Decidability of theories of modules over tubular algebras

We show that the common theory of all modules over a tubular algebra (over a recursive algebraically closed field) is decidable. This result supports a long standing conjecture of Mike Prest which says that a finite-dimensional algebra (over a recursively given field) is tame if and only its common theory of modules is decidable. Moreover, as a corollary, we are able to confirm this conjecture for the class of concealed canonical algebras over algebraically closed fields. These are the first examples of non-domestic algebras which have been shown to have decidable theory of modules.


Introduction
The study of the decidability and undecidability of theories of modules over finite-dimensional algebras began with papers of Baur which showed that the 4-subspace problem is decidable [Bau80] and that the 5-subspace problem is undecidable [Bau75] (also see [SF75]). For a given ring R, the theory of R-modules is said to be decidable if there is an algorithm that decides whether a given first order sentence in the language of R-modules is true in all R-modules. It follows easily from the results of Baur that the Date: June 17, 2019. The author acknowledges the support of EPSRC through Grant EP/K022490/1. theory of modules over the path algebras of D 4 (in subspace orientation) is decidable and that the theory of modules over the path algebra of D 5 (in subspace orientation) is undecidable.
Geisler [Gei94] and Prest [Pre85] showed that the theory of modules over all tame hereditary algebras (over recursive fields with splitting algorithms) is decidable. In the converse direction, Prest showed that the theory of modules of strictly wild algebras is undecidable [Pre96] and thus all wild finite-dimensional hereditary algebras have undecidable theories of modules. Improving this result, the author, together with Prest, has shown that, over an algebraically closed field, all (finitely) controlled wild algebras have undecidable theories of modules [GP16]. Note that at this time there is no known finite-dimensional algebra (over an algebraically closed field) which is wild but not (finitely) controlled wild. Indeed, Ringel has conjectured that all wild algebras (over algebraically closed fields) are controlled wild.
Toffalori and Puninski [PT09] have worked on the problem of classifying finite commutative rings which have decidable theories of modules, which of course includes all commutative finite-dimensional algebras over finite fields.
The main result of this paper is the following.
Theorem. Let R be a tubular algebra over a recursive algebraically closed field. The common theory of R-modules is decidable.
Our result supports the following long standing conjecture of Mike Prest.
Conjecture. [Pre88] Let R be a finite-dimensional algebra over a suitably recursive field. The theory of R-modules is decidable if and only if R is of tame representation type.
Tubular algebras are finite-dimensional non-domestic tame algebras of linear growth (see [Sko90,3.6] where tubular algebras are referred to as Ringel algebras). These are the first examples of non-domestic algebras which have been shown to have decidable theory of modules.
A finite-dimensional k-algebra is of tame representation type if for each dimension d, almost all d-dimensional modules are in µ(d) ∈ N many 1parameter families (for a precise definition see [SS07,3.3]). An algebra is of domestic representation type if there is a finite bound on µ(d). So, the module categories of non-domestic algebras are significantly more complex than those of domestic representation type. A finite-dimensional kalgebra R is of wild representation type if there is an exact k-linear functor F : fin-k x, y → mod-R which preserves indecomposability and reflects isomorphism type, where fin-k x, y denotes that category of finitedimensional right modules over the free k-algebra in two non-commuting variables. Drozd, [Dro79], showed that all finite-dimensional algebras over algebraically closed fields are either wild or tame and not both. Tubular algebras, introduced by Ringel in [Rin84], belong to a wider class of algebras called the concealed canonical algebras. According to [LdlPn99], the concealed canonical algebras are exactly those algebra which admit a sincere separating tubular family of stable tubes. Equivalently, they are exactly the endomorphism rings of tilting bundles in categories of coherent sheaves on Geigle-Lenzing weighted projective lines. Moreover, [LM96,3.6], the tubular algebras are exactly the tame non-domestic concealed canonical algebras; this perspective gives a geometric interpretation of the category of finite-dimensional modules over a tubular algebra akin to Atiyah's description of the category for coherent sheaves on an elliptic curve. As a corollary to our main theorem, we are able to conclude, see 9.3, that Prest's conjecture is true for all concealed canonical algebras.
Our methods for proving our main result are inspired by results of Harland and Prest in [HP15], an understanding of the Ziegler topology for modules of a fixed rational slope, decidability for tame hereditary algebras and decidability of Presburger arithmetic. For most of the paper we will work with general tubular algebras. However, in section 6, we will mainly deal with canonical algebras of tubular type.
We also show that corollary 8.8 of [HP15] is false and provide, see 7.7, a best possible replacement for that result.

Background
If R is a ring then we write mod-R for the category of finitely presented right R-modules, Mod-R for the category of all right R-modules and ind-R for the set of isomorphism classes of finitely presented indecomposable right R-modules. If M, N ∈ Mod-R then we will frequently write (N, M ) for Hom R (N, M ).
If X is a class of modules then we write (M, X ) = 0 (respectively (X , M ) = 0) to mean that (M, X) = 0 (respectively (X, M ) = 0) for all X ∈ X .
We will usually assume that finite-dimensional algebras are basic, connected and over an algebraically closed base field. Note, however, that every finite-dimensional algebra is (k-linearly) Morita equivalent to a basic algebra and every basic finite-dimensional algebra is isomorphic to a finite product of basic connected algebras. So, for the main results of this article, restricting to basic connected finite-dimensional algebras is only a cosmetic restriction.
2.1. Grothendieck groups and the Euler characteristic. Let R be a finite-dimensional algebra, let S 1 , . . . , S n be the simple modules over R and for 1 ≤ i ≤ n, let P i be the projective cover of S i . If M is a finite-dimensional module over R then we call dimM = (dim Hom R (P 1 , M ), . . . , dim Hom R (P n , M )), the dimension vector of M . The Grothendieck group, K 0 (R), of a finite-dimensional algebra R is the abelian group generated by the isomorphism classes [X] of modules X ∈ mod-R and subject to the relation [Y ] − [X] − [Z] = 0 whenever there exists a short exact sequence Note that K 0 (R) ∼ = Z n . Moreover, we identify K 0 (R) with Z n via the unique isomorphism which for all M ∈ mod-R sends [M ] to dimM .
We say a vector x = (x 1 , . . . , x n ) ∈ K 0 (R) is positive if x i ≥ 0 for 1 ≤ i ≤ n and x = 0. Note that x is positive if and only if x = dimM for some non-zero M ∈ mod-R.
The assumption that R is basic implies, [ASS06,II.2], that there exists a quiver Q, with vertices corresponding to the simple R-modules, such that R is isomorphic to the quotient of the path algebra kQ by an admissible ideal. We say x ∈ K 0 (R) is connected if its support is connected in the underlying quiver Q of R.
The Grothendieck group of a finite-dimensional algebra R of finite global dimension can be equipped with a bilinear form −, − , called the Euler characteristic, such that for all M, N ∈ mod-R, The Euler quadratic form of R is defined as χ R (x) := x, x . We call an element x ∈ K 0 (R) radical if χ R (x) = 0 and a root if χ R (x) = 1. We denote the set of radical vectors radχ R .
2.2. Tubular algebras. We will not give the definition of a tubular algebra in terms of branch extensions of tame concealed algebras, for this see [Rin84,Chapter 5] or [SS07,XIX 3.19], instead we will describe their module categories.
As mentioned in the introduction, another route to tubular algebras is via coherent sheaves on Geigle-Lenzing weighted projective lines. We will use this perspective in section 7 and briefly in section 9. Introductory material and references on this topic are contained in 7.2.1.
The Euler quadratic form of a tubular algebra is positive semi-definite. It follows from [Rin84, 1.1.1], that x ∈ radχ R if and only if x, y + y, x = 0 for all y ∈ K 0 (R). So, in particular, (1) radχ R is a subgroup of K 0 (R), (2) if x, y ∈ radχ R then x, y = − y, x and (3) if x ∈ radχ R and y ∈ K 0 (R) then χ R (x + y) = χ R (y).
If R is a tubular algebra then there exists a canonical pair of linearly independent radical vectors h 0 , h ∞ which generate a subgroup of radχ R of finite index [Rin84, Section 5.1].
For finite-dimensional indecomposable modules M over R we define the slope of M to be , let T q be the set of isomorphism classes of indecomposable finite-dimensional modules of slope q. Let P 0 be the preprojective component (the indecomposable finite-dimensional modules M with h 0 , dimM < 0 and h ∞ , dimM ≤ 0) and let Q ∞ be the preinjective component (the indecomposable finite-dimensional modules M with h 0 , dimM ≥ 0 and h ∞ , dimM > 0).
Let R be a tubular algebra. The set of indecomposable finite-dimensional modules is where each T q is a tubular family separating P q : That T q separates P q from Q q means that 0 = (T q , P q ) = (Q q , T q ) = (Q q , P q ) and, that for every tube T (ρ) ∈ T q and every homomorphism from L ∈ P q to M ∈ Q q , factors through a direct sum of modules in T (ρ).
All finite-dimensional modules over tubular algebras have injective dimension ≤ 2 and projective dimension ≤ 2. This means that dim Ext n (−, −) terms of the Euler characteristic are zero for n > 2. All finite-dimensional indecomposable modules of strictly positive rational slope have projective and injective dimensions less than or equal to 1 [Rin84, 3.1.5]. Thus for those modules Theorem 2.2. [Rin84, 5.2.6, pg 278] Let R be a tubular algebra.
(1) For any indecomposable finite-dimensional R-module X, dimX is either a connected positive root or a connected positive radical vector of χ R . (2) For any positive connected root vector x ∈ K 0 (R), there is a unique indecomposable module X ∈ mod-R with dimX = x.
(3) For any positive connected radical vector x ∈ K 0 (R) there is an infinite family of indecomposable modules with dimX = x. We will refer to the properties asserted in this theorem as mod-R is controlled by χ R .
Remark 2.3. Let R be a tubular algebra. If M is a finite-dimensional indecomposable with slope a then the slope of M * is 1/a, here we read 1/0 as ∞ and 1/∞ as 0. If M is preprojective (respectively preinjective) then M * is preinjective (respectively preprojective).
