Representation embeddings, interpretation functors and controlled wild algebras

We establish a number of results which say, roughly, that interpretation functors preserve algebraic complexity. First we show that representation embeddings between categories of modules of finite-dimensional algebras induce embeddings of lattices of pp formulas and hence are non-decreasing on Krull-Gabriel dimension and uniserial dimension. A consequence is that the category of modules of any wild finite-dimensional algebra has width $\infty$ and hence, if the algebra is countable, there is a superdecomposable pure-injective representation. It is conjectured that a stronger result is true: that a representation embedding from ${\rm Mod}\mbox{-}S$ to ${\rm Mod}\mbox{-}R$ admits an inverse interpretation functor from its image and hence that, in this case, ${\rm Mod}\mbox{-}R$ interprets ${\rm Mod}\mbox{-}S$. This would imply, for instance, that every wild category of modules interprets the (undecidable) word problem for (semi)groups. We show that the conjecture holds for finitely controlled representation embeddings. Finally we prove that if $R,S$ are finite dimensional algebras over an algebraically closed field and $I:{\rm Mod}\mbox{-}R\rightarrow{\rm Mod}\mbox{-}S$ is an interpretation functor such that the smallest definable subcategory containing the image of $I$ is the whole of ${\rm Mod}\mbox{-}S$ then, if $R$ is tame, so is $S$ and similarly, if $R$ is domestic, then $S$ also is domestic.

For arbitrary rings R and S, by a representation embedding we mean a functor − ⊗ S B R : Mod-S → Mod-R, which is exact and restricts to a functor (which we also refer to as a representation embedding) mod-S → mod-R, and this restricted functor preserves indecomposability and reflects isomorphism. Representation embeddings are examples of interpretation functors. We are particularly interested in the case where R and S are finite-dimensional algebras, where the standard definition coincides with ours after restriction to the finite-dimensional modules. A representation embedding in our sense is the extension to arbitrary modules, which is exact and commutes with direct sums (see [23,Subsection XIX.1]).
The intuition behind this definition is that it indicates that the category of (finitedimensional) R-modules is at least as complex, in the sense of obtaining some classification of modules, as that of (finite-dimensional) S-modules. One might expect, or even conjecture, that this particular meaning of 'complexity' will correspond to other ways of measuring the complexity of an abelian category. Our results support this expectation, and give a partial answer to a long-standing conjecture of the second author [14, p. 350] that 'wild implies undecidable'. This conjecture has been proved for strictly wild (and somewhat more general [15]) algebras and has been verified in many particular cases, both within and outwith the context of finite-dimensional algebras (see [18, p. 651ff.] for a relatively recent summary and references); here we prove it for controlled wild algebras. Our results on non-decreasing complexity apply to any dimension which can be defined in terms of the lattice of pp formulas, equivalently, to any dimension which can be defined in terms of successive localizations of functor categories (mod-R, Ab) fp (the category of finitely presented additive functors from the category, mod-R of finitely presented R-modules to the category, Ab, of abelian groups).
In Section 2, we show that a representation embedding induces an embedding of pp-lattices, hence these functors are non-decreasing on dimensions as above. The method that we use there is very explicit, though we will see, in the second half of Section 3, that using deeper results from the general theory allows a faster proof at least for finite-dimensional algebras. We do this by showing that if R, S are finite-dimensional k-algebras and I : Mod-R → Mod-S is an interpretation functor whose image generates Mod-S as a definable subcategory, then I induces an embedding of pp-lattices. The right adjoint I : Mod-S → Mod-R of a representation embedding F : mod-R → mod-S (extended to the whole of Mod-R) is an interpretation functor. From a result of Nagase [12,Proposition 2.3], it follows that the image of I generates Mod-R as a definable subcategory.
In Section 4, we show that finitely controlled representation embeddings are reversible; this allows us to conclude that over any finitely controlled wild algebra the theory of modules interprets the word problem for groups, hence is undecidable.
Interpretation functors are much more general than representation embeddings, but there is a corresponding underlying idea in that if there is an interpretation functor I from a definable category D to another C such that the definable subcategory generated by the image of I is all of C, then, roughly, D 'contains' C in a model-theoretic sense (this is made precise using the model-theoretic terminology of sorts and imaginaries). So if D is a definable subcategory of Mod-R, and there is an interpretation functor from D to Mod-S such that the definable subcategory generated by the image of I is all of Mod-S, then we may regard D, and hence Mod-R, as 'containing' Mod-S in a model-theoretic sense. Therefore, if model-theoretic and algebraic measures of complexity are to align, then it should not be possible for a tame category of modules to interpret one which is wild, or for a domestic category of modules to interpret one which is non-domestic. We prove this, in the case that D = Mod-R, for finite-dimensional algebras over algebraically closed fields, in Sections 5 and 6, respectively.
Throughout, k is a field, all algebras are k-algebras and all functors are assumed to be additive and, where it makes sense, k-linear (so, below, we could as well write (mod-R, Mod-k) fp in place of the equivalent functor category (mod-R, Ab) fp ).

