On wild algebras and super-decomposable pure-injective modules
aa r X i v : . [ m a t h . R T ] O c t On wild algebras and super-decomposablepure-injective modules
Grzegorz Pastuszak ∗ Abstract
Assume that k is an algebraically closed field and A is a finite-dimensionalwild k -algebra. Recently, L. Gregory and M. Prest proved that in this case thewidth of the lattice of all pointed A -modules is undefined. Hence the resultof M. Ziegler implies that there exists a super-decomposable pure-injective A -module, if the base field k is countable. Here we give a straightforward proofof the fact that there exists a special family of pointed A -modules, called anindependent pair of dense chains of pointed A -modules. This also yields theexistence of a super-decomposable pure-injective A -module. The remarkable tame and wild dichotomy of Yu. Drozd [4] states that the class offinite-dimensional algebras over algebraically closed fields divides into two disjointclasses: tame algebras and wild algebras . The class of wild algebras properly containsthe class of strictly wild algebras . We refer the reader to [33] for definitions of theseclasses. Furthermore, A. Skowroński introduced in [34] a concept of the growth of atame algebra. This yields a stratification of the class of tame algebras into domestic , linear and polynomial growth algebras. Tame algebras which are not of polynomialgrowth are called non-polynomial growth algebras.Understanding various aspects of representation types is still one of the centraltopics of the representation theory of finite-dimensional algebras over algebraicallyclosed fields. A good example supporting this fact is provided by the tame self-injective algebras. Indeed, the representation theory of these algebras is well de-veloped for the polynomial growth (see [36, 37]), but much less is known for thenon-polynomial growth. Recently, K. Erdmann and A. Skowroński introduced in ∗ Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina12/18, 87-100 Toruń, Poland, [email protected] 2010: Primary 16G20; Secondary 03C60.Key words and phrases: wild algebras, super-decomposable pure-injective modules, string al-gebras, pointed modules.
15] (and study in a huge ongoing project, see the introduction of [6] for more de-tails) a prominent class of weighted surface algebras . These algebras are some spe-cial representation-infinite tame symmetric algebras (and hence self-injective). Sincemost of them is of non-polynomial growth, they play a significant role in understand-ing the representation theory of all tame self-injective algebras.The representation type is studied by using various concepts and methods. Onthe level of finite-dimensional modules we have, in particular, results on the shapeof connected components of the Auslander-Reiten quiver or the component quiver,see for example [35], [39] or [14]. On the level of infinite-dimensional modules, afundamental characterization of the representation type is given in [3] in terms of generic modules . The paper [11] studies representation type in terms of matrix prob-lems and matrix reduction algorithms. It is conjectured by M. Prest (see [24], [25])that a finite-dimensional algebra A over an algebraically closed field is of domesticrepresentation type if and only if the Krull-Gabriel dimension KG ( A ) of A is finite(see [9, 10] for definitions and [22] for a list of results supporting this conjecture).The second conjecture due to Prest states that an algebra A is of domestic repre-sentation type if and only if there is no super-decomposable pure-injective A -module(see for example [26]). Such a module is some special infinite-dimensional module.This paper is related to the second conjecture of Prest.Assume that R is a ring with a unit. By an R -module we mean a left R -module.An R -module M is super-decomposable if and only if M = 0 and M has no indecom-posable direct summands. We refer to [19], [13] and [15, Chapter 7] for the conceptof pure-injectivity .The problem of the existence of super-decomposable pure-injective R -modulesis stated in [42]. In this paper M. Ziegler proves a fundamental criterion for suchmodules to exist, asserting that if the ring R is countable, then R possesses a super-decomposable pure-injective module if and only if the width of the lattice of all pp-formulas is undefined, see [24] or [25] for the definitions. The later statementcan be formulated in terms of the lattice of all pointed finitely-presented R -modules (these lattices are isomorphic, see [29] and [17] for more details).The case when R is a finite-dimensional algebra over a field k is studied inmany papers. We