The definition of slope on finite-dimensional indecomposable modules is extended to infinite-dimensional modules by Reiten and Ringel in [RR06] as follows.
. We say a module M is of slope r if (M, P ) = 0 and (Q, M ) = 0 for all P ∈ P r and Q ∈ Q r . This is equivalent to saying that M ⊗ P * = 0 and (Q, M ) = 0 for all P ∈ P r and Q ∈ Q r .
Theorem 2.5. [RR06, 13.1] All indecomposable modules, except for the finite-dimensional preprojectives and preinjectives, over a tubular algebra have a slope.
2.3. pp-formulas. We now recall some concepts and results from model theory of modules; the necessary background can be found in [Pre09] or [Pre88].
A pp-n-formula is a formula in the language L R = (0, +, (·r) r∈R ) of (right) R-modules of the form ∃y(x, y)H = 0 where x is a n-tuple of variables and H is an appropriately sized matrix with entries in R. If ϕ is a pp-formula and M is a right R-module then ϕ(M ) denotes the set of all elements m ∈ M n such that ϕ(m) holds. Note that for any module M , ϕ(M ) is a subgroup of M n equipped with the addition induced by addition in M . We identify two pp-formulas if they define the same subgroup in every R-module, equivalently in every finitely presented R-module [Pre09, 1.2.23]. After apply this identification, for each n ∈ N the set of (equivalence classes of) pp-n-formulas forms a lattice with the order given by implication, i.e. ψ ≤ ϕ if and only if ψ(M ) ⊆ ϕ(M ) for all R-modules M . The meet of two pp-n-formulas ϕ, ψ is given by ϕ ∧ ψ and the join is given by ϕ + ψ.
If M is finitely presented module and m ∈ M n then there is a pp-nformula ϕ which generates the pp-type of m in M , that is, for all ppformulas ψ, ψ ≥ ϕ if and only if m ∈ ψ(M ). Conversely, if ϕ is a ppn-formula, then there exists a finitely presented module M and m ∈ M n such that ϕ generates the pp-type of m in M . We call M together with m a free-realisation of ϕ. For proofs of these assertions and more about free-realisations, see [Pre09, Section 1.2.2].
A pair of pp-formulas ϕ/ψ 3 is a pp-n-pair if for all R-modules M , ϕ(M ) ⊇ ψ(M ). We say that a pp-pair ϕ/ψ is open on M if ϕ(M ) = ψ(M ) and closed on M if ϕ(M ) = ψ(M ). 2 By R ∞ 0 we mean the non-negative reals together with a maximal element ∞ 3 This notation for a pair of pp-formulae, which is standard in the area, is used to indicate that we are interested in the quotient groups ϕ(M )/ψ(M ) for R-modules M .
A functor from Mod-R to Ab is said to be coherent if it is additive and if it commutes with products and direct limits. Every pp-pair gives rise to a coherent (additive) functor ϕ/ψ : Mod-R → Ab by sending M ∈ Mod-R to ϕ(M )/ψ(M ). All coherent functors arise in this way. Moreover these are exactly the functors F such that there exist A, B ∈ mod-R and f : B → A such that For example, the functors (M, −) and − ⊗ M are coherent when M is finitely-presented and hence equivalent to functors defined by pp-pairs. 2.4. Definable subcategories and Ziegler spectra. A definable subcategory of Mod-R is a subcategory which is closed under pure-submodules, taking direct limits and products. Equivalently, see [Pre09,3.47], a full subcategory D of Mod-R is a definable subcategory if there is a set of pp-pairs Ω such that M ∈ D if and only if ϕ(M ) = ψ(M ) for all ϕ/ψ ∈ Ω. If X ⊆ Mod-R then we will write X for the smallest definable subcategory containing X.
Let R be a tubular algebra and r ∈ R ∞ 0 . Since − ⊗ P * is a coherent functor for each finite-dimensional P and (Q, −) is a coherent functor for each finite-dimensional Q, the class of all modules of slope r is a definable subcategory which we denote D r .
Let R be a ring. An embedding of R-modules f : The (right) Ziegler spectrum of a ring R is a topological space with set of points, pinj R , the isomorphism classes of indecomposable pure-injectives and basis of open sets given by The open sets (ϕ/ψ) are exactly the compact open sets of Zg R . Note that this means that Zg R itself is compact. We should mention here that the open sets of the form (ϕ/ψ) where ϕ and ψ are pp-1-formulas are also a basis for Zg R .
There is a correspondence between closed subsets of Zg R and definable subcategories of Mod-R given by taking a closed subset C to the smallest definable subcategory C containing C and in the opposite direction taking a definable subcategory D to D ∩ pinj R .
The following is an explanation of the above correspondence. If D 1 and D 2 are definable subcategories of Mod-R then D 1 = D 2 if and only if D 1 ∩ pinj R = D 2 ∩ pinj R [Pre09, 5.1.5]. Thus, if D is a definable subcategory then D = D ∩ pinj R . Conversely, if C is a closed subset of Zg R and N ∈ C ∩ pinj R then for all pp-pairs ϕ/ψ, ϕ(N ) = ψ(N ) implies ϕ(M ) = ψ(M ) for some M ∈ C. Since C is closed and the basis of Zg R is given by pp-pairs, N ∈ C. Thus C = C ∩ pinj R .
Let R be a tubular algebra and r ∈ R ∞ 0 . We denote the set of all indecomposable pure-injectives of slope r by C r . So C r = D r ∩ pinj R .
For each rational q ∈ Q +4 the indecomposable pure-injective modules in D q have been completely described.
Lemma 2.6. [Har11, Lemma 50] The following is a complete list of the indecomposable pure-injective modules in D q : (1) The modules in T q (2) A unique Prüfer module S[∞] for each quasi-simple S in T q (3) A unique adic module S for each quasi-simple S in T q (4) The unique generic module G q In section 4 we will describe the topology on C q for q ∈ Q + . For r ∈ R +5 irrational this has already been done in [HP15].
Theorem 2.7. [HP15, Theorem 8.5] Let R be a tubular algebra and r ∈ R + irrational. The definable subcategory D r has no non-trivial proper definable subcategories.
Two points x, y in a topological space are said to be topologically indistinguishable if for all open sets U , x ∈ U if and only if y ∈ U . Since the closed subsets of Zg R correspond to definable subcategories of Mod-R this means that, after identifying topologically indistinguishable points, there is exactly one point in C r for r ∈ R + irrational.
In section 6 we will use Prest's elementary duality for pp-formulas and Herzog's elementary duality for the Ziegler spectrum in order to transfer results about P 0 ∪ C 0 to results about C ∞ ∪ Q ∞ .
A duality between the lattice of right pp-n-formulae and the lattice of left pp-n-formulae was first introduced by Prest [Pre88, Section 8.4] and then extended by Herzog [Her93] to give an isomorphism between the lattice of open set of the left Ziegler spectrum of a ring and the lattice of open sets of the right Ziegler spectrum of a ring.
Definition 2.8. Let ϕ be a pp-n-formula in the language of right R-modules of the form ∃ȳ(x,ȳ)H = 0 wherex is a tuple of n variable,ȳ is a tuple of l variables, H = (H ′ H ′′ ) T and H ′ (respectively H ′′ ) is a n × m (respectively l×m) matrix with entries in R. Then Dϕ is the pp-n-formula in the language of left R-modules ∃z Similarly, let ϕ be a pp-n-formula in the language of left R-modules of the form ∃ȳ H x y = 0 wherex is a tuple of n variable,ȳ is a tuple of l variables, H = (H ′ H ′′ ) and H ′ (respectively H ′′ ) is a m × n (respectively 4 By Q + we mean the strictly positive rationals 5 By R + we mean the strictly positive reals m×l) matrix with entries in R. Then Dϕ is the pp-n-formula in the language of right R-modules ∃z (x,z) Note that the pp-formula a|x for a ∈ R is mapped by D to a formula equivalent to xa = 0 and the pp-formula xa = 0 for a ∈ R is mapped by D to a formula equivalent to a|x. It is unknown whether this lattice isomorphism always comes from a homeomorphism or even if this map always comes from a homeomorphism between Zg R and R Zg after identifying topologically indistinguishable points in both spaces.
The lattice isomorphism between open sets in Zg R and open sets in R Zg gives rise to a lattice isomorphism between the lattices of closed sets.
Remark 2.11. Under the lattice isomorphism D, C 0 in Zg R is sent to C ∞ in R Zg. This follows from the proofs of 3.8 and 3.9 in section 3 and 2.3. 2.5. Baur-Monk and decidability. Let ϕ/ψ be a pp-n-pair and n ∈ N. There is a sentence, denoted by |ϕ/ψ| ≥ n in the language of (right) Rmodules, which expresses in every R-module M that the quotient group ϕ(M )/ψ(M ) has at least n elements. Such sentences will be referred to as invariant sentences.
Theorem 2.13 (Baur-Monk Theorem). [Pre88] Let R be a ring. Every sentence χ ∈ L R is equivalent to a boolean combination of invariant sentences.
If R is an algebra over an infinite field then for all pp-pairs ϕ/ψ and all R-modules M , |ϕ(M )/ψ(M )| is either equal to one or infinite. This is because if M is a module over a k-algebra then ϕ(M ) and ψ(M ) are k-vector subspaces of M n and thus so is ϕ(M )/ψ(M ).
A recursive field is a field k together with a bijection with N such that addition and multiplication in the field induce recursive functions on N via this bijection. If k is a countable algebraically closed field then there exists a bijection f : k → N so that k together with f is a recursive field. With a bit of work, this follows from [EGN + 98,5.1] together with the fact, [Mar02, 2.2.9], that the theory of algebraically closed fields of a specified characteristic is decidable.
We will frequently use the word "effectively" followed by an operation in this paper. For example, "effectively calculate", "effectively decide" or as in the next paragraph "effectively list". This is just short hand for there exists an algorithm which performs that operation.
If R is a finite-dimensional algebra over a recursive field then the theory of R-modules is recursively axiomatisable i.e. we can effectively list axioms for the theory of R-modules. In this situation we may use the so called proof algorithm, which lists all sentences that are true in all R-modules by listing all formal proofs in first order logic from the axioms for the theory of R-modules. 6 .