Lattice embeddings
We will express the results here using the language and techniques of pp formulas (in n free variables) for R-modules and the lattice, pp n R , that they form. There are many sources that explain this, for instance, [18]. We recall that the (pre-)ordering † on pp formulas (in the same free variables) is implication, equivalently inclusion of their solution sets: ψ φ if ψ(M ) φ(M ) for every (finitely presented) module M . The lattice pp n R is naturally isomorphic to the lattice of finitely generated subfunctors of the nth power, (R n , −), of the forgetful functor from mod-R to Ab [18, 10.2.33], the isomorphism being given by φ → F φ , where the latter is the functor that assigns to every finitely presented module M the solution set φ(M ). Every functor F ∈ (mod-R, Ab) has an essentially unique extension to a functor from Mod-R to Ab which commutes with direct limits; we denote this extension by − → F , or just F , and we will use the term coherent functor for such extensions. Since the functor M → φ(M ) commutes with direct limits, we can safely use F φ to denote the solution set functor on all modules or its restriction to finitely presented modules. In view of this identification of pp formulas and finitely presented subfunctors of powers of the forgetful functor, Theorem 2.1 below says that a representation embedding Mod-S → Mod-R induces an embedding from the lattice of finitely generated subfunctors of (S, −) to that of (R n , −) for a suitable n.
Suppose that R and S are rings and that − ⊗ S B R : Mod-S → Mod-R sends finitely presented S-modules to finitely presented R-modules (equivalently B R is finitely presented). Choose a finite tuple t = (t 1 , . . . , t n ) from B which generates the module B R . We use this tuple to define a map β : pp 1 S → pp n R , where n is the length of t, as follows.
is generated by φ, meaning that it is the set of all pp formulas ψ φ (for example, see [18,Paragraph 1.2.2]). Consider the tuple c ⊗ t = (c ⊗ t 1 , . . . , c ⊗ t n ) from C ⊗ B: this module is finitely presented over R, so we define βφ = β t φ to be any pp formula in pp n R which generates the pp-type of c ⊗ t in C ⊗ B. The choice of formula is not literally unique, but up to equivalence on R-modules, that is, up to equality in pp n R , it is. So this map β is well-defined, given (C, c). This map also is independent of choice of free realization of φ since if (D, d) is another free realization of φ, then there are morphisms f : C → D and g : D → C such that gf c = c and fgd = d, and hence then, applying − ⊗ B, we obtain morphisms The above construction is based on Harland's explicit proofs on the effect of tilting functors, these are interpretation functors, in [6,Paragraph 4.7.1].
Recall that a ring is said to be Krull-Schmidt if every finitely presented (right or leftit makes no difference, for example, see [18,Paragraph 4.3.8]) module is a finite direct sum of indecomposable modules, each with local endomorphism ring; so the isomorphism types of these summands, including their multiplicities, are uniquely determined. If R, S are Krull-Schmidt and − ⊗ S B R is a representation embedding, then β t is an embedding of lattices.
The map β preserves the lattice operation + (that is, sup): take φ, ψ ∈ pp 1 S ; then, with notation as above, which is generated by βφ + βψ.
The map β preserves the lattice operation ∧ (that is, inf): with notation as above, consider the pushout where by the map c φ we mean that which takes 1 S to c φ and similarly for c ψ . Recall [18, 1.2.28] that (C φ∧ψ , c), where c = fc φ = gc ψ , is, as the notation indicates, a free realization of φ ∧ ψ. Since the functor − ⊗ B is a left adjoint, and hence preserves colimits, the diagram is a pushout in mod-R. Since morphisms preserve pp formulas, both βφ and βψ, and hence βφ ∧ βψ, are in pp C φ∧ψ ⊗B (c ⊗ t), which is, by definition, generated by β(φ ∧ ψ). Therefore βφ ∧ βψ β(φ ∧ ψ). We now show the reverse inequality. Enlarge the above diagram as shown below where if t = (t 1 , . . . , t n ), then t also denotes the morphism, which by assumption is a surjection, which takes e i = (0, . . . , 1, 0, . . . , 0) (1 in the ith position) to t i and where P together and so (P, p) is a free realization of βφ ∧ βψ. Since the morphism t is surjective, we have h · c φ ⊗ 1 B = k · c ψ ⊗ 1 B , so there is a morphism l as shown. Since l(c ⊗ t) = p, we conclude that β(φ ∧ ψ) βφ ∧ βψ, as required.
That shows that we always have a lattice homomorphism. Now suppose that the functor − ⊗ B is a representation embedding and that both R and S are Krull-Schmidt. We must show that if φ > ψ in pp 1 S , then βφ > βψ. First, we deal with the case that C φ is indecomposable. Choose a morphism f : Since , and hence f ⊗ 1 B is a split embedding. Decompose C ψ as a direct sum, C 1 ⊕ · · · ⊕ C m , of indecomposables, decomposing c ψ , and hence ([18, 1.2.27]) ψ accordingly, say ψ = ψ 1 + · · · + ψ m . Since φ > ψ, also φ > ψ i for each i. By hypothesis, and using the already-established properties of β, we have βφ = βψ = i βψ i . Since − ⊗ B is a representation embedding, C φ ⊗ B is indecomposable, and hence [18, 1.2.32]βφ is +-irreducible in the lattice pp n R , so βφ = βψ i for some i. Replacing our original choice of ψ by this ψ i , we may, therefore, without loss of generality, suppose that C ψ also is indecomposable. Therefore f ⊗ 1 B is actually an isomorphism. So, since − ⊗ B is a representation embedding and therefore exact and faithful, f is an isomorphism, contradicting φ > ψ.
