstr¨om order of infinite lattice type. By definition, D is a tame hereditary algebra. Note that, even if we assume that Λ is indecomposable, D is not necessarily indecomposable (see [RR79, Example 4]). Let D = Qin=+11 Di be such that Dn+1 is of finite representation type and that for 1 ≤ i ≤ n, Di is of infinite representation type and indecomposable as a ring.
We briefly recall some information about the category of finite-dimensional D-modules (see [DR74] and [Rin79a]) and about the Ziegler spectrum of D (see [Pre98] and [Rin98]). Let τ denote the Auslander-Reiten translate which, since D is hereditary, is an endofunctor on mod-D. We call the finite-dimensional D-modules of the form τ -nP where P is a projective module and n ∈ N0 prepro jective; finite-dimensional modules of the form τnI where I is an injective module and n ∈ N0 preinjective; and finite-dimensional modules with no prepro jective or preinjective direct summands regular. Note that an indecomposable finite-dimensional D-module is both prepro jective and preinjective if and only if it is a Dn+1-module.
The (full) subcategory of regular modules is an abelian category and its inclusion into mod-D is exact. A D-module is quasi-simple if it is simple in the category of regular modules. We call a quasi-simple type i if it is a Di-module. For each natural number n and quasi-simple module E, there exists a unique regular module E[n] which as an object in the category of regular modules is uniserial, has socle E and length n. Dually, there exists a unique regular module [n]E which as an 15

object in the category of regular modules is uniserial, has top E and length n. All indecomposable regular modules are of the form E[n] (and dually of the form [n]E) for some quasi-simple E and n ∈ N.
For each quasi-simple E, the direct limit along a chain of (irreducible) embeddings
E[1] ,→ E[2] ,→ E[3] ,→ .. .
is denoted E[∞]. Dually, for each E, the inverse limit along a chain of (irreducible) epimorphisms

is denoted E. The modules E[∞] and E are both pure-injective and indecomposable. There are

[1]E ← [2]E ← [3]E ← . . . finitely many remaining infinite dimensional indecomposable pure-injective D-modules. Namely, for each 1 ≤ i ≤ n, there is a unique indecomposable generic Di-module.
By 3.16, all infinite dimensional indecomposable pure-injective D-modules are in the image of F. Thus F gives a bijective correspondence between the R-reduced modules in ZgtΛf which are not Λ-lattices and the infinite dimensional points in ZgD . For each 1 ≤ i ≤ n, let Gi denote the unique pseudogeneric module in Mod-Λ such that FGi is the unique generic Di-module 3.15.
The topology on ZgD for D an indecomposable tame hereditary algebra is described in [Pre98] and [Rin98]. We use the formulation from [Rin98].
Theorem 3.19. A subset C ⊆ ZgtΛf is closed if and only if the following conditions hold:
(1)		If FS is quasi-simple and if there exist infinitely many indecomposable X ∈ LattΛ with HomD (FS, FX) ̸= 0 then F-1((FS)[∞]) ∈ C.
(2)		If FS is quasi-simple and if there exist infinitely many indecomposable X ∈ LattΛ with HomD (FX, FS) ̸= 0 then F-1(FS) ∈ C.
(3)		If there are infinitely many indecomposable X ∈ LattΛ of type i in C or if there exists at least one module of the form F-1 ((FS)[∞]) or F-1 (FS) where FS is a quasi-simple of type i in C, then Gi ∈ C.
(4)		For each simple QΛ-module M , if there exists N ∈ C such that M is a direct summand of NQ then M ∈ C.
Proof. If C is closed then that (1), (2) and (3) are true of C follows from the description of the Ziegler spectra of tame hereditary algebras given in [Rin98]. Property (4) holds by [GLT20, 2.7].
Now suppose that C is a subset of ZgtΛf and (1)-(4) are true of C. Then, since (1)-(3) are true of C, the set V of FN such that N ∈ C and N is not R-divisible is a closed subset of ZgD by 3.6. It now follows from (4) that C is
{N ∈ (x=x/π|x) | FN ∈ V}∪{S ∈ ZgQΛ | there exists N ∈ (x=x/π|x) with S|NQ and FN ∈ V}∪W where W is some subset of ZgQΛ. So by 3.8, C is closed.	□
We now give a description of the modules F-1 ((FS)[∞]) as Λ-modules. We are not able to give a description of the modules F-1 (FS) but one might guess that this module is isomorphic to the inverse limit along the irreducible epimorphisms FS[i + 1] → FS[i]. However, it is unclear whether this module is indecomposable.
Roggenkamp, in [Rog83], showed how to obtain the Auslander-Reiten quiver of LattΛ from the Auslander-Reiten quiver of mod-D using F. It is a consequence of [Rog83, 1.11(ii)] that each of the stable tubes in the Auslander-Reiten quiver of D lifts to a stable tube in the Auslander-Reiten quiver of LattΛ. If S ∈ LattΛ is such that FS is quasi-simple then we write S[n] (respectively [n]S) for the L ∈ LattΛ with FL = (FS)[n] (respectively FL = [n](FS)).
For M ∈ Mod-Λ, we write M⋆ for the inverse limit along the surjective maps M/Mπn+1 → M/Mπn for n ∈ N. Note that if M is R-reduced then so is M⋆ .
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Proposition 3.20. Let S ∈ LattΛ be such that FS is quasi-simple and let S[∞] denote the direct limit along a ray of irreducible maps S[n] → S[n + 1]. Then S[∞]⋆ is an R-torsionfree R-reduced indecomposable pure-injective Λ-module and F(S[∞]⋆) is isomorphic to (FS)[∞].
Proof. It is explained in [Gre23], see [Gre23, 4.9] and the discussion above the question on page 596 of that article, that S[∞]⋆ is the pure-injective hull of S [∞]. Since F commutes with direct limits and the irreducible maps S[n] → S[n+ 1] are sent to irreducible maps (FS)[n] → (FS)[n+1] [Rog83, 1.12], FS[∞] ∼= (FS)[∞]. Since F is full on pure-injectives and hence, [Pre97, 3.15 & 3.16] preserves pure-injective hulls, FS[∞]⋆ ∼= (FS)[∞]. Finally, if S[∞]⋆ = N ⊕M then either FN = 0 or FM = 0. Hence, by 3.4, either N or M is R-divisible. This contradicts that fact that S[∞]⋆ is R-reduced. □
Acknowledgements
The work in this article was started while the author was employed by Universit´a di Camerino when Gena Puninski suggested looking at B¨ackstr¨om orders. Work continued while the author was employed at Universit´a della Campania “Luigi Vanvitelli” and the article was finally completed at the University of East Anglia. I wish to thank the anonymous referee for carefully reading this article and for making suggestions which have improved the text.
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School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, UK
Email address: Lorna.Gregory@gmail.com