With the proof algorithm in hand, we may then compute, for each sentence Θ ∈ L R , a Boolean combination χ of invariant sentences that is equivalent to Ω as follows: In the list of formal proofs we search for entries of the form Ω ↔ χ for some Boolean combination of invariant sentences χ. By Baur-Monk the search terminates.
Thus, given a finite-dimensional algebra R over an algebraically closed recursive field k, in order to show that the theory of R-modules is decidable it is enough to show that there is an algorithm which, given a boolean combination χ of invariant sentences of the form |ϕ/ψ| > 1, answers whether there is an R-module in which χ is true.
If χ is a boolean combination of invariants sentences, we can put it into disjunctive normal form n i=1 χ i where each χ i is a conjunction of invariants sentences and negations of invariants sentences. It is therefore enough to be able to check whether one of the χ i is true in some R-module.
Suppose χ is of the form where ϕ i /ψ i and σ j /τ j are pp-1-pairs. Since every module is elementary equivalent to a (possibly infinite) direct sum of indecomposable pure-injective modules [Pre88,4.36] and solution sets of pp-formulas commute with direct sums, there is an R-module M which satisfies χ if and only if there are indecomposable pure-injective R-modules M 1 , . . . , M n such that M i satisfies for each 1 ≤ i ≤ n. 6 The existence of such an algorithm may be found in any standard source on first order logic, e.g. [End01] Thus, it is enough to show that there is an algorithm which given pp-1pairs, ϕ/ψ, ϕ 1 /ψ 1 , . . . , ϕ n /ψ n answers whether An interpretation functor, I : Mod-R → Mod-S, is specified (up to equivalence) by giving a pp-m-pair ϕ/ψ and, for each s ∈ S, a pp-2m-formula ρ s such that, for all M ∈ Mod-R, the solution set ρ s (M, M ) ⊆ M m × M m defines an endomorphism of ϕ(M )/ψ(M ) as an abelian group, and such that ϕ(M )/ψ(M ), together with the ρ s actions, is an S-module (see [Pre97] or [Pre09,18.2.1]).
An interpretation functor I : Mod-R → Mod-S gives rise to a mapping χ → χ ′ from the set of sentences in the language of S-modules to the set of sentences in the language of R-modules such that for any R-module M , χ is true for IM if and only if χ ′ is true for M . In particular, χ is true for all S-modules in the image of I if and only if χ ′ is true for all R-modules.
If R and S are finite-dimensional k-algebras and I : Mod-R → Mod-S is a k-linear interpretation functor, then together with ϕ/ψ, it is enough to specify pp-formulas ρ s 1 , . . . , ρ sn where s 1 , . . . , s n are a k-basis for S and then extend k-linearly. Moreover, if k is a recursive field then the induced mapping on sentences in the previous paragraph is effective.

Basic calculations
Let k be a recursive field. In this section we list basic operations that can be carried out effectively over k which we will need later in the paper. We will sketch proofs of some of the less trivial operations.
Remark 3.1. Given a finite subset S of k n and v ∈ k n we can effectively calculate a basis for SpanS, decide whether v ∈ SpanS and find a basis for a complement of SpanS. In particular, we can effectively calculate the dimension of SpanS.
Remark 3.2. Given a matrix M ∈ M n×l (k) we can effectively find a basis for the kernel of M in k n and the image of M in k l when considering M as a linear map from k n to k l . Hence we can calculate the rank of M .
Let R be a finite-dimensional algebra with k-basis r 1 = 1, . . . , r s . Let α k ij ∈ k be such that r i r j = s k=1 α k ij r k . These relations and that r 1 , . . . , r s is a k-basis for R completely define R as a k-algebra. An Rmodule M is now given by a k-vector space V together with linear maps

. , A s ) is a presentation of an Rmodule if and only if
for the R-module it represents.
Remark 3.3. Let R be a finite-dimensional k-algebra with k-basis r 1 , . . . , r s . Given presentations of R-modules (k n , A 1 , . . . , A s ) and (k l , B 1 , . . . , B s ) we can effectively calculate a basis for the subspace Note that this set is Hom(M (A 1 , . . . , A s ), M (B 1 , . . . , B s )) in terms of matrices with respect to the standard basis.
From now on we will assume that k is a recursive algebraically closed field and that R is a finite-dimensional algebra over k given in terms of a k-basis r 1 , . . . , r s and relations.
Lemma 3.4. Given a presentation (k n , A 1 , . . . , A s ) of an R-module M , we can effectively decide whether M is indecomposable or not. If M is decomposable then we can effectively find non-zero pair-wise disjoint A 1 , . . . , A s -invariant subspaces V 1 , . . . , V m of k n such that each V i with the restricted action of A 1 , . . . A s is indecomposable as an R-module and such that V 1 + . . . + V m = k n .
Proof. First we effectively find a basis for End R (M ). That is we find a basis T 1 , . . . , T l for the subspace We may assume T 1 is the identity matrix. Now M is indecomposable if and only if End R (M ) has no idempotents apart from 0 and 1. For a 1 , . . . , a l ∈ k, the condition that a 1 T 1 + . . . + a l T l is idempotent is equivalent to a = (a 1 , . . . , a l ) being a root of a particular system of polynomial equations with coefficients in k (and we can find this system effectively). Using effective quantifier elimination for algebraically closed fields, we can thus decide whether there exists (a 1 , . . . , a l ) ∈ k such that a = 0, a = (1, 0, . . . , 0) and a 1 T 1 + . . . + a l T l is idempotent. Thus, given a presentation of a finite-dimensional R-module, we can effectively decide if it is indecomposable or not.
Supposing that we know that M is not indecomposable we may now search for an idempotent e represented by a 1 T 1 + . . . + a l T l in End R (M ) which is not the identity or zero. We know we will eventually find one because M is not indecomposable. Now M = eM ⊕ (e − 1)M and we can easily use our presentation of M to get presentations of eM and (e − 1)M . If either eM or (e−1)M is decomposable then we may repeat the process eventually stoping when we get a decomposition of M into indecomposable summands.
Lemma 3.5. There is an algorithm which lists the indecomposable finitedimensional representations of R.
Proof. Given 3.4, it is enough to be able to effectively decide if, given two presentations (k n , A 1 , . . . , A s ) and (k n , B 1 , . . . , B s ) of indecomposable R-modules M and N , M is isomorphic to N .
We can compute a basis T 1 , . . . , T l for and S 1 , . . . , S l for . . , A n ) and M (B 1 , . . . , B n ) are isomorphic if and only if there exist t 1 , . . . , t l ∈ k and s 1 , . . . , s l ∈ k such that (t 1 T 1 + . . . + t l T l )(s 1 S 1 + . . . + s l S l ) = 1. This can be expressed as a system of polynomial equations over k in t 1 , . . . , t l , s 1 , . . . , s l and thus we may check, using effective quantifier elimination for algebraically closed fields, whether it has a solution or not.
Lemma 3.6. Given a presentation of a finitely presented R-module M and a pp-pair ϕ/ψ we can effectively decide whether ϕ/ψ is open on M or not.
Proof. Given a pp-n-formula ϕ we can calculate the dimension as a k-vector space of the solution set of ϕ in M n as follows. Suppose ϕ is Lemma 3.7. Given a presentation of a finite-dimensional module M we can effectively find a pp-n-formula generating the pp-type of a generating tuple for M .
Proof. Let (k n , A 1 , . . . , A s ) be a presentation of M and let e = (e 1 , . . . , e n ) be the n-tuple of standard basis vectors for k n . We need to write down finitely many linear equations over R which describe the linear relations over R which e satisfies. We can do this by describing a system of finitely many linear equations over k which describe the linear relations between the vectors e i A j in k n .
Lemma 3.8. Given a presentation of a finite-dimensional module M we can effectively find a pp-n-pair ϕ/x = 0 such that the functor (M, −) is equivalent to the functor defined by ϕ/x = 0.
Proof. Given a presentation (k n , A 1 , . . . , A s ) of M , by lemma 3.7, we can effectively find a pp-n-formula generating the pp-type of the standard basis for k n in M . Lemma 3.10. Given a presentation of a finitely presented R-module M we can calculate its dimension vector. Hence, given a presentation of a finitely presented indecomposable R-module M over a tubular algebra R, we can calculate the slope of M .
Proof. That we can calculate the dimension vector of a module now follows directly from 3.3.
Corollary 3.11. We can list presentations of the finite-dimensional indecomposable modules of slope q. We can list the quasi-simples of slope q.
Proof. The quasi-simples of slope q are just those modules of slope q with 1-dimensional endomorphism ring.
Lemma 3.12. Given a quasi-simple S of slope q, we can list the finitedimensional modules in the ray starting at S and in the coray starting at S Proof. Look for M indecomposable of slope q with Hom(S, M ) = 0 (respectively Hom(M, S) = 0). Lemma 3.13. There is an algorithm, which given a presentation of a finitedimensional indecomposable R-module M , outputs a pp-1-pair ϕ/ψ isolating M in Zg R .
Proof. Given a finite-dimensional indecomposable module M over a finitedimensional algebra, [But80] gives a method of effectively constructing an almost split sequence Pick m ∈ M non-zero and calculate ϕ generating the pp-type of m and ψ generating the pp-type of f (m). Then ϕ/ψ isolates M (see [Pre09,Theorem 5.3.31]). By 3.7 we can effectively find a pp-formula ψ(x) generating the pp-type of the standard k-basis (e 1 , . . . , e n ) of M . The pp-type of m is ∃y Thus we can effectively find a pp-formula generating the pp-type of m.
Lemma 3.16. There is an algorithm which, given a presentation of a module M and an element m, outputs a presentation of M/mR.
Proof. We are using the fact that given a finite subset L of k n we can algorithmically find a basis for SpanL and a basis for a complement of SpanL. Let (k n , A 1 , . . . , A s ) be a presentation for M and identify m with its image in k n with respect to this presentation.
Find a basis e 1 , . . . , e t for Span{m, mA 1 , . . . , mA s } and a basis for a complement f 1 , . . . , f l of Span{m, mA 1 , . . . , mA s }. By considering the action of A 1 , . . . , A s on f 1 , . . . , f l we get a presentation of M/mR.