For the general case, decompose C φ , and hence φ as above: say φ = φ 1 + · · · + φ p with each φ i +-irreducible. Some φ i , say φ 1 , is not contained in ψ, so we have φ 1 > φ 1 ∧ ψ. By the case just dealt with we must have βφ 1 > β(φ 1 ∧ ψ) = βφ 1 ∧ βψ. By modularity βφ 1 + βψ > βψ, but βψ = βφ βφ 1 + βψ, a contradiction as required. For example, the categories of modules over the domestic algebras Λ n considered in [1,22] form a strictly increasing chain with respect to 'containment' via representation embeddings. There are obvious representation embeddings from mod-Λ m to mod-Λ n if m n, but the fact that KG(Λ n ) = n + 1 precludes there being representation embeddings in the other direction. Since the width of the lattice of pp formulas for the archetypal wild algebra k X, Y , as well as its finite-dimensional avatars such as k • ( ( / / 6 6 • , is undefined (for example, [18, 7.3.27]), we get the next result, where the second statement follows from a result of Ziegler ([25, 7.8(2)]).
Corollary 2.4. If R is a finite-dimensional algebra of wild representation type, then the width of the lattice of pp formulas for R-modules is undefined. In particular, if R is countable, then there will be a superdecomposable pure-injective module.

Interpretation functors
The notion of interpretation comes from model theory. Roughly, a structure interprets another if the first 'definably contains' the second. This is made precise by using the notion of definable sorts (or 'imaginaries'). In the additive context we usually want the additive structure to be preserved by any interpretation, and this means that the formulas used to specify the interpretation should be pp formulas. This also has the result that an interpretation is functorial. It has been proved [8, 7.2, 19, 12.9] that the interpretation functors between definable categories of modules are exactly those which preserve direct products and direct limits, examples are representation embeddings, representable functors (A, −) when A is a finitely presented module, and Ext n (A, −) when A is FP n+1 [17,Section 4]. But we also need the original definition from [17], repeated below † , not least because this is often how interpretation functors are, in practice, specified.
An interpretation functor I : D → Mod-S, where D ⊆ Mod-R is a definable subcategory, is specified (up to equivalence) by giving a pp-m-pair φ/ψ, that is a pair of pp formulas each with m free variables and with φ ψ, and, for each s ∈ S, a pp-2m-formula (one with 2m free variables) ρ s such that, for all M in the definable subcategory D, the solution set as an abelian group, and such that, together, these definable actions of elements of S give φ(M )/ψ(M ) the structure of an S-module with the action of s ∈ S being given by ρ s (see [17] or [18, 18.2.1]).
In the following remark, we spell out the conditions on a pp-2m-formula ρ which ensure that it defines an endomorphism of the solution set, Hence ρ defines a well-defined map (necessarily additive since ρ is pp).
Each condition may be expressed by saying that an implication between pp formulas holds in M , namely So to say that these hold for every M in a definable category D is equivalent to the condition that the corresponding pp-pairs be closed on D, where a pp- definable subcategories are exactly those obtained by specifying that some set of pp-pairs be closed [18, 3.4.7]).
There is the following algebraic characterization of interpretation functors. 19, 25.3]). An (additive as always) functor I : C → D between definable categories is an interpretation functor if and only if it commutes with direct products and direct limits. † Though it is not the most general form since definable categories may be definable subcategories of rings with many objects; for the general form see [19,Chapter 25].
In particular, representation embeddings are examples; more generally, since tensoring with a finitely presented module preserves both direct products [11,Sätze 1,2] and direct limits, we have the following.
Note that when R and S are both k-algebras and s 1 , . . . , s n is a basis for S over k, then we need only define ρ s1 , . . . , ρ sn and extend k-linearly. Indeed, in order to specify an interpretation functor, it is enough that the actions of a generating set of S (as a k-algebra) be specified and if S is finitely presented as an algebra, then only finitely many conditions (along the lines of those seen in the next proof) need be added to ensure that we have an S-action. Definition 1. The Ziegler spectrum, Zg R of a ring R is the topological space which has, for its points, the isomorphism classes of indecomposable pure-injective (right) R-modules and with a basis of open sets being the sets where φ > ψ ranges over pairs of pp formulas; equivalently this basis consists of the sets We use the fact that each of these basic open sets is compact (in the sense that every open covering is finite). Furthermore, if R is a finite-dimensional algebra, then each indecomposable finite-dimensional R-module is pure-injective, hence a point of Zg R , indeed an isolated point and, together, the finite-dimensional points are dense in Zg R (see the background references, for instance, [18,Subsection 5.3] for all this).

Definition 2.
We say that a definable subcategory D ⊆ Mod-R is finitely axiomatizable if there exist finitely many pp-pairs φ 1 /ψ 1 , . . . , φ t /ψ t such that Equivalently, a definable subcategory is finitely axiomatizable if the corresponding closed subset of the Ziegler spectrum is the complement of a compact open subset [25, 4.9].