refer to the introduction of [18] for an up-to-date list of resultsin this direction (except for the most recent one, stating that representation-infinitedomestic standard self-injective algebras over algebraically closed fields do not havesuper-decomposable pure-injective modules, see [22, Theorem 8.3]). In most of thesepapers it is assumed that the base field k is countable. This yields R is countableand hence one can apply the criterion of Ziegler. All the known results support theconjecture of Prest concerning super-decomposable pure-injective modules.The first result on the existence of super-decomposable pure-injective modules2or finite-dimensional algebras is proved by M. Prest in [24, Theorem 13.7]. It statesthat these modules do exist over strictly wild algebras. For a very long time it wasnot known whether this holds for wild algebras. Recently, L. Gregory and M. Prestprove in [12] that this is the case. Indeed, they show in [12, Theorem 2.1] that anyrepresentation embedding functor induces an embedding of lattices of pp-formulas.This implies that the width of the lattice of all pp-formulas over a wild algebra A is undefined, and so there exists a super-decomposable pure-injective A -module, ifthe base field is countable (see Corollary 2.4 of [12]). We stress that the paper [12]contains many more interesting results.Recall that an independent pair of dense chains of pointed modules is some spe-cial family of pointed modules that allows to formulate a handy sufficient conditionfor the existence of a super-decomposable pure-injective module (see Section 2 forthe details). This notion is introduced in [29] and generalized in [17]. It is success-fully used in the series of papers [16, 17, 23, 18] which is devoted to the problemof the existence of super-decomposable pure-injective modules for strongly simplyconnected algebras , see [38, 21].This paper is devoted to show that if A is a wild k -algebra over an algebraicallyclosed field k , then there exists an independent pair of dense chains of pointed A -modules. This result is proved in Theorem 4.2. The proof of Theorem 4.2 is ratherstraightforward. It is based on some facts from [16] on the existence of independentpairs of dense chains of pointed modules for string algebras of non-polynomial growthand the very definition of a wild algebra.Let us clarify the overlap of [12] and the present paper. It is proved in Corollary2.4 of [12] that the width of the lattice of all pp-formulas over a wild algebra is un-defined. We prove in Theorem 4.2 the existence of independent pairs of dense chainsof pointed modules for wild algebras. Thus Theorem 4.2 implies [12, Corollary 2.4](see Theorem 2.4), but the converse is not known (however unexpected, see Theo-rem 2.3). In this sense, Theorem 4.2 is stronger than [12, Corollary 2.4]. However,we observe that, after some additional work, the assertion of Theorem 4.2 could bederived from [12, Theorem 2.1]. Nevertheless, the paper [12] of Gregory and Presttakes a different perspective than the point of view presented here. Therefore we seeour results as another, simple and independent of [12], proof of the fact that wildalgebras possess super-decomposable pure-injective modules.The paper contains four sections. In Section 2 we collect basic information onpointed modules, recall the definition of an independent pair of dense chains ofpointed modules and formulate the Ziegler criterion. We also introduce some specialindependent pairs of dense chains of pointed modules which we call strong . Section3 is devoted to show the existence of a strong independent pair of dense chains ofpointed modules (with some additional properties) over any string algebra of non-3olynomial growth, see Theorem 3.4. This result is only implicitly contained in [16],so we decided to include here its detailed proof. This makes the present paper moreconvenient to the reader. In Section 4 we present our main results. Indeed, Theorem4.1 shows that representation embedding functors, under some mild assumptions,preserve strong independent pairs of dense chains of pointed modules. Then Theorem4.2, stating the existence of an independent pair of dense chains of pointed modulesand a super-decomposable pure-injective module for any wild algebra (if the basefield is countable), is a direct consequence of Theorem 4.1 and Theorem 3.4.Throughout, k is a fixed algebraically closed field. By an algebra we mean afinite-dimensional associative basic k -algebra with a unit. If A is an algebra, thenby an A -module we mean a left A -module. We denote by A -mod the category of allfinitely-generated (hence