Ziegler Spectra of tubes of rational slope
In this section we describe the Ziegler spectrum of D q where q ∈ Q + . That is we describe the induced topology on C q := Zg R ∩ D q by describing the closed subsets of C q .
Recall the complete list of indecomposable pure-injectives of rational slope q from 2.6.
We essentially follow Ringel's proof from [Rin98] for tame hereditary algebras.
Proposition 4.1. A subset X of C q is closed if and only if the following hold: (1) If S is a quasi-simple in T q and if there are infinitely many finite length modules M ∈ X with Hom(S, M ) = 0 then S[∞] ∈ X.
(2) If S is a quasi-simple in T q and if there are infinitely many finite length modules M ∈ X with Hom(M, S) = 0 then S ∈ X.
(3) If there are infinitely many finite length modules in X or X contains an infinite length module then G q ∈ X.
Lemma 4.2. If X is a closed subset of C q then (1) from 4.1 holds.
Proof. This is clear since the Prüfer module is a direct union of such modules.
Lemma 4.3. If X is a closed subset of C q then (2) from 4.1 holds.
Proof. Suppose X contains infinitely many finite-dimensional modules M with Hom(M, S) = 0. Then each of these M is in the coray starting at S. Therefore S is an inverse limit of these M . So by [BP02, 2.3], S is in the closure of these M .
Lemma 4.4. If X is a closed subset of C q then (3) from 4.1 holds.
Proof. Since X is closed it is compact. Therefore if X contain infinitely many isolated points then X must contain a non-isolated point. By [Pre09,5.3.31], all finite-dimensional points are isolated in the Ziegler spectrum of a finite-dimensional algebra and hence, all finite-dimensional points are isolated in C q . Therefore, if X contains infinitely many finite-dimensional indecomposable modules then X must contain an infinite-dimensional module i.e. either an adic, Prüfer or generic module. By [Kra98,8.10] we know that the generic is in the closure of every adic and every Prüfer.
Proof of 4.1. The proof of proposition 4.1 now is the same as Ringel's proof for tame hereditary algebras but working inside D q .
Definition 4.5. Let E be a quasi-simple of slope q and i ∈ N. Let In this section, we present an algorithm which, given n + 1 pp-pairs ϕ/ψ, ϕ 1 /ψ 1 , . . . , ϕ n /ψ n and q ∈ Q + , answers whether Note that D q is axiomatised by saying that for each finite-dimensional indecomposable module Q of slope strictly greater than q, the functor (Q, −) is zero on D q and for each finite-dimensional indecomposable module P of slope strictly less than q, the functor − ⊗ P * is zero on D q . Given a presentation of a module M , by lemma 3.8 we can effectively find a ppn-pair σ/τ such that the functor defined by σ/τ is equivalent to (M, −) and by lemma 3.9 a pp-n-pair σ/τ such that the functor defined by σ/τ is equivalent to − ⊗ M * . By 3.5 and 3.10 we can list the indecomposable finite-dimensional modules of slope < q and those of slope > q. Thus, given q ∈ Q + , we can recursively list sentences which axiomatise D q . Let Σ q be the recursive list of sentences axiomatising D q .
Remark 5.1. Let q ∈ Q + and ϕ/ψ, ϕ 1 /ψ 1 , . . . , ϕ n /ψ n be pp-pairs. Then By compactness, this means that there is some finite subset of Σ ⊆ Σ q such that We now use the results of the previous section to give canonical forms for compact open subsets of C q .
Lemma 5.2. Each compact open subset U of C q is unique of the form: ( Proof. Proposition 4.6 gives us a description of the compact open subsets of C q . We just need to observe that the list above contains no repeats.
Lemma 5.3. There is an algorithm, which given q ∈ Q + and pp-pairs ϕ/ψ, ϕ 1 /ψ 1 , . . . , ϕ n /ψ n , answers whether Proof. By 3.14, 3.13, 3.9 and 3.8, there is an algorithm which lists pp-pairs defining the open sets of the form F ({X 1 , . . . , X n }), {X}, C(X) and R(X) where X 1 , . . . , X n , X ∈ C q . Thus, since D q is recursively axiomatised, there is an algorithm which given a pp-pair ϕ/ψ, finds a compact open set U such that (ϕ/ψ)∩C q = U ∩C q and such that U is in the canonical form given in 5.2.
We now need to take each of the compact open sets of the form F ({X 1 , . . . , X n }), {X}, C(X) and R(X) and write an algorithm which determines whether it is contained in a finite union, W 1 ∪ . . . ∪ W n of some specified other open sets of the form F ({X 1 , . . . , X n }), {X}, C(X) and R(X). . . , Y l }\{X 1 , . . . , X m }. We now just need to check whether the finite subset {Y 1 , . . . , Y l }\{X 1 , . . . , X m } is contained in W 1 ∪ . . . ∪ W n . This is case 0.

One-point extensions and coextensions
Let T n 1 ,...,nt be the star quiver with t arms of length n 1 , . . . , n t in the "subspace" configuration. A canonical algebra of tubular type is a onepoint extension of the tame hereditary path algebra of T n 1 ,...,nt by a quasisimple module X at the base of a tube such that (n 1 , . . . , n t ) is in the set {(3, 3, 3), (2, 4, 4), (2, 3, 6), (2, 2, 2, 2)} [Rin84,pg161]. These algebras may equally well be viewed as one-point coextensions of star path algebras with the "cosubspace" configuration by a quasi-simple module X at the base of a tube.
Throughout this section A will be the path algebra of a star quiver in subspace configuration as above, X ∈ mod-A will be a quasi-simple at the base of a tube and A[X] will be the one-point extension of A by X i.e. the 2 × 2-matrix algebra The category Mod-A[X] is equivalent to Rep(X), the category of representation of the bimodule k X A and also to, Rep(X), the k-category with objects

This gives us the following two embeddings of Mod-A into Mod-A[X]
; We also have a functor r : This functor is right adjoint to F 0 and left adjoint to F 1 . In this section we will see that F 0 and F 1 are interpretation functors whose images are finitely axiomatisable and that every indecomposable pure-injective in P 0 ∪ D 0 is in the union of the images of F 0 and F 1 .
Remark 6.1. Let A be a k-algebra and X a right A-module. The assignment F 0 which sends a right A-module M to the right A[X]-module (0, M, 0) and sends a morphism g : M → N to (0, g) is clearly a full and faithful exact interpretation functor whose image is a (finitely axiomatisable) definable subcategory of Mod-A[X].
Proposition 6.2. Let A be a k-algebra and X a finitely presented right A-module. The functor F 1 is a full and faithful (left exact) interpretation functor whose image is a definable subcategory (after closing under isomor- Proof. It is straightforward to see that F 1 is indeed a functor and that it is full, faithful and left exact (see for instance [SS07, 1.4]).
The functor F 1 is an interpretation functor if and only if it commutes with direct limits and products [Pre11,25.3]. In order to check that F 1 commutes with direct limits and products, it is enough to check that its composition with the forgetful functor from Mod-A[X] to Mod-k commutes with direct limits and products. This follows since Hom(X, −) commutes with direct limits and products.
We now show that the image of F 1 is a (finitely axiomatisable) definable subcategory of Mod-A[X]. First note that L = (L 0 , L 1 , Γ L ) is in the (essential) image of F 1 if and only if Γ L is an isomorphism. Let t 1 , . . . , t n generate X as an A-module. Note that for any δ ∈ L 0 and γ ∈ Hom(X, L 1 ) we have that Γ L (δ) = γ if and only if Γ L (δ) Let ψ ∈ pp n R be the pp-formula Let ϕ generate the pp-type of (t 1 , . . . , t n ) viewed as a tuple from (0, X, 0). Now Let E 0 be the image of the functor F 0 and E 1 be the image of the functor F 1 . We will show, 6.8, that every indecomposable finite-dimensional module of slope 0 is either contained in E 0 or E 1 . Thus, if the finite-dimensional modules of slope 0 are dense in the Ziegler closed subset corresponding to the definable subcategory of slope 0 then all indecomposable pure-injective modules of slope zero are either contained in E 0 or E 1 . Proposition 6.3. Let R be a tubular algebra. The finite-dimensional indecomposable R-modules of slope zero are dense in the Ziegler closed subset of indecomposable pure-injective modules of slope zero.
Proof. By an argument exactly as the first paragraph of [Pre88, Theorem 13.6] we know that every open set containing a module of slope zero contains a finite-dimensional module of slope greater than or equal to zero.
Suppose N is an infinite-dimensional indecomposable module of slope zero and that F is a coherent functor with F N = 0. Let P 2 , P 1 ∈ mod-R be preprojective, T 2 , T 1 ∈ mod-R of slope zero, Q 2 , Q 1 ∈ mod-R of slope greater than zero and f : P 1 ⊕ T 1 ⊕ Q 1 → P 2 ⊕ T 2 ⊕ Q 2 be such that be canonical projections and embeddings for i = 1, 2. Let G be a coherent functor such that Suppose that M is an indecomposable module of slope zero. We show that (π 2 f µ 1 , M ) is surjective if and only if (f, M ) is surjective. That is, we show that F M = 0 if and only if GM = 0.
Since there are no non-zero maps from modules of slope greater than zero to M , for all g ∈ (P 2 ⊕ T 2 ⊕ Q 2 , M ), g = gµ 2 π 2 . So the following diagram commutes. Now in order to show that there is some L ∈ ind-R of slope zero such that F L = 0, it is enough to show that GL = 0.
By the first paragraph, there exists L ∈ mod-R of slope greater than or equal to zero, such that GL = 0. Suppose that the slope of L is greater than zero and let h ∈ (P 1 ⊕ T 1 , L) be such that it doesn't factor through g. Since the finite-dimensional modules of slope zero separate the preprojective modules from those of slope greater than zero, h factors through some direct sum of finite-dimensional modules of slope zero. One of these modules T is such that GT = 0.