In connection with proving undecidability results, given an interpretation functor I : D → Mod-R, it is useful to be able to replace D by a finitely axiomatizable subcategory D ⊇ D and I by an interpretation functor I which extends I. For then, if the theory of D interprets the word problem for groups, so does the theory of Mod-R (since the conditions for membership of D can be said as a single sentence). Proposition 3.3. Let S, R be k-algebras with S finite-dimensional, more generally [2, 2.6.4], with S finitely presented as a k-algebra. Suppose that D ⊆ Mod-R is a definable subcategory and that I : D → Mod-S is an interpretation functor. Then I may be extended to an interpretation functor from D to Mod-S for some finitely axiomatizable definable subcategory D ⊇ D of Mod-R.
Proof. Suppose that data defining I (in the sense above) are: a pp-m-pair φ/ψ which gives the object part of I; and pp-2m-formulas ρ s1 , . . . , ρ sn which define the actions of a chosen finite generating set s 1 , . . . , s n of S as a k-algebra. We will show how to write down finitely many conditions which express that these data define an interpretation functor. Each condition will be an implication between pp formulas, that is, closure of a pp-pair. So, together, these will cut out a finitely axiomatizable subcategory D of Mod-R which contains D, and on which these data define a functor I extending I as required.
The conditions that each ρ si defines a well-defined function on φ(−)/ψ(−) have already been observed, above, to be implications of pp-pairs, so it remains to say that, together, they do define an action of S on φ(−)/ψ(−). The point is that each relation between the s i is the condition that a certain non-commutative polynomial p over k is such that p(s 1 , . . . , s n ) = 0, and the translation of this to the same condition on the actions of the ρ si on φ modulo ψ is expressible by an implication between pp formulas. We illustrate this with the case where the s i form a k-basis for S; the general case uses the same ideas, but is messy to write down.
So suppose that the s i form a basis for S.
Then an implication of pp-m-formulas which expresses that the composition of the action defined by ρ i followed by that of ρ j does indeed give the correct linear composition of actions of ρ s1 , . . . , ρ sn is If we take D to be the finitely axiomatized subcategory of Mod-R defined by the finitely many pp-pairs whose closures are exactly these implications and those mentioned earlier, then the data (φ/ψ; ρ s1 , . . . , ρ sn ) define an interpretation functor I from D to Mod-S which extends I.
Associated to each definable category D there is a skeletally small abelian category fun(D) which may be obtained as the category of functors from D to Ab, which commute with direct products and direct limits and which, alternatively, is obtained as the category, L eq+ (D), of ppsorts on D. The latter has, for its objects, the sorts φ/ψ corresponding to pp-pairs ψ φ and, for its morphisms, the pp-definable-with-respect-to-D maps between sorts. In the case that D is a module category Mod-R, we write fun-R = fun(Mod-R) and have that fun-R = (mod-R, Ab) fp is equivalent to the category of coherent functors on Mod-R; since every finitely presented functor on mod-R is a subquotient of a power of the forgetful functor, one sees why the objects also correspond to pp-pairs. We also set Fun-R = (mod-R, Ab) to be the locally coherent Grothendieck abelian category which has fun-R for its category of finitely presented objects. If D is a definable subcategory of Mod-R, then fun(D) (respectively, Fun(D)) is the quotient category of fun-R (respectively, Fun-R) by the Serre (respectively, torsion) subcategory S D of those functors which are 0 on D (that is, those corresponding to pp-pairs which are closed on D). Furthermore, the relation between fun(D) and Fun(D) is just as in the case D = Mod-R. For the equivalence between the category (mod-R, Ab) fp of finitely presented functors on finitely presented modules and the category L eq+ R = L eq+ (Mod-R) of pp-sorts (also called ppimaginaries), see [18, 10.2.30] and, for further details about this background material, see [18] and also [19,20].
To any interpretation functor I : C → D, we associate the functor I 0 : fun(D) → fun(C), which is given on objects by sending B ∈ fun(D) to the composition BI and by sending a natural transformation τ : B 1 → B 2 to the natural transformation whose component at C ∈ C is given by τ IC . We use the basic formula for C ∈ C and G ∈ fun(D). See [19,Chapter 13] (or [18,Chapter 18]) for more detail, including the facts that this is exact and that all exact functors from fun-S to fun-R arise in this way.
Indeed, [20,21] this gives an equivalence between the 2-categories of definable categories and small abelian categories. Proof. This follows from the correspondence between definable subcategories of Mod-S and Serre subcategories of (that is, kernels of exact functors from) fun-S; see [18, 12.4.1]. Since In the remainder of this section, we investigate lattice homomorphisms between pp-lattices induced by interpretation functors.
Let A be an abelian category and A be an object of A. Recall that a subobject of A is an equivalence class of monomorphisms i : The collection of subobjects Sub(A) of A is ordered by defining i j if there is k with i = jk. It has meets given by pullbacks and joins given by pushouts.
We note the following. Proof. Since F is exact, it maps subobjects of A to subobjects of F A and preserves pullbacks and pushouts (the operations which define meet and join, respectively).