finite-dimensional) left A -modules. In this section we recall some basic facts on pointed modules and related concepts. Inparticular, we present a sufficient condition for the existence of a super-decomposablepure-injective module in terms of independent pairs of dense chains of pointed mod-ules.Assume that R is a ring with a unit. We denote by R -mod the category of allfinitely-presented left R -modules. Assume that Θ ∈ R -mod. A Θ -pointed R -module is a pair ( M, χ M ) where M is a finitely-presented R -module and χ M : Θ → M is an R -module homomorphism.Assume that ( M, χ M ) and ( N, χ N ) are Θ -pointed R -modules. By a Θ -pointed R -homomorphism from ( M, χ M ) to ( N, χ N ) we mean an R -homomorphism f : M → N such that f χ M = χ N . If f : M → N is a Θ -pointed R -homomorphism from ( M, χ M ) to ( N, χ N ) , we write f : ( M, χ M ) → ( N, χ N ) . If f : M → N is an isomorphism,we call f : ( M, χ M ) → ( N, χ N ) a Θ -pointed isomorphism and the corresponding Θ -pointed modules ( M, χ M ) and ( N, χ N ) Θ -isomorphic .Assume that t ∈ N , t ≥ , Θ = R t and ( M, χ M ) is a Θ -pointed R -module. Assumethat e , . . . , e t form the R -base of the module Θ . The homomorphism χ M is uniquelydetermined by the elements χ ( e ) , . . . , χ ( e t ) ∈ M . This yields that any Θ -pointed R -module can be identified with a tuple ( M, m , . . . , m t ) where M is an R -module and m , . . . , m t ∈ M . Moreover, a Θ -pointed R -homomorphism from ( M, m , . . . , m t ) to ( N, n , . . . , n t ) can be identified with an R -homomorphism f : M → N such that f ( m i ) = n i , for i = 1 , . . . , t . In case Θ = R , we simply speak about pointed modulesand pointed homomorphisms.Let P Θ R be the set of all Θ -isomorphism classes of Θ -pointed R -modules. Let4 be a binary relation on P Θ R defined by ( M, χ M ) ≡ ( N, χ N ) if and only if thereexist pointed homomorphisms f : ( M, χ M ) → ( N, χ N ) and g : ( N, χ N ) → ( M, χ M ) .Then ≡ is an equivalence relation and the quotient set P Θ R = P Θ R / ≡ is a poset withrespect to the relation ≤ defined by ( M, χ M ) ≤ ( N, χ N ) if and only if there exists apointed homomorphism f : ( N, χ N ) → ( M, χ M ) . We denote by ( S, χ S ) the ≡ -classof a Θ -pointed R -module ( S, χ S ) .The poset P Θ R is a modular lattice with respect to the operations ⊕ and ∗ definedbelow, see [24] for details.Assume that ( M, χ M ) , ( N, χ N ) are Θ -pointed R -modules. A Θ -pointed R -module ( M ⊕ N, χ M ⊕ N ) where χ M ⊕ N ( l ) = ( χ M ( l ) , χ N ( l )) for any l ∈ Θ is the pointed directsum of ( M, χ M ) and ( N, χ N ) . We set ( M, χ M ) ⊕ ( N, χ N ) = ( M ⊕ N, χ M ⊕ N ) .Assume that M ∗ N is the pushout of χ M and χ N , that is, M ∗ N = M ⊕ N/ { ( χ M ( l ) , − χ N ( l )); l ∈ Θ } . Moreover, let ǫ M : M → M ∗ N , ǫ N : N → M ∗ N be the R -module homomorphismsgiven by ǫ M ( m ) = ( m, , ǫ N ( n ) = (0 , n ) for any m ∈ M , n ∈ N . A Θ -pointed R -module ( M ∗ N, χ M ∗ N ) where χ M ∗ N = ǫ M χ M = ǫ N χ N is the pointed pushout of ( M, χ M ) and ( N, χ N ) . We set ( M, χ M ) ∗ ( N, χ N ) = ( M ∗ N, χ M ∗ N ) .It is easy to see that sup { ( M, χ M ) , ( N, χ N ) } = ( M ⊕ N, χ M ⊕ N ) , inf { ( M, χ M ) , ( N, χ N ) } = ( M ∗ N, χ M ∗ N ) . Recall that if
Θ = R t , then the lattice P Θ R is equivalent to the lattice of all pp-formulas with t free variables ( t ≥ ), see [24, 25].We recall definitions of wide lattices of pointed modules and independent pairs ofdense chains of pointed modules. Moreover, we present in Theorem 2.3 the relationbetween these notions.We say that a lattice L ⊆ P Θ R of Θ -pointed R -modules is wide if and only if forany ( M p , χ M p ) < ( M q , χ M q ) ∈ L there are incomparable elements ( M, χ M ) , ( N, χ N ) of L such that ( M p , χ M p ) < ( M, χ M ) , ( N, χ N ) < ( M q , χ M q ) , ( M p , χ M p ) ≤ ( M ∗ N, χ M ∗ N ) < ( M ⊕ N, χ M ⊕ N ) ≤ ( M q , χ M q ) . In case the lattice P Θ R contains a wide sublattice L , we say that the width of P Θ R is undefined . The above definition is a special case of a general definition of a widelattice, see [24, 25] or Section 3 of [17].Assume that C is a set. A family { ( M q , χ M q ); q ∈ C } of Θ -pointed R -modules isdenoted by ( M q , χ M q ) q ∈ C . Let Q be the set of rational numbers viewed as a posetwith respect to the natural ordering ≤ . Recall that a poset P is a Q -chain if and5nly if it is a dense chain without end points. It is well known that any Q -chain isisomorphic as a poset with the set Q .Assume that R is a ring with a unit and Θ is a finitely-presented R -module. Thefollowing definitions were introduced in [29] and generalized in [17]. Definition 2.1.