We gather together the facts we need in order to show that every finitedimensional module of slope zero over a canonical algebra of tubular type is either in the image of F 1 : Mod-A → Mod-A[X] or in the image of From now on, let m be the rank of the tube T containing X. The quasisimples at the mouth of T are X, τ −1 X, . . . , τ −(m−1) X. Note that if 1 ≤ p < m and i ∈ N then Hom A (X, Proof. Suppose P is a preprojective A-module, so Hom(X, P ) = 0. Then Hom A (rM, P ) ∼ = Hom A[X] (M, F 1 P ) = 0 since F 1 P = (0, P, 0) ∈ P 0 .
Note that if Q is preinjective over A then (0, Q, 0) has slope greater than zero. This is because dimX, dimQ = dim Hom A (X, Q)−dim Ext A (X, Q) = dim Hom A (X, Q) = 0, since X is a regular module. Then So for all preprojective A-modules P , Hom A (rM, P ) = 0 and for all prein- Lemma 6.5.
(1) When m ≥ 2, for each i ∈ N and 1 ≤ p < m, Proof. Apply [SS07, XV 1.6] to the almost split sequence 0 The following lemma is most likely well known but since we couldn't find a reference, we include a proof.
Lemma 6.6. For each i ∈ N, is an almost split exact sequence.
Proof. We prove this by induction on i. Suppose i = 1. First note that the embedding of X[1] into X[2] remains irreducible in A[X] after applying F 0 and the canonical embedding of F 0 X[1] into F 1 X[1] is irreducible since it is the embedding of the radical of an indecomposable projective.
Suppose that N is an indecomposable non-projective module and that f : Now suppose that we have proved the assertion of the lemma for all i ≤ n. Suppose that N is an indecomposable non-projective module and that f : F 0 X[n + 1] → N is irreducible. Then, as before, there is an irreducible map from τ N to It remains now to note that every map from F 0 X[n + 1] to F 1 X[n + 1] factors though the canonical embedding and that the spaces of irreducible morphisms Irr A[X] (F 0 X[n + 1], F 0 X[n + 2]) and Irr A (X[n + 1], X[n + 2]) are isomorphic. Now suppose i = 1. Then F 1 X[1] is an indecomposable projective and F 0 X[1] is its radical. Thus either f : The second possibility can't occur since f 0 = 0. Proposition 6.9. Every indecomposable pure-injective module of slope zero is either in the image of F 0 or in the image of F 1 . Note that all preprojective modules are in the image of F 0 .
Proof. The pure-injective modules of slope zero form a closed subset C 0 of the Ziegler spectrum. By 6.3, the finite-dimensional indecomposable modules of slope zero are dense in this set. We have shown, 6.1, 6.2, that the images of F 0 and F 1 are definable subcategories. Let A 0 and A 1 be their images in Zg A[X] intersected with C 0 , note that both A 0 and A 1 are closed. Since all finite-dimensional points of slope zero are contained in either A 0 or A 1 the closure of A 0 ∪ A 1 is C 0 . Thus A 0 ∪ A 1 = C 0 as required.
We now use the above to provide an algorithm which given pp-pairs ϕ/ψ, ϕ 1 /ψ 1 , . . . , ϕ n /ψ n answers whether there is a preprojective or module of slope zero in (ϕ/ψ) but not in n i=1 (ϕ i /ψ i ). Proposition 6.10. Let A be a tame hereditary algebra and A[X] be a canonical algebra of tubular type both over a recursive algebraically closed field. There is an algorithm which given pp-pairs ϕ/ψ, ϕ 1 /ψ 1 , . . . , ϕ n /ψ n answers whether there is an indecomposable pure-injective module N such We now describe how to effectively check whether there is an N ∈ E 0 such that N ∈ (ϕ/ψ) and N / ∈ n i=1 (ϕ i /ψ i ). Given a pp-pair ϕ/ψ we can effectively translate this to a pp-pair σ/τ in the language of A-modules via F 0 such that N ∈ (σ/τ ) if and only if F 0 N ∈ (ϕ/ψ). Since the common theory of modules over a tame hereditary algebra is decidable ( [Gei94]) we can answer whether one Ziegler basic open set over A is contained in a finite union of other specified Ziegler open sets over A.
The argument is exactly the same for E 1 .
We now deal with the preinjective component and modules of slope ∞. If R is a canonical algebra of tubular type (respectively a tubular algebra) then R op is also a canonical algebra of the same tubular type (respectively a tubular algebra) as R. For canonical algebras, this can be easily seen from the original definition of canonical algebra [Rin84, pg 161] and for tubular algebras is [Rin84, 5.2.3].
We have shown 6.3 that, for a tubular algebra R, the finite-dimensional indecomposable R-modules of slope zero are dense in C 0 . Using elementary duality, this implies the same result for C ∞ .
Proposition 6.11. Let R be a tubular algebra. The finite-dimensional indecomposable R-modules of slope infinity are dense in the definable subcategory of modules of slope infinity.
If E is a definable subcategory of Mod-R such that E := {N ∈ Mod-R | ϕ i (N ) = ψ i (N ) for all i ∈ I} then let DE be the definable subcategory of Mod-R op such that DE : Lemma 6.12. Let A be a tame hereditary algebra and R := A[X] be a canonical algebra of tubular type. Every indecomposable pure-injective module of slope infinity and every indecomposable preinjective module over R op is in Proof. This is true for all finite-dimensional modules by 2.3. So by 6.11, this is also true for all indecomposable pure-injectives of slope infinity.
If R is a canonical algebra of tubular type then let E ′ 0 (respectively E ′ 1 ) be DE 1 (respectively DE 0 ) where E 0 and E 1 are the images of F 0 (respectively F 1 ) as functors to Mod-R op .
Lemma 6.13. Let A be a tame hereditary algebra and R := A[X] be a canonical algebra of tubular type. Let ϕ/ψ, ϕ 1 /ψ 1 , . . . , ϕ n /ψ n be pp-pairs over R. There exists an indecomposable pure-injective R-module N ∈ E 0 ∪E 1 such that N ∈ (ϕ/ψ) and N / ∈ (ϕ i /ψ i ) if and only if there exists a inde- . Then Hom(L, k) is in the definable category E 0 ∪ E 1 and Hom(L, k) opens Dψ/Dϕ but not Dψ i /Dϕ i for any 1 ≤ i ≤ n. Therefore there is an indecomposable pure-injective module M over R op in E ′ 0 ∪ E ′ 1 which is in (Dψ/Dϕ) but not in n i=1 (Dψ i /Dϕ i ). The reverse direction is proved symmetrically.
Corollary 6.14. Let A be a tame hereditary algebra and A[X] be a canonical algebra of tubular type over a recursive algebraically closed field. There is an algorithm which given pp-pairs ϕ/ψ, ϕ 1 /ψ 1 , . . . , ϕ n /ψ n answers whether there is an indecomposable pure-injective module N such that N ∈ (ϕ/ψ)

Corrections to a paper of Harland and Prest
Throughout this section, unless explicitly indicated, R will be a tubular algebra.
The main work of this section is to show, 7.15, that Corollary 8.8 of [HP15] is false for all tubular algebras and to provide, 7.7, a best possible replacement. Although the replacement of corollary 8.8 will not be used in later sections, many statements in this section will be needed.
We start by correcting some statements in section 3 of [HP15].
In [HP15], it is claimed that for a, b ∈ R + the set of modules which are direct limits of finite-dimensional modules with slope in the interval (a, b) is a definable subcategory of Mod-R, in [HP15] this is called the set of modules supported on (a, b). This is false for a, b ∈ Q + . The problem is, that although the set of modules supported on (a, b) is a definable category by [Len83, 2.1], it is not a definable subcategory of Mod-R.
Using the terminology of [HP15], the set of modules lying over (a, b), i.e. those modules M such that M ⊗ P * = 0 for all finite-dimensional P of slope less than or equal to a and (P, M ) = 0 for all finite-dimensional P of slope greater than or equal to b, is a definable subcategory by definition. However, the description of the indecomposable pure-injectives lying over (a, b) given in [HP15] is not correct. The following proposition corrects this.
Proposition 7.1. Let R be a tubular algebra and let a, b ∈ R + . The smallest definable subcategory, D + (a,b) , containing all finite-dimensional indecomposable modules with slope in (a, b) contains exactly all indecomposable pureinjectives with slope in (a, b) plus, (1) the Prüfer and generic modules of slope a if a ∈ Q + (2) the adic and generic modules of Before we prove the proposition, we need a few lemmas and to recall a few facts. The following remark will hold for general rings if Hom(M, k) is replaced an appropriate notion of dual module (see [Pre09] for notions of dual modules in the general context). We will only need it for finitedimensional k-algebras. Proof of 7.1. We give proofs for the case a, b ∈ Q + and the case a, b / ∈ Q + . Note that the definable subcategory D + (a,b) is contained in the definable subcategory of modules M such that (P, M ) = 0 for all finite-dimensional P of slope greater than or equal to b and M ⊗ P * = 0 for all finite-dimensional P of slope less than or equal to a. We will now show that this definable subcategory is in fact D + (a,b) . By 7.3, every indecomposable pure-injective module with slope in (a, b) is in D + (a,b) . If b ∈ Q + , ǫ ∈ R + and a definable subcategory D of Mod-R contains all finite-dimensional indecomposable modules with slope in (b − ǫ, b) then D contains the generic at b and all adic modules at b. This is because any module of slope b is a direct union of direct sums of indecomposable finite-dimensional modules with slope in (b − ǫ, b] and no finite-dimensional indecomposable module of slope b is a submodule of the generic at b or any adic module at b by 7.4. If a ∈ Q + , ǫ ∈ R + and a definable subcategory D of Mod-R contains all finite-dimensional indecomposable modules with slope in (a, a + ǫ) then D contains the generic at a and all Prüfer modules at a. This follows from the above paragraph, since each left adic module at 1/a is equal to the k-dual of some right Prüfer module at a, so, by [Pre09,1.3.16], every right Prüfer module at a is a pure-submodule of the k-dual of a left adic module at 1/a. Note now that, by 4.1, if any definable subcategory contains either a Prüfer module at slope c or an adic module at slope c then it also contains the generic module at slope c.