If F is faithful and i : X → A, j : Y → A are comparable subobjects, say there is a monomorphism g : X → Y with i = jg, but F i is equal to F j as a subobject of F A, then F must annihilate the cokernel of g. So, since F is faithful, that cokernel is 0. Hence i and j are equal as subobjects of A. The converse is clear since, F being exact, in order to check that F is faithful, it is enough to show that if F A = 0, then A = 0. Now suppose that ψ φ is a pp-pair for R-modules. Then (see the comments at the beginning of Section 2) the subobject lattice of the corresponding object φ/ψ in L eq+ R ( fun-R) is naturally isomorphic to the interval, [φ, ψ], between ψ and φ in the pp-lattice pp n R .  If S is a finite-dimensional algebra, then, for every proper definable subcategory D, there is a finite-dimensional S-module N such that N / ∈ D (because the set of finite-dimensional indecomposables is dense in the Ziegler spectrum of S; see [18, 5.3.36]). So 3.4 implies that, in the situation of the above proposition, the definable subcategory generated by the image of (B R , −) : Mod-R → Mod-S is the whole of Mod-S (in the finitely controlled case (see Section 4) this will also follow from Theorem 4.2).
This gave us another route to Theorem 2.1 (at least for finite-dimensional algebras). In fact, as we now show, the explicit construction of a lattice homomorphism β from pp 1 S to pp n R in Section 2 is essentially the same as the lattice homomorphism defined above by the interpretation functor ( S B R , −) : Mod-S → Mod-R.
First note that both constructions are dependent upon picking an n-tuple t generating B R . This is clear in Section 2. In this section, such a choice is implicit in the proof of Corollary 3.8, where we identify Hom R ( S B R , −) composed with the forgetful functor from Mod-S to Ab with a pp-pair. Indeed, if t generates B R , then this induces an embedding (t, −) : Hom R (B, −) → Hom R (R n , −), the image of which, as a subfunctor of Hom R (R n , −), is given by a (quantifierfree) pp formula ψ which generates the pp-type of t in B R .
Now let (C, c) be a free realization of φ ∈ pp 1 S . So im(c, −) = F φ ⊆ (S, −). Set I = Hom R ( S B R , −); since I 0 : fun-S → fun-R is exact, I 0 sends im(c, −) to the image of I 0 (c, −). Using ( * ) and the Hom-tensor adjunction, we have that I 0 applied to Hom S (C, −) is Hom R (C ⊗ S B R , −) and, moreover, that That is, in terms of formulas, φ is sent to a generator, that is, βφ, of the pp-type of c ⊗ t in C ⊗ S B R .

Controlled wild algebras
Let S be a k-algebra, not necessarily finite-dimensional. Denote by fin-S the category of finite-dimensional right S-modules. Let R be a finite-dimensional k-algebra. A faithful exact functor F : fin-S → mod-R is a controlled representation embedding if there exists a full subcategory C of mod-R, closed under direct sums and direct summands, such that, for all M, N ∈ fin-S, where Hom R (F M, F N) C denotes the set of morphisms from F M to F N which factor through some C ∈ C. We say that F is controlled by C, or C-controlled. We say that F is finitely controlled if it is controlled by add(C) for some C ∈ mod-R. That a controlled representation embedding F : mod-S → mod-R, where S, R are finite-dimensional algebras, is a representation embedding, can be extracted directly from the proof of [5, Proposition 2.2].
We say that a finite-dimensional k-algebra R is (finitely) controlled wild if there is a (finitely) controlled representation embedding F : fin-k X, Y → mod-R.

Remark 3.
We recall a simplification from [16]. If S B R is a bimodule such that S B is a finitely generated projective generator of Mod-S, then the functor F = (− ⊗ S B R ) : Mod-S → Mod-R can be factored as a Morita equivalence followed by restriction of scalars. Namely, let T = End( S B); so the right action of R on B induces a homomorphism λ : R → T and F = GH where H = (− ⊗ S B T ) : Mod-S → Mod-T is the Morita equivalence and G : Mod-T → Mod-R is the restriction of scalars map induced by λ. It is immediate that G is a representation embedding if and only if F is. Also, if F is controlled by C, then so is G, since H induces an equivalence between fin-S and fin-T .
Therefore, we may assume that B is S S R and the functor F is just restriction of scalars from S to a subring R, in which case the first formula above becomes   . . . , f n ). It will be enough to consider morphisms from M to powers of C. Suppose g : M → C m is given coordinatewise as g = (g 1 , . . . , g m ) T . For each i = 1, . . . , m, let λ ij ∈ k be such that g i = n j=1 λ ij f j . Define the map h : C m → C n to be that given by the matrix (λ ij ) ij . Then hΔ = g, as required. More generally, we have this for a C-controlled representation embedding, − ⊗ S B R , provided B R has a C-preenvelope in mod-R.
Proof. In order to define I, we must find a pp-pair φ/ψ of pp formulas such that, given an R-module N , the quotient φ(N )/ψ(N ) will be the underlying group of IN and, for each element s ∈ S, we have to show that multiplication by s on that underlying group can be defined by a pp formula.