Assume that C is a Q -chain. A dense chain of Θ -pointed R -modules is a family ( M q , χ M q ) q ∈ C of Θ -pointed R -modules such that:(a) the endomorphism ring End R ( M q ) is local and χ M q = 0 for any q ∈ C ,(b) there exist Θ -pointed homomorphisms µ q,q ′ : ( M q , χ M q ) → ( M q ′ , χ M q ′ ) forany q < q ′ ∈ C ,(c) the pointed modules ( M q , χ M q ) and ( M q ′ , χ M q ′ ) are not Θ -isomorphic for any q = q ′ ∈ C . ✷ Definition 2.2. An independent pair of dense chains of Θ -pointed R -modules isa pair (( M q , χ M q ) q ∈ C , ( N t , χ N t ) t ∈ C ) of dense chains of Θ -pointed R -modules suchthat:(a) the endomorphism ring End R ( M q ∗ N t ) is local for any q ∈ C , t ∈ C where ( M q ∗ N t , χ M q ∗ N t ) = ( M q , χ M q ) ∗ ( N t , χ N t ) ,(b) the pointed module ( M q , χ M q ) ∗ ( N t , χ N t ) is not Θ -isomorphic to ( M q ′ , χ M q ′ ) ∗ ( N t , χ N t ) nor to ( M q , χ M q ) ∗ ( N t ′ , χ N t ′ ) for any q = q ′ ∈ C , t = t ′ ∈ C . ✷ Independent pairs of dense chains of pointed modules generate wide lattices ofpointed modules in the following way.
Theorem 2.3.
Assume that the pair (( M q , χ M q ) q ∈ C , ( N t , χ N t ) t ∈ C ) is an indepen-dent pair of dense chains of Θ -pointed R -modules. Then the lattice Gen (( M q , χ M q ) q ∈ C ∪ ( N t , χ N t ) t ∈ C ) , which is the smallest sublattice of P Θ R containing sets ( M q , χ M q ) q ∈ C and ( N t χ N t ) t ∈ C ,is a wide lattice. Therefore the width of the lattice P Θ R is undefined. Proof.
The assertion is a direct consequence of [17, Theorem 3.4]. ✷ It is not known whether the existence of a wide sublattice of P Θ R (or, equivalently,a wide sublattice of the lattice of all pp-formulas over R ) implies the existence of anindependent pair of dense chains of Θ -pointed R -modules.The assertion (1) of the following theorem is the Ziegler’s criterion, see [42], and (2) is a handy version of this criterion. Observe that (2) follows directly from (1) and Theorem 2.3. Theorem 2.4.
Assume that R is a countable ring with a unit and Θ is a finitelypresented R -module. If the lattice P Θ R of Θ -pointed R -modules has width undefined, then there existsa super-decomposable pure-injective R -module. (2) If there exists an independent pair of dense chains of Θ -pointed R -modules, thenthere exists a super-decomposable pure-injective R -module. ✷ We apply the above theorem only when R is a finite-dimensional k -algebra.Note that in this case any finitely-presented R -module M is finite-dimensional andEnd R ( M ) is local if and only if M is indecomposable. Moreover, it is easy to seethat R is countable if and only if the field k is countable.Assume that ( M, χ M ) and ( N, χ N ) are Θ -pointed R -modules. Note that if wehave ( M, χ M ) ∼ = ( N, χ N ) , then M ∼ = N , but the converse does not hold in general.This justifies the following special version of Definitions 2.1 and 2.2. Definition 2.5.