Thus if a, b are both rational then D + (a,b) contains all pure-injective indecomposables with slope in (a, b), the adic and generic modules at b and the Prüfer and generic modules at a. These are exactly the pure-injective indecomposable modules M such that (P, M ) = 0 for all finite-dimensional P of slope greater than or equal to b and M ⊗ P * = 0 for all finite-dimensional P of slope less than or equal to a.
If both a, b are irrational then the situation is much simpler. All indecomposable modules of slope b are direct unions of direct sums of finitedimensional modules with slope in (a, b). In order to deal with the a irrational case we use 7.5, which says that if ϕ/ψ is open on some indecomposable pure-injective module of slope a then it is open on all homogeneous tubes with slope in (a, a + ǫ) for some ǫ > 0. Thus if ϕ/ψ is open on some indecomposable pure-injective module of slope a then it is open on some finite-dimensional indecomposable module with slope in (a, b). So the indecomposables pure-injectives of slope a are in the definable subcategory generated by the finite-dimensional indecomposable modules with slope in (a, b).
We say a pp-pair ϕ/ψ is uniformly open at q ∈ Q + if ϕ/ψ is open on all finite-dimensional indecomposable modules of slope q. We say that ϕ/ψ is uniformly closed at q ∈ Q + if ϕ/ψ is closed on all finite-dimensional indecomposable modules of slope q. We say that q ∈ Q + is a non-uniform slope for ϕ/ψ if ϕ/ψ is neither uniformly open or closed at q.
Corollary 8.8 of [HP15] states that if ϕ/ψ is a pp-pair over a tubular algebra then for all but finitely many r ∈ R + , ϕ/ψ is either ϕ/ψ is open on all indecomposable pure-injective modules of slope r or closed on all indecomposable pure-injective modules of slope r. It further states that the set of r ∈ R + for which ϕ/ψ is open on all indecomposable pure-injective modules of slope r is the union of finitely many rational points and intervals with rational endpoints. We will show in 7.15 that for all tubular algebras this is not the case and in fact, that for all p ∈ Q ∞ 0 there exists a pp-pair ϕ/ψ such that p is an accumulation point of the set of slopes q ∈ Q + where ϕ/ψ is neither uniformly open nor uniformly closed at q.
We first prove the following which is a best possible replacement for corollary 8.8 of [HP15].
Theorem 7.7. Let ϕ/ψ be a pp-pair and S be the set of slopes q ∈ Q + where ϕ/ψ is neither uniformly open nor uniformly closed at q. The set S has finitely many accumulation points in R, and all these accumulation points are in Q.
The following series of lemmas will be used in the proof of 7.7.
Lemma 7.8. If q ∈ Q + and ϕ/ψ is open on all finite-dimensional modules of slope q in homogeneous tubes then ϕ/ψ is closed on at most finitely many X ∈ ind-R of slope q.
Proof. This follows directly from 4.6.
Lemma 7.9. Suppose that q ∈ Q + , ϕ/ψ is a pp-pair and v ∈ K 0 (R) is such that dim ϕ/ψ(X) = v · dimX for all X ∈ ind-R of slope q. Then ϕ/ψ is either open on all modules in homogeneous tubes of slope q or closed on all modules in homogeneous tubes of slope q.
Proof. Let w be the dimension vector of a finite-dimensional quasi-simple in a homogeneous tube of slope q. Then for all finite-dimensional indecomposable modules X of slope q lying in homogeneous tubes, dimX = n · w for some n ∈ N. Since dim ϕ/ψ(X) = v ·dimX for all X ∈ ind-R of slope q, ϕ/ψ is open on all modules in homogeneous tubes of slope q if v · w > 0 and ϕ/ψ is closed on all modules in homogeneous tubes of slope q if v · w = 0.
Proposition 7.10. Suppose that q ∈ Q + , ϕ/ψ is a pp-pair and v ∈ K 0 (R) is such that dim ϕ/ψ(X) = v · dimX for all X ∈ ind-R of slope q. If ϕ/ψ is closed on all modules of slope q in homogeneous tubes then ϕ/ψ is closed on all modules of slope q.
Proof. Let E 1 , . . . , E n be the quasi-simples at the mouth of an inhomogeneous tube T (ρ) of slope q. Then dimE 1 +. . .+dimE n is the dimension vector of an indecomposable module in a homogeneous tube with slope q. Thus ϕ/ψ is closed on all modules with dimension vector dimE 1 + . . .
Thus v · dimX = 0 for every finitedimensional module in T (ρ). Thus ϕ/ψ is closed on all finite-dimensional indecomposable modules of slope q. So, by 4.1, ϕ/ψ is closed on all modules of slope q.
Lemma 7.11. Let a < b ∈ Q ∞ 0 , ϕ/ψ be a pp-pair and suppose there is a v ∈ K 0 (R) such that dim ϕ/ψ(X) = v · dimX for all X ∈ ind-R of slope q ∈ (a, b). If ϕ/ψ is closed on all homogeneous tubes of slope q for some rational q ∈ (a, b) then ϕ/ψ is uniformly closed on all rational slopes in (a, b).
Proof. Let q ∈ (a, b) be such that ϕ/ψ is closed on all homogeneous tubes of slope q. By 7.10, we may assume that ϕ/ψ is uniformly closed at q. Thus So, since the closed sets C (q−ǫ,q+ǫ) form a chain and (ϕ/ψ) is compact, there exists a δ > 0 such that (ϕ/ψ) ⊆ Zg R \C (q−δ,q+δ) .
There is an algorithm, 3.4, which given a presentation of M outputs presentations of its indecomposable factors with multiplicity. From this we can compute the dimension vectors of the indecomposable factors of M and coker(m) with multiplicity. We may now, by 3.10, compute the slope of each of the indecomposable factors of M and coker(m). Let q 1 < . . . < q n be the slopes of those indecomposable factors which have slope greater than zero and less than infinity.
For 0 ≤ i ≤ n, let w i (respectively u i ) be the sum of the dimension vectors of all indecomposable factors of M (respectively of coker(m)) with slope strictly smaller than q i+1 (equivalently have slope less than or equal to q i ).
Since all indecomposable factors of M are either preprojective or have slope less than or equal to q i , . . , S m are the simple modules over R.
Corollary 7.14. Let ϕ/ψ be a pp-pair. There is an algorithm which outputs n ∈ N, q 0 = 0 < q 1 < q 2 < . . . < q n < q n+1 = ∞ ∈ Q ∞ 0 and v 0 , . . . , v n ∈ K 0 (R) such that for all for all finite-dimensional indecomposable modules N with slope in (q i , q i+1 ), Proof of theorem 7.7. By 7.14, it is enough to show that if a < b ∈ Q ∞ 0 and there exists v ∈ K 0 (R) such that for all M ∈ ind-R, then there are only finitely many accumulation points of non-uniform slopes for ϕ/ψ in (a, b).
Lemmas 7.8, 7.9, 7.10 and 7.11 show that either ϕ/ψ is uniformly closed on all rational q ∈ (a, b) or for each rational q ∈ (a, b), ϕ/ψ is open on all but finitely many points of slope q, all of which are finite-dimensional.
If ϕ/ψ is uniformly closed on all rational q ∈ (a, b) then ϕ/ψ is closed on all indecomposable pure-injectives with slope in (a, b). This is because the finite-dimensional indecomposable modules are dense in C (a,b) .
If for each rational q ∈ (a, b), ϕ/ψ is open on all but finitely many points of slope q, all of which are finite-dimensional then there are no rational accumulation points in the set of non-uniform slopes for ϕ/ψ between a and b by 7.12.
It remains to show that if for each rational q ∈ (a, b), ϕ/ψ is open on all but finitely many points of slope q, all of which are finite-dimensional then there are no irrational accumulation points of non-uniform slopes for ϕ/ψ inside (a, b). Note that if ϕ/ψ is closed on X ∈ ind-R then X is in an inhomogeneous tube. We refer forward to 8.3, which states that there is a finite set Ω of roots of χ R such that for all X ∈ ind-R lying in inhomogeneous tubes dimX = y + w where y ∈ Ω and w ∈ radχ R . Let g 1 , g 2 generate radχ R . Note that, since ϕ/ψ is open on all homogeneous tubes with slope in (a, b) either vg 1 = 0 or vg 2 = 0. So if dimX = y + αg 1 + βg 2 for y ∈ Ω and α, β ∈ Z, X has slope in (a, b) and ϕ/ψ is closed on X then 0 = v · dimX = αv · g 1 + βv · g 2 + v · y.
For fixed y ∈ Ω, we now consider the set of α, β ∈ Z such that αv · g 1 + βv · g 2 + v · y = 0. If v · g 1 = 0 then β is a fixed integer and if v · g 1 = 0 then α = σβ + µ for some fixed σ, µ ∈ Q. In the first case there are fixed rationals c, d, e, f such that the slope of y + αg 1 + βg 2 is of the form (cα + d)/(eα + f ). So as α tends to ±∞, the slope of y + αg 1 + βg 2 tends to a rational or ±∞. In the second case there are fixed rationals c, d, e, f such that the slope of y + αg 1 + βg 2 is of the form (cβ + d)/(eβ + f ). So as β tends to infinity, the slope of y + αg 1 + βg 2 tends to a rational or ±∞. Therefore, since Ω is finite, there are no irrational accumulation points of non-uniform slope for ϕ/ψ in (a, b).
In the above proof we could have replaced the final argument with the following argument using 2.7. We know that if r ∈ (a, b) is irrational and an accumulation point of non-uniform slopes for ϕ/ψ then ϕ/ψ is open on all points in C r . Thus ǫ>0 C (r−ǫ,r+ǫ) = C r ⊆ (ϕ/ψ). So Zg R \(ϕ/ψ) ⊆ 7.2. Accumulation points of non-uniform slopes. The rest of this section will be spent proving the following result.
Proposition 7.15. Let R be a tubular algebra. For any q ′ ∈ Q + ∪ {0} there exists L ∈ ind-R of slope q ′ such that q ′ is an accumulation point of the set of non-uniform slopes for Hom(L, −).