First note that it is enough to consider the case where F is the restriction of scalars from a ring S to a subring R: use Remark 3 combined with the easily checked facts that Morita equivalences are interpretation functors (originally from [13, 1.1]), as is the inclusion of Mod-R into Mod-R whenever R → R is a surjection of rings.
Next, for every R-module N , if s ∈ S and f : S R → N factors through C, then so does f · s, where f · s means left multiplication by s on S followed by f , the right S-module structure on Hom R (S, N ). Thus the vector space decomposition Let C ∈ mod-R be such that C = add(C). Since S R is finite-dimensional, by Lemma 4.1, there exists Δ : S R → C n such that, for all f : S R → N which factor through C, there is h : C n → N such that f = hΔ. Thus the image of (Δ, −) : Hom R (C n , −) → Hom R (S, −) is exactly Hom R (S, −) C . The latter, therefore, is a finitely presented functor (precisely, the lim − → -commuting extension of a finitely presented functor on mod-R to one on Mod-R), hence [18, 10.2.43] is pp-definable. Therefore, the quotient Hom R (S, −)/Hom R (S, −) C also is finitely presented, equivalently is given by a pp-pair. Furthermore, each action of an element of S on this quotient is, as we have just seen, an endomorphism in the functor category, hence [ Thus we have shown that if M is an S-module, then its image N = F M under F still contains a copy of M in the model-theoretic structure N eq+ , which is N expanded by all the pp-definable sorts and pp-definable functions between these sorts (see [10] or [18,Appendix B2] for a definition of that structure). That is, applying the representation embedding has hidden the original module, but not lost it.
By [5,Lemma 2.4], a finitely controlled wild finite-dimensional algebra R has a finitely controlled representation embedding from the modules over any strictly wild finite-dimensional algebra. So we deduce that, in this case, Mod-R interprets the module category over any strictly wild finite-dimensional algebra. Recall that F = (− ⊗ S B R ) is said to be a strict representation embedding if it is a full representation embedding, equivalently, if the induced R → End( S B) is a ring epimorphism; in this case, the image of F is a definable subcategory of Mod-R [18, 5.5.4] equivalent to Mod-S. It follows easily (for example, use results in [23, Section XIX.1]) that if R is a strictly wild finite-dimensional algebra, then, for every finitely generated algebra T , there is a definable subcategory of Mod-R equivalent to Mod-T .
For finitely controlled wild algebras, the category of modules 'definably contains' the category of modules over any finitely generated algebra in the following weaker sense.
Corollary 4.4. Let R be a finitely controlled wild finite-dimensional k-algebra. Then, for every finitely generated k-algebra S, there is an interpretation functor from some definable subcategory C of Mod-R to Mod-S whose image is the whole of Mod-S.
Proof. Let T be a strictly wild finite-dimensional algebra. Then, as observed above, there is a definable subcategory D of Mod-T which is equivalent to Mod-S. There is also a finitely controlled representation embedding from mod-T to mod-R, and hence, by 4.2, there is an interpretation functor I : Mod-R → Mod-T whose image is the whole of Mod-T . By [19, 13.3], I −1 D is a definable subcategory of Mod-R, and the restriction of I to this definable subcategory is as required.
In particular, we can take S = k X, Y .

Corollary 4.5. Let R be a finitely controlled wild finite-dimensional k-algebra. Then there is an interpretation functor from a definable subcategory
Corollary 4.6. Let k be a countable recursively given field and R be a finitely controlled wild k-algebra. Then the theory of Mod-R is undecidable.
Proof. The algebra S = k • ( ( / / 6 6 • is finite-dimensional and has undecidable theory of modules (see [14, 17.3] for a proof and references). So there can be no decision procedure for determining, given a sentence σ in the theory of S-modules, whether there is an S-module which satisfies σ. There is a finitely controlled representation embedding of S-modules into R-modules, so let I be as in 4.2, giving an interpretation of S-modules in R-modules. Then, using I, there is a sentence σ in the language of R-modules such that an R-module N satisfies σ if and only if IN satisfies σ. Therefore, there can be no decision procedure for determining, given a sentence in the language of R-modules, whether there is an R-module satisfying that sentence, as required.
Examples of finitely controlled wild algebras include, at least in the case that k is algebraically closed, all local wild k-algebras and wild k-algebras with square zero radical [5, Proposition p. 290, Theorem p. 291], so the hypotheses above apply for all these. We would like to extend the conclusions to all wild algebras; so far as we are aware, there are, currently, no known wild algebras which are not (even finitely) controlled wild. Since (M, −) is projective, any endomorphism of F lifts to one of (M, −) and one sees that EndF is a subquotient of End(M ) op , and hence is a finite-dimensional k-algebra.
Suppose that I : Mod-R → Mod-S is an interpretation functor, with effect on objects being given by the coherent functor extending F ∈ (mod-R, Ab) fp and with the definable subcategory generated by the image being all of Mod-S. Then, by 3.4, I 0 : L eq+ S → L eq+ R is faithful and sends the forgetful functor in L eq+ S to F . Thus we have a ring embedding from S to EndF , and hence S must be a finite-dimensional k-algebra.