Assume that R is a ring with a unit.(1) A dense chain ( M q , χ M q ) q ∈ C of Θ -pointed R -modules is strong if and only if M q and M q ′ are not isomorphic (as R -modules), for any q < q ′ ∈ C .(2) An independent pair (( M q , χ M q ) q ∈ C , ( N t , χ N t ) t ∈ C ) of dense chains of Θ -pointed R -modules is strong if and only if dense chains ( M q , χ M q ) q ∈ C and ( N t , χ N t ) t ∈ C are strong and the module M q ∗ N t is not isomorphic (as an R -module) to M q ′ ∗ N t nor to M q ∗ N t ′ , for any q = q ′ ∈ C , t = t ′ ∈ C . ✷ In this section we recall Theorem 5.7 from [16] (see Theorem 3.3) and derive itsimportant special case (see Theorem 3.4). This theorem (which is some refinementof [27, Theorem 4.1]) states that if A is a string algebra of non-polynomial growth,then there exists and independent pair of dense chains of pointed A -modules. Westress that this pair is strong, see Definition 2.5. This fact does not play a significantrole in [16, Theorem 5.7] itself, but is a crucial ingredient in proofs of our mainresults.The content of this section is faithfully based on Sections 4 and 5 of [16]. Forconvenience, some concepts related with string algebras are recalled.Let Q = ( Q , Q ) be a finite quiver with the set Q of vertices and the set Q ofarrows. Given an arrow α ∈ Q with the starting point s ( α ) and the terminal point t ( α ) we denote by α − its formal inverse . We set s ( α − ) = t ( α ) , t ( α − ) = s ( α ) and ( α − ) − = α . The set of all formal inverses of the arrows from Q is denoted by Q − .The elements of Q are called direct arrows whereas of Q − - inverse arrows .7y a walk from x to y of length n ≥ in Q we mean a sequence c . . . c n in Q ∪ Q − such that s ( c n ) = x ∈ Q , t ( c ) = y ∈ Q , s ( c i ) = t ( c i +1 ) and c − i = c i +1 ,for all ≤ i < n . We agree that ( c . . . c n ) − = c − n . . . c − . A walk c . . . c n is a path provided c i ∈ Q , for ≤ i ≤ n . Furthermore, to each vertex x ∈ Q we associatethe stationary path e x of length , with s ( e x ) = t ( e x ) = x .Given a finite quiver Q = ( Q , Q ) we denote by kQ the path algebra of thequiver Q . The k -basis of kQ is the set of all paths in Q and the multiplication in kQ is induced by the concatenation of paths. For example, if α, β ∈ Q and s ( α ) = t ( β ) ,then αβ is the path β −→ α −→ .A two-sided ideal I in kQ is called admissible if h Q i n ⊆ I ⊆ h Q i , for some n ∈ N , n ≥ . If I is an admissible ideal in kQ , then the pair ( Q, I ) is called the bound quiver and the associated quotient algebra kQ/I the bound quiver algebra .The fundamental result of P. Gabriel [7, 8] states that any finite-dimensional basicassociative k -algebra over algebraically closed field k is isomorphic to some boundquiver k -algebra, see also Chapter II of [1].A bound quiver ( Q, I ) and the corresponding bound quiver k -algebra kQ/I is special biserial [40] if an only if the following conditions are satisfied: • any vertex of Q is the starting point of at most two arrows and the terminalpoint of at most two arrows, • given an arrow β there is at most one arrow α with s ( β ) = t ( α ) and βα / ∈ I and at most one arrow γ with s ( γ ) = t ( β ) and γβ / ∈ I .A string algebra is a special biserial algebra kQ/I such that I is generated bypaths. By a string in the string algebra kQ/I we mean a walk c . . . c n in Q suchthat neither c i . . . c i + t nor c − i + t . . . c − i belongs to I , for ≤ i < i + t ≤ n . Moreover,by a band in kQ/I we mean a string S = c . . . c n such that: • all powers of S are defined, i.e. t ( c ) = s ( c n ) and S m is a string, for all m ∈ N , • c is a direct arrow and c n is an inverse arrow.Assume that S = c . . . c n is a string in the string algebra kQ/I . We denote by M ( S ) the associated string module . This is some ( n + 1) -dimensional module with k -linear basis { z , . . . , z n } , called