If X is finite-dimensional then the functor Hom(X, −) is equivalent to a functor given by a pp-pair ϕ/ψ (see 3.8). So 7.15 implies that for all q ′ ∈ Q + ∪ {0} there is a pp-pair ϕ/ψ such that q ′ is an accumulation point of the set of non-uniform slopes of ϕ/ψ thus contradicting corollary 8.8 of [HP15].
Corollary 7.16. Let R be a tubular algebra. For all q ∈ Q ∞ 0 there exists a pp-pair ϕ/ψ such that q is an accumulation point of the set of non-uniform slopes for ϕ/ψ.
Proof. For all q ∈ Q + ∪ {0}, this follows directly from 7.15. The result for q = ∞ follows by combining duality for pp-formulas with 2.12 and 2.3. 7.2.1. Coherent sheaves on weighted projective lines. We will use categories of coherent sheaves on a weighted projective lines of tubular type to prove 7.15. In order to introduce notation and for the readers convenience, we briefly review various features of categories of coherent sheaves on weighted projective lines. Our main references are [GL87] and [LM92].
Let t ∈ N be greater than 2, p = (p 1 , . . . , p t ) be a t-tuple of strictly positive integers p i , called a weight sequence, and λ = (λ 1 , . . . , λ t ) a ttuple of pairwise distinct elements of P 1 (k), called a parameter sequence, normalised so that λ 1 = ∞, λ 2 = 0 and, if it exists λ 3 = 1. For 1 ≤ i ≤ t, we will refer to the point λ i ∈ P 1 (k) as an exceptional point (of weight p i ) and all other points in P 1 (k) as ordinary.
For every pair (p, λ), Geigle and Lenzing define, [GL87, 1.1,1.5], a weighted projective line X := X(p, λ). We will not give this definition but instead define the category of coherent sheaves on X(p, λ) purely in terms of (p, λ).
Given a weight sequence p, let L := L(p) be the abelian group on generators x 1 , . . . , x t with the relations The degree homomorphism δ : L → Z is defined on generators by δ(x i ) := p i /p where p is the lowest common multiple of p 1 , . . . , p t .
The L(p)-graded k-algebra S := S(p, λ) is the quotient of k[X 1 , . . . , X t ] by the ideal generated by with the L(p)-grading given by assigning X i , for 1 ≤ i ≤ t, degree x i . Let mod L -S denote the category of finitely generated L-graded S-modules with morphisms given by S-linear maps of degree zero. Let mod L 0 -S denote the full subcategory of all graded modules of finite length. The category of coherent sheaves, coh(X), on the weighted projective line X, is equivalent to the category mod L -S/mod L 0 -S (see [GL87, Serre's theorem] and [GL91,7.4]). The structure sheaf O is the image of S in mod L -S/mod L 0 -S. The group L acts on mod L -S by grading shift i.e. for x ∈ L and M ∈ mod L -S, M (x) is defined to be the L-graded S-module such that for all y ∈ L, the homogeneous component of degree y, M (x) y , is equal to M x+y . Since the L-action on mod L -S fixes mod L 0 -S as a subcategory, L acts on coh(X).
The category coh(X) is a hereditary hom-finite k-category with Serre duality. In particular, [GL87,2.2], for all X, Y ∈ coh(X), x i is called the dualising element. Moreover, coh(X) has almost-split sequences and the Auslander-Reiten translate τ X of X ∈ coh(X) is X(ω).
The Grothendieck group of coh(X), denoted K 0 (X) := K 0 (coh(X)), is equipped with the Euler form −, − : K 0 (X) × K 0 (X) → Z which is given on sheaves X, Y ∈ coh(X) by The torsion-free objects in coh(X), i.e. those without non-zero subobjects of finite length, are called vector bundles. Every object in coh(X) decomposes as V ⊕ F where V is a vector bundle and F is finite length.
The subcategory, coh 0 (X), of finite length objects is uniserial and decomposes into connected components as λ∈P 1 (k) U λ where for each λ ∈ P 1 (k)\{λ 1 , . . . , λ t }, U λ is a homogeneous tube and for 1 ≤ i ≤ t, U λ i is a stable tube of rank p i . We will refer to the sheaves in U λ for λ ∈ P 1 (k) as being sheaves concentrated at λ.
There are linear forms rk : K 0 (X) → Z [GL87, 1.8.2], called rank, and deg : K 0 (X) → Z [GL87, 2.8], called degree. The linear form rk is determined by rkO(x) = 1 for all x ∈ L and the linear form deg is determined by degO(x) = δ(x) for all x ∈ L. For all X ∈ coh(X), rkX ≥ 0. A coherent sheaf X has rank zero if and only if X is finite length. If X ∈ coh(X) is finite length and non-zero then deg X > 0.
The vector bundles of rank 1 are called line bundles and they are all isomorphic to vector bundles of the form O(x) where x ∈ L. Moreover, [GL87, 2.6], every vector bundle F has a filtration by line bundles and the number of line bundles occurring in such a filtration is equal to the rank of F .
The Riemann-Roch formula, given here as in [LM92,2.4], relates the degree and rank of x, y ∈ K 0 (X) with the Euler form: The (GL-)slope of a non-zero vector bundle X is given by µ(X) := deg X/rkX. If X is a finite length sheaf then we set µ(X) := ∞.
A vector bundle X ∈ coh(X) is semistable if for each Y ⊆ X, µ(Y ) ≤ µ(X). For q ∈ Q ∪ {∞}, let W q denote the full subcategory of all semistable sheaves of slope q.
When X is of tubular type the following theorem describes coh(X).
7.2.2. Concealed canonical algebras. A vector bundle Σ is said to be a tilting bundle if Ext 1 (Σ, Σ) = 0 and D b (X) := D b (coh(X)) is the smallest triangulated subcategory of D b (X) containing Σ (see [GL87,3.1]). A concealed canonical algebra is the endomorphism ring of a tilting bundle in coh(X) for some weighted projective line X. If X is of tubular type and Σ ∈ coh(X) is a tilting bundle then End(Σ) is a tubular algebra and all tubular algebras occur in this way [LM96,3.6].
For any weighted projective line X(p, λ), Geigle and Lenzing defined a tilting bundle Σ can := ⊕ 0≤x≤c O(x), called the canonical tilting bundle. The endomorphism ring of Σ can is the canonical algebra Λ(p, λ) in the sense of Ringel [GL87,§4].
Suppose that R is a concealed canonical algebra of tubular type, i.e. a tubular algebra, and Σ ∈ coh(X) is a tilting vector bundle such that End(Σ) = R.
Let T be the torsion class of Σ, that is, the full subcategory of coh(X) generated by Σ (or equivalently, the full subcategory of objects X ∈ coh(X) such that Ext(Σ, X) = 0) and let F be the torsion-free class of Σ, that is, the full subcategory of objects X ∈ coh(X) such that Hom(Σ, X) = 0.
Since coh(X) is hereditary, the objects of the bounded derived category, D b (X), are of the form ⊕ i∈I X i [i] where I ⊆ Z is finite and X i ∈ coh(X) for all i ∈ I. For all X, Y ∈ coh(X) and i, j ∈ Z, The right derived functor of Hom(Σ, −) gives equivalence of bounded derived categories and mod-R is equivalent to the subcategory T ∨ F[1] of D b (X) consisting of objects of the form X ⊕ Z[1] where X ∈ T and Z ∈ F (see [GL87,§3]). Moreover, RHom(Σ, −) induces an Euler form preserving isomorphism of Grothendieck groups In what follows, we will use this equivalence to identify D b (X) and D b (R) (and hence K 0 (X) and K 0 (R) equipped with their Euler forms).
We now recall, see [LM92] and [Kus97,4.9], how the various parts of mod-R sit in T ∨ F[1]. This will allow us to link slope in the sense of Geigle and Lenzing and slope in the sense of Ringel.
Throughout this section, let µ max (respectively µ min ) be the maximal (respectively minimal) GL-slope of any indecomposable direct summand of Σ. Decompose the tilting bundle Σ = Σ 0 ⊕ Σ max = Σ ∞ ⊕ Σ min where Σ max (respectively Σ min ) is the sum of the indecomposable direct summands of Σ with GL-slope µ max (respectively µ min ). Note that µ min < µ max since if all indecomposable direct summands of Σ had the same GL-slope then Σ would not generate D b (X).
(3) The indecomposable projective R-modules are the indecomposable direct summands of Σ and the preprojective component of R is equal to those X ∈ coh(X) with µ(X) < µ max and Ext X (Σ, X) = 0. So, in particular, the indecomposable direct summands of Σ max are exactly the projective R-modules of Ringel slope zero. (4) The indecomposable injective R-modules are (τ X)[1] where X is an indecomposable direct summand of Σ and the preinjective component of R is equal to Z[1] such that Z ∈ coh(X), µ(Z) > µ min and Hom(Σ, Z) = 0. Following [LM92], let u j := [τ j O] and u := p−1 j=0 u j and w := [S] where S ∈ coh(X) is a simple sheaf concentrated at an ordinary point. Then u, x = degx and w, x = −rkx. By [LM92, 2.6], w and u generate rad(K 0 (X)) as an abelian group.
Suppose X ∈ T . Then Lemma 7.19. With the notation as in the rest of this section, the following hold: Proof. (i) If X is an indecomposable direct summand of Σ max then X ∈ T and X is a projective R-module not in the preprojective component by 7.18 (3). Hence X has Ringel slope 0. Therefore 0 = h 0 , [X] = rkX(α ∞ µ(X) − β ∞ ). So, since rkX > 0, β 0 = µ max α 0 . Note that we can also conclude from this that α 0 = 0.
Proof. Let E be a simple sheaf concentrated at an exceptional point of weight p. By definition, see [GL87, 2.8], degE = 1 and rkE = 0. For all q ∈ Q there is an equivalence, defined in [LM92], called a telescopy functor, Φ q,∞ : C ∞ → C q . These functors are compositions of shift functors S : coh(X) → coh(X) given on objects by SX = X(x i ) where 1 ≤ i ≤ t is such that p i = p, inverses of shift functors and right mutations R q : C q → C q/1+q for 0 < q ≤ ∞. If X ∈ coh(X) then rkSX = rkX and degSX = degX + rkX. If X ∈ coh(X) and 0 < µ(X) ≤ ∞ then rkRX = rkX + degX and degRX = degX. So S and R, and hence Φ q,∞ preserve coprimeness of rank and degree.