Tame does not interpret wild
In this section, we show that, for finite-dimensional k-algebras R, S, where k is algebraically closed, if I : Mod-R → Mod-S is an interpretation functor such that IMod-R = Mod-S, then S wild implies that R is wild. Here we use X to denote the definable subcategory generated by the modules in X ; when X is the image of an interpretation functor, it is easy to see that this is the closure of X under pure subobjects (see [20, 3.8]). Our result, together with 3.7, gives us the following new definition of a wild algebra in terms of interpretation functors. Of course, we could replace k • ( ( / / 6 6 • in the above theorem by any other finite-dimensional wild algebra. To prove this, we show (Corollary 5.5) that if I : Mod-R → Mod-S is an interpretation functor such that IMod-R = Mod-S, then, for each d ∈ N, there is n d ∈ N such that, for all M ∈ mod-S of dimension less than or equal to d, there is N ∈ mod-R such that M |IN and dim N n d . A result of Krause, [9, 10.8], then shows that if S is wild, then R is also wild.
Note that if I : Mod-R → Mod-S is an interpretation functor and R is a finite-dimensional k-algebra, then, since I is given on objects by a subquotient of a finite power of the forgetful functor, for any finite-dimensional module M ∈ Mod-R, IM is finite-dimensional.
Definition 3. Let R be a k-algebra. We write ind-R for the set of (isomorphism classes of) indecomposable finite-dimensional R-modules and, for each d ∈ N, ind d -R will denote the set of those of dimension d.
Suppose that M ∈ ind-R, a ∈ M is non-zero and f : M → N is left minimal almost split. Let φ generate the pp-type of a in M and let ψ generate the pp-type of f (a) in N . Then, [14, 13.1], (φ/ψ) isolates M in Zg R . We recall the argument here for the convenience of the reader. Suppose that L ∈ mod-R and b ∈ φ(L). There exists a map g : M → L such that g(a) = b. Now either g is a section, and hence M is a direct summand of L or there is a map h : N → L such that hf = g. So either M is a direct summand of L or hf (a) = b, and hence b ∈ ψ(L). Thus, either the pp-type of b is generated by φ or b ∈ ψ(L). So, for any pp-1-formula σ < φ, we have σ ψ. Now suppose that L ∈ Mod-R and b ∈ φ(L). Either b ∈ ψ(L) or the pp-type of b is generated by φ. Thus, by [18, 4.3.48], M is a direct summand of L.
So, for any module M ∈ ind-R, there is a coherent functor − −− → F φ/ψ : Mod-R → Ab (with φ/ψ chosen as in the previous paragraph) such that, for any L ∈ Mod-R, We now show that, as M varies over indecomposables of bounded dimension, the 'size' of φ, chosen to be part of an isolating pp-pair for M as above, also is bounded above. Here, by the size of φ we could mean just the number of symbols in φ or, more meaningfully, the number of variables and equations appearing in φ. Indeed, these bounds arise from a minimal projective presentation of M , with the number of variables needed being determined by the number of generators needed for M and the number of equations being determined by the number of relations on those generators needed to present M .
Proof. Suppose that M ∈ ind d -R. We saw in the comments preceding Proposition 5.2 that φ may be taken to be a pp formula generating the pp-type of any chosen non-zero element a ∈ M .
Let b 1 , . . . , b d generate M as a k-vector space. Let H be a matrix with entries from R such that if N is a module with generators e 1 , . . . , e d and eH = 0, then there is a surjective map t : M → N sending b i to e i for 1 i d (that is, the columns of H generate the R-linear relations on the b i ). Note that H is a d × e matrix where e dim M × dim R.
Given a ∈ M, we have a = bG for some d × 1 matrix G. The pp-type of a in M is generated by the pp formula See [18, p. 21] for details. In the proof of Theorem 5.4, we take a pp-pair γ/δ over S and pull it back using the interpretation functor I : Mod-R → Mod-S to a pp-pair σ/τ over R, so that if M ∈ Mod-R opens σ/τ, then IM opens γ/δ. We do this in such a way that c(σ) is only dependent on c(γ) and d(γ). Each variable will be pulled back to an m-tuple of variables, indicated in the proof by overlines.
Let N ∈ ind d -S. We know from 5.3 that there exists a pp-1-pair γ/δ isolating N with γ of the form y j a ij = 0 with e dp + 1.
For any N ∈ Mod-S, N |N if and only if γ(N ) δ(N ). For 1 i e and 1 k p, let β k i ∈ k be such that b i = p k=1 s k β k i . For 1 i e, 1 j d and 1 k p, let α k ij ∈ k be such that a ij = p k=1 s k α k ij . Let σ(X) be the pp-m-formula over R given by Thus IM opens γ/δ if and only if M opens σ/τ . Each of the tuples Y j , Z k , W i and U jk is of length m. Thus the pp-m-formula σ(X) has m(d + p + e + dp) bound variables. Set n d := m(d + p + (dp + 1) + dp) c(ρ s k ) + (dp + 1)c(ψ).
Now since e dp + 1, we have n d m(d + p + e + dp) and n d is only dependent on d.