the canonical basis of M ( S ) and denoted as a tuple ( z , . . . , z n ) . Recall that any string module is indecomposable and M ( S ) ∼ = M ( S ) if and only if S = S or S = S − . We refer to [2] and [40] and for more details onstring algebras and string modules.Assume that A = kQ/I is a fixed string algebra. Given an arrow a ∈ Q , wedefine S ( a ) to be the set of the strings over A that start with a . We recall from [2]8hat there exists some linear ordering on S ( a ) . We denote this ordering by ≤ andset S < T if and only if S ≤ T and S = T .A pair ( U, V ) of two different bands over A starting with the same direct arrowand ending with the same inverse arrow is Q -generating provided U < V and U isnot a prolongation of V or vice versa. This means that U = V X and V = U Y , forany strings X and Y . Moreover, assume that Σ( U, V ) is the set of all finite wordsover the alphabet { U, V } , including the empty word φ . Theorem 3.1. [16, Theorem 5.3]
Assume that ( U, V ) is a Q -generating pair ofbands over A and S, T ∈ Σ( U, V ) . Then the set L TS ( U, V ) = { SXT U ; X ∈ Σ( U, V ) } is a dense chain without end points. ✷ Assume that S = s . . . s n is a string over the string algebra A , M ( S ) is theassociated string module and z S ∈ M ( S ) is the first element of the canonical k -basisof M ( S ) . We call the pointed A -module ( M ( S ) , z S ) the canonical pointed stringmodule associated with S .Assume that T , S are strings over A such that T = t . . . t k , S = s . . . s m and T S is also a string. We denote by z ( T,S ) the element z T Sk +1 of the canonical basis ( z T S , . . . , z T Sk +1 , . . . , z T Sk + m +1 ) of M ( T S ) .The following fact is proved in [28, 3.1,3.2]. Lemma 3.2.
Asume that A = kQ/I is a string algebra. (1) Assume that a ∈ Q , S, T ∈ S ( a ) and S < T . There exists a pointed A -homomorphism f ( T,S ) : ( M ( T ) , z T ) → ( M ( S ) , z S ) of the canonical pointed stringmodules ( M ( T ) , z T ) and ( M ( S ) , z S ) . (2) Assume that
T, S are strings over A such that T − S is also a string. Thepointed module ( M ( T − S ) , z ( T − ,S ) ) is the pointed pushout of the pointed modules ( M ( S ) , z S ) and ( M ( T ) , z T ) . ✷ The following theorem is proved in [16]. Here we stress the fact that Q -generatingpairs of bands over string algebras induce independent pairs of dense chains ofpointed modules that are strong. Theorem 3.3. [16, Theorem 5.7]
Assume that ( U, V ) and ( U − , V − ) are Q -generatingpairs of bands over the string algebra A . Let S, T ∈ Σ( U, V ) and S ′ , T ′ ∈ Σ( U − , V − ) .Then the pair (( M ( X ) , z X ) X ∈L TS ( U,V ) , ( M ( Y ) , z Y ) Y ∈L T ′ S ′ ( U − ,V − ) ) is a strong independent pair of dense chains of pointed modules in A -mod . roof. Set C = L TS ( U, V ) , C = L T ′ S ′ ( U − , V − ) . We prove that ( M ( X ) , z X ) X ∈ C and ( M ( Y ) , z Y ) Y ∈ C are strong dense chains of pointed A -modules. First observethat Theorem 3.1 yields the sets C , C are dense chains without end points.The modules M ( X ) and M ( Y ) are indecomposable, for any strings X ∈ C and Y ∈ C , since they are string modules over A .The existence of pointed homomorphisms in ( M ( X ) , z X ) X ∈ C , ( M ( Y ) , z Y ) Y ∈ C follows from Lemma 3.2 (1).The modules M ( X ) and M ( X ) are not isomorphic, for any X , X ∈ C suchthat X = X . Indeed, X = X − since X − starts with a different direct arrow than X and X = X by the assumption. Similarly, M ( Y ) ∼ = / M ( Y ) , for any Y , Y ∈ C such that Y = Y .Consequently, ( M ( X ) , z X ) X ∈ C and ( M ( Y ) , z Y ) Y ∈ C are strong dense chains ofpointed modules in A -mod. Now we prove that they form a strong independent pair.Assume that X, X , X ∈ C , Y, Y , Y ∈ C . The pointed pushout of ( M ( X ) , z X ) and ( M ( Y ) , z Y ) is indecomposable since it is isomorphic with ( M ( X − Y ) , z ( X − ,Y ) ) ,by Lemma 