Proposition 7.22. Let X be a weighted projective line of tubular type.
(a) Let q ∈ Q and Y be a quasi-simple in a tube of rank p with µ(Y ) = q. There exist • (q n ) n∈N a strictly decreasing sequence with q n ∈ Q such that q n → q as n → ∞, and • X n , Z n ∈ coh(X) with µ(X n ) = µ(Z n ) = q n , Hom(Y, X n ) = 0 and Hom(Y, Z n ) = 0. (b) There exist • Y ∈ coh(X) with µ(Y ) = ∞, and • X n , Z n ∈ coh(X) with µ(X n ) = µ(Z n ) = −n, Hom(Z n , Y ) = 0 and Hom(Z n , Y ) = 0.
Proof. (a) Let q ∈ Q. By 7.21, r := rkY > 0 and d := degY are coprime and d/r = q. Since r and d are coprime, there exists a 0 ∈ Z and b 0 ∈ N such that ra 0 − b 0 d = 1. For all n ∈ N, let a n = a 0 + nd and b n = b 0 + nr. Then ra n − b n d = 1 for all n ∈ N 0 and hence a n and b n are coprime. Moreover q n := a n /b n is a strictly decreasing sequence of rational numbers such that q n → d/r = q as n → ∞.
Since a n and b n are coprime, by 7.21, for each n ∈ N, there exists a quasisimple W in a tube of rank p such that rk(W ) = b n and deg(W ) = a n . By the Riemann-Roch equation and since q n > q, p−1 j=0 dim Hom(Y, τ j W ) = ra n − b n d = 1. Therefore dim Hom(Y, τ j W ) = 0 for exactly one 0 ≤ j ≤ p − 1.
(b) The argument is similar to part (a) and left to the reader.
Proof of 7.15. Let X be a weighted projective line and Σ ∈ coh(X) a tilting bundle such that End(Σ) ∼ = R. We keep the notation as in the rest of this section. First suppose that q ′ ∈ (0, −α 0 /α ∞ ). Let q ∈ (µ max , ∞) be such that γ(q) = q ′ . Let Y, X n , Z n ∈ coh(X) and q n ∈ Q be as in 7.22(a). Since µ(Y ), µ(X n ), µ(Z n ) > µ max , Y, X n , Z n ∈ T . So Hom R (Y, X n ) = 0 for all n ∈ N and Hom R (Y, Z n ) = 0 for all n ∈ N. Let q ′ n := γ(q n ). Then slopeY = q ′ , slopeX n = slopeZ n = q ′ n and q ′ n → q as n → ∞. So q ′ is an accumulation point of the set of non-uniform slopes for Hom R (Y, −).
Suppose q ′ = 0. The description of the tilting objects of coh(X) given in [LM96, 3.1 & 3.5] means that if X is of tubular type and if T is inhomogeneous tube of slope µ max then if Σ max has a direct summand in T then Σ max has a quasi-simple from T as a direct summand. Let Y be a quasisimple of slope µ max in a tube T of rank p. If Σ max has a direct summand from T then assume that Y is a direct summand of Σ max . In either case, Ext(Σ max , Y ) = 0 and hence Ext(Σ, Y ) = 0. Now, arguments as in 7.22 imply that there exist a strictly decreasing sequence q n ∈ Q such that q n → q as n → ∞ and X n , Z n ∈ coh(X) indecomposable of slope q n such that Hom X (Y, X n ) = 0 and Hom X (Y, Z n ) = 0. Since µ(X n ) = µ(Z n ) > µ max for n ∈ N, Y, X n , Z n ∈ T . The argument now proceeds as in the previous cases.

Almost all slopes and Presburger arithmetic
The language of Presburger arithmetic is L Pr := (+, <, 0) where + is a binary function symbol, < is a binary relation symbol and 0 is a constant symbol. Presburger arithmetic is the theory of Z in L Pr where + is interpreted as the usual addition on Z, < is interpreted as the usual order on Z and 0 is interpreted as the additive unit in Z. Presburger arithmetic is decidable. For more information about Presburger arithmetic see [Mar02] (see [Mar02,3.1.21] for the proof of decidability).
We start this section by showing that for a tubular algebra R, the set of x ∈ Z n ∼ = K 0 (R) such that x is the dimension vector of some indecomposable X ∈ mod-R is a definable subset of Z n in the language of Presburger arithmetic 8.4. In order to do this, we will use the fact that mod-R is controlled by χ R , in particular that the dimension vectors of indecomposable finite-dimensional R-modules correspond exactly to the positive connected radical and root vectors of χ R . Note, however, that if we add a function symbol χ to Presburger arithmetic and interpret it as any non-zero quadratic form on Z then we can define multiplication in Z and hence, the theory becomes undecidable. So instead we argue that for χ R the Euler quadratic form on K 0 (R), the set of x ∈ Z n such that χ R (x) = 0 or χ R (x) = 1 is already a definable subset of Z n in the language of Presburger arithmetic. For any pure subgroup G of Z n there is an n-formula ∆(x 1 , . . . , x n ) in the language of Presburger arithmetic such that (g 1 , . . . , g n ) is in G if and only if ∆(g 1 , . . . , g n ) holds in Z.
Proof. Let V be the Q-linear span of G as a subset of Q n . Since G is pure, V ∩Z n = G. Since V is a subspace of Q n there is a matrix A with entries from Q such that v ∈ V if and only if vA = 0. By multiplying A by some integer, we may assume that A has integer entries. Now, for any g ∈ Z n , g ∈ G if and only if gA = 0. Let ∆(x 1 , . . . , x n ) be the formula (x 1 , . . . , x n )A = 0. Note that ∆ is a formula without parameters.
Corollary 8.2. Let R be a tubular algebra. The group radχ R ⊆ Z n ∼ = K 0 (R) is definable in the language of Presburger arithmetic.
Proof. Recall, that, when R is a tubular algebra, χ R is positive semi-definite and hence radχ R is a subgroup of K 0 (R). If x ∈ K 0 (R) and nx ∈ radχ R for some n ∈ Z\{0} then n 2 χ R (x) = χ R (nx) = 0 and hence x ∈ radχ R . So radχ R is pure in K 0 (R).
Similar results to the following have been obtained purely K-theoretically in [Kus00,2.3]. However we require exactly the formulation of 8.3. Lemma 8.3. Let R be a tubular algebra. There is a finite subset Ω ⊆ K 0 (R) such that for all x ∈ K 0 (R) with χ R (x) = 1, there exists y ∈ Ω such that x − y ∈ radχ R .
Proof. Suppose that no such finite set Ω ⊆ K 0 (R) exists. Then there are infinitely many y with χ R (y) = 1 all in pairwise distinct cosets of radχ R . Note that if λ, µ ∈ Z then h 0 , y + λh 0 + µh ∞ = h 0 , y + µ h 0 , h ∞ and h ∞ , y + λh 0 + µh ∞ = h ∞ , y + λ h ∞ , h 0 . Let a, b ∈ N be such that a = h 0 , h ∞ and −b = h ∞ , h 0 . Thus, there are infinitely many y with χ R (y) = 1 in pairwise different cosets of radχ R such that 0 < h 0 , y ≤ a and −b ≤ h ∞ , y < 0. Therefore, there exists e, f ∈ N such that there are infinitely many y with χ R (y) = 1 in pairwise different cosets of radχ R such that h 0 , y = e and h ∞ , y = −f .
Let x = f h 0 + eh ∞ . Note that by [Rin84, 5.1.1] x is sincere i.e. x i > 0 for all 1 ≤ i ≤ n where x = (x 1 , . . . , x n ). A quick calculation gives that Lemma 8.6. There is an algorithm which given w, v 1 , . . . , v n ∈ Z m and a < b ∈ Q ∞ 0 answers whether there is an indecomposable finite-dimensional module X with slope in (a, b) such that w · dimX > 0 and for 1 ≤ i ≤ n, v i · dimX = 0.
Proof. Note that there are vectors g 0 and g ∞ such that for all x ∈ Z m , h 0 , x = g 0 · x and h ∞ , x = g ∞ · x.
Thus x ∈ Z m has "slope" in (a, b) if and only if −(g 0 · x)/(g ∞ · x) ∈ (a, b). This statement can be easily rewritten in the language of Presburger arithmetic.
In 8.4, we showed that set of dimension vectors of indecomposable finitedimensional modules over R is definable in Presburger arithmetic. Thus, since Presburger arithmetic is decidable, there is an algorithm which decides whether there is an x ∈ Z m such that x is the dimension vector of an indecomposable finite-dimensional module over R, x has slope in (a, b), w · x > 0 and for 1 ≤ i ≤ n, v i · x = 0.

Decidability for theories of modules over tubular algebras
In this section we combine the results of the previous sections in order to prove that if R is a tubular algebra over a recursive algebraically closed field then R has decidable theory of modules.
Theorem 9.1. Let R be a tubular algebra over a recursive algebraically closed field. The common theory of R-modules is decidable.
We now extend the above result to tubular algebras. Note that, since the results of sections 5, 7 and 8 are for general tubular algebras, the only part of the proof missing is an algorithm which given pp-pairs ϕ/ψ, ϕ 1 /ψ 1 , . . . , ϕ n /ψ n answers yes or no such that (1) if the algorithm answers yes then there is an (indecomposable pureinjective) R-module N such that N ∈ (ϕ/ψ) and N / ∈ n i=1 (ϕ i /ψ i ) and (2) if the algorithm answers no then there does not exist N ∈ H such that N ∈ (ϕ/ψ) and N / ∈ n i=1 (ϕ i /ψ i ) where H := P 0 ∪C 0 ∪C ∞ ∪Q ∞ . Using Herzog's duality, as in 6.13, it is sufficient to replace H in the above by P 0 ∪ C 0 .
Let Γ be a finite-dimensional algebra. A finite-dimensional Γ-module T is a tilting module if the following three conditions are satisfied: (T1) T has projective dimension less than or equal to 1,