Suppose that θ(x, y) is a quantifier-free pp formula over R (that is, a conjunction of R-linear equations), that x is an m-tuple of variables and that y is an n-tuple of variables. Let σ(x) := ∃yθ(x, y). If σ/τ is a pp-pair and M opens σ/τ , say a ∈ σ(M ) \ τ (M ), then choose a tuple b of elements from M , such that θ(a, b) holds in M, and let M be the R-submodule of M generated by the entries of a and b. Then clearly M opens σ/τ and dim M (m + n) dim R.

Corollary 5.5. Suppose I : Mod-R → Mod-S is an interpretation functor such that
Proof. Suppose I : Mod-R → Mod-S is an interpretation functor given by (φ/ψ; ρ s1 , . . . , Suppose L ∈ Mod-R opens σ/τ . Since σ is a pp-m-formula with c(σ) n d , there is, by the argument directly preceding this corollary, a submodule N of L opening σ/τ of dimension less than or equal to b d . Thus there is an R-module N with dim N b d such that M is a direct summand of IN .
In [9], a noetherian algebra is said to be endofinitely tame if, for each n ∈ N, the Ziegler closure of the set of indecomposable finitely presented modules of endolength at most n contains only finitely many non-finitely presented points. A finite-dimensional algebra over an algebraically closed field is tame if and only if it is endofinitely tame [9, Theorem 10.13]. Proof. Suppose that R is tame and I : Mod-R → Mod-S is an interpretation functor such that IMod-R = Mod-S. Since R is tame, R is endofinitely tame. By Corollary 5.5, for each n ∈ N there is a b n such that if L ∈ ind n -S, then there is an M ∈ bn i=1 ind i -R such that L is a direct summand of F M. Thus, by Theorem 5.6, S is endofinitely tame. Hence S is tame.

Domestic does not interpret non-domestic
In this section, we show that if R, S are finite-dimensional k-algebras with k an algebraically closed field and I : Mod-R → Mod-S is an interpretation functor such that IMod-R = Mod-S, then if R is tame domestic, so also is S. We allow 'finite type' to be included in 'tame' since, if R is of finite representation type, so is S, for example by the following general result. Proof. We know that if M is finite endolength, then IM is also finite endolength (see for instance [9,Corollary 9.7]). If G is generic and R is tame, then End R G/radEnd R G ∼ = k(x) [3, 4.4] and the embedding of radEnd R G into End R G is split. Thus IG is a quotient of two k(x)-modules, so is either zero or infinite-dimensional. Lemma 6.3 (cf. [7, 9.6]). Let R be a finite-dimensional k-algebra. Any compact subset of Zg R not containing any generic module contains only finitely many modules of each finite dimension over k.
Proof. For any topological space, the intersection of a compact subset with a closed subset is compact. Thus, if Y is a compact subset of Zg R , then, for any n ∈ N, the intersection of Y with the closed subset of points of finite endolength n is compact. If it were infinite, then it would contain a non-isolated point, but any non-isolated point of finite endolength is generic. Thus Y contains only finitely many points of each finite dimension over k.
We now recall some basic information about Ziegler spectra of Dedekind domains, all of which is contained in [18, section 5.2.1].
Let R be a Dedekind domain. The points of Zg R are as follows: the field of fractions, Q, of R; for each prime p, a Prüfer point R p ∞ and an adic point R p ; for each prime p and for each positive integer n, the finite-length module R/p n . Each point R/p n is clopen and Q belongs to the closure of each infinite-dimensional point. This follows since the complement of U , being closed, can contain only finite-dimensional, hence isolated, points (for the generic is in the closure of each adic and Prüfer point); each adic and Prüfer point is in U . But this complement also is compact, and hence must be finite.
We will heavily use the following result of Crawley-Boevey.  (2) The functor − ⊗ B G from k[x, f −1 ]-modules to R-modules reflects isomorphism and preserves indecomposability and Auslander-Reiten sequences.
(3) For each d ∈ N, all but finitely many indecomposable R-modules of dimension d have the form k[x, f −1 ]/ g ⊗ B G for some generic G and some g ∈ k[x, f −1 ]. Theorem 6.5. Suppose that k is an algebraically closed field and let R, S be finitedimensional k-algebras with R domestic. If there is an interpretation functor I : Mod-R → Mod-S such that IMod-R = Mod-S, then S must be domestic.
Proof. Since R is domestic, by Theorem 5.7, S must be tame. Let G 1 , . . . , G n be the generic R-modules. As noted already, each IG i is of finite endolength, hence the closed subset X := IG 1 , . . . , IG n ∩ Zg S is finite. Thus, since S is non-domestic, there is a generic S-module H of endolength strictly greater than the endolength of any module in X. Since X is closed and H / ∈ X, there is a basic Ziegler-open set containing H and not intersecting X, so there is a pp-defined = lim − → commuting extension, F , of a finitely presented functor from Mod-S to Ab such that F H = 0 and F M = 0 for all M ∈ X. To complete the proof of the theorem, let d ∈ N be such that the set {C ∈ ind-S|F C = 0} contains infinitely many modules of dimension d. Let b d ∈ N be such that, for all C ∈ ind-S with dim C d, there exists N ∈ ind-S with dim N b d such that C|IN (5.5). By Claim 1, there are only finitely many N ∈ ind-S with dim N b d and F IN = 0, hence only finitely many direct summands of these, which is a contradiction.