3.2 (2).If X = X , then M ( X − Y ) ∼ = / M ( X − Y ) , because X − Y = X − Y (since X = X ) and X − Y = ( X − Y ) − = Y − X (since X − starts with a different directarrow than Y − ). Similarly, if Y = Y , then M ( X − Y ) ∼ = / M ( X − Y ) .The above arguments show that (( M ( X ) , z X ) X ∈ C , ( M ( Y ) , z Y ) Y ∈ C ) is a strongindependent pair of dense chains of pointed A -modules. ✷ We recall that if the base field k is algebraically closed and A is a string algebra,then A is tame of non-polynomial growth if and only if there exist Q -generatingpairs of bands ( U, V ) , ( U − , V − ) , see [30, 31, 32, 41].In Section 4 we apply the following refinement of Theorem 3.3. Theorem 3.4.
There exists a string algebra Λ and a strong independent pair P =(( M q , χ M q ) q ∈ C , ( N t , χ N t ) t ∈ C ) of dense chains of Θ -pointed Λ -modules such that Θ is indecomposable and χ M q , χ N t are monomorphisms, for any q ∈ C , t ∈ C . Proof.
We give a concrete example of a string algebra satisfying the thesis ofthe theorem. This algebra plays an important role in [16] (see also [17] and [23]).Assume that Q = x β (cid:11) (cid:11) α (cid:19) (cid:19) x γ (cid:11) (cid:11) δ (cid:19) (cid:19) x Λ = kQ/I , where I = h δα, γβ i . It is easy to see that Λ is a string algebra.Moreover, direct calculations show that if ( U, V ) = ( γαβ − δ − , γδ − ) , then ( U, V ) and ( U − , V − ) are Q -generating pairs of bands over Λ .We set C = L φφ ( U, V ) and C = L φφ ( U − , V − ) . Then Theorem 3.3 yields thatthe pair P = (( M ( X ) , z X ) X ∈ C , ( M ( Y ) , z Y ) Y ∈ C ) is a strong independent pair of dense chains of pointed modules in Λ -mod.Let P ( x ) = Λ e x be the simple projective Λ -module associated to the vertex x . We recall from [16] (see particularly Corollary 7.1 (a)) that z X ∈ e x M ( X ) and z Y ∈ e x M ( Y ) , for any X ∈ C , Y ∈ C . Observe that if ( M, m ) is a pointed Λ -module such that m ∈ e x M , then there is a homomorphism χ M : P ( x ) → M such that χ M ( e x ) = m , so ( M, m ) can be identified with ( M, χ M ) . These argumentsimply that P induces a strong independent pair P = (( M ( X ) , χ M ( X ) ) X ∈ C , ( M ( Y ) , χ M ( Y ) ) Y ∈ C ) of dense chains of Θ -pointed Λ -modules where Θ = P ( x ) (see Lemma 3.10 of [17]for the details). Observe that Θ is a simple module, so homomorphisms χ M ( X ) , χ M ( Y ) are monomorphisms, for any X ∈ C , Y ∈ C . Therefore, the thesis holds for thepair P = P . ✷ This section is devoted to prove our main results. In Theorem 4.1 we show that rep-resentation embeddings preserve strong independent pairs of dense chains of pointedmodules which additionally satisfy conditions from the thesis of Theorem 3.4. It fol-lows directly from Theorem 4.1 and Theorem 3.4 that any wild algebra A possessesan independent pair of dense chains of pointed modules. Hence there exists a super-decomposable pure-injective A -module, if the base field is countable. These facts arestated in Theorem 4.2.Throughout the section, A, B are k -algebras. All functors considered are co-variant functors. Assume that F : B -mod → A -mod is a functor and ( M, χ M ) is a Θ -pointed B -module, for some Θ ∈ B -mod. The F (Θ) -pointed A -module ( F ( M ) , F ( χ M )) is denoted by F ( M, χ M ) . If ( M q , χ M q ) q ∈ C is a family of Θ -pointed B -modules, then F ( M q , χ M q ) q ∈ C denotes the family ( F ( M q ) , F ( χ M q )) q ∈ C of F (Θ) -pointed A -modules.Recall that a functor F : B -mod → A -mod is a representation embedding ifand only if F is exact, respects the isomorphism classes (that is, F ( X ) ∼ = F ( Y ) implies X ∼ = Y , for any B -modules X, Y ) and carries indecomposable modules toindecomposable ones. An algebra A is of wild representation type (or wild ) if and11nly if there exists a representation embedding functor F : C -mod → A -mod, forany k -algebra C . Theorem 4.1.
Assume that
A, B are k -algebras and F : B -mod → A -mod isa representation embedding. Assume that (( M q , χ M q ) q ∈ C , ( N t , χ N t ) t ∈ C ) is a strongindependent pair of dense chains of Θ -pointed B -modules such that the module Θ isindecomposable and homomorphisms χ M q , χ N t are monomorphisms, for any q ∈ C , t ∈ C . Then ( F ( M q , χ M q ) q ∈ C , F ( N t , χ N t ) t ∈ C ) is an independent pair of dense chains of F (Θ) -pointed A -modules. Proof.
We show that F ( M q , χ M q ) q ∈ C is a dense chain of F (Θ) -pointed A -modules (similar arguments show that F ( N t , χ N t ) t ∈ C is a dense chain of F (Θ) -pointed A -modules as well). Indeed, assume that q ∈ C . The module F ( M q ) isindecomposable, because M q is indecomposable. Since F ( χ M q ) : F (Θ) → F ( M q ) isa monomorphism such that F (Θ) indecomposable, we get F ( χ M q ) = 0 .Assume that q < q ′ and µ q,q ′ : ( M q , χ M q ) → ( M q ′ , χ M q ′ ) is a Θ -pointed homo-morphism. Since µ q,q ′ χ M q = χ M q ′ , we get F ( µ q,q ′ ) F ( χ M q ) = F ( χ M q ′ ) , so F ( µ q,q ′ ) is a F (Θ) -pointed homomorphism from F ( M q , χ M q ) to F ( M q ′ , χ M q ′ ) .Furthermore, F (Θ) -pointed modules F ( M q , m q ) and F ( M q ′ , m q ′ ) are not isomor-phic, for any q = q ′ . Indeed, we have F ( M q ) ∼ = / F ( M q ′ ) , because M q ∼ = / M q ′ . This showsthat F ( M q , χ M q ) q ∈ C is a dense chain of F (Θ) -pointed A -modules.We show that that dense chains F ( M q , χ M q ) q ∈ C and F ( N t , χ N t ) t ∈ C form anindependent pair. Indeed, the module F ( M q ∗ N t ) is indecomposable, because M q ∗ N t is indecomposable, for any q ∈ C , t ∈ C .Observe that the functor F : mod ( B ) → mod ( A ) preserves finite colimits (andfinite limits), since it is exact, see [20]. This implies that F (( M q , χ M q ) ∗ ( N t , χ N t )) ∼ = F ( M q , χ M q ) ∗ F ( N t , χ N t ) , for any q ∈ C , t ∈ C . In particular, we get F ( M q ∗ N ) ∼ = F ( M q ) ∗ F ( N t ) . Assumethat F ( M q , χ M q ) ∗ F ( N t , χ N t ) is isomorphic with F ( M q , χ M q ) ∗ F ( N t ′ , χ N t ′ ) , for some q ∈ C and t = t ′ ∈ C . Then we get F ( M q ∗ N t ) ∼ = F ( M q ) ∗ F ( N t ) ∼ = F ( M q ) ∗ F ( N t ′ ) ∼ = F ( M q ∗ N t ′ ) , which yields M q ∗ N t ∼ = M q ∗ N t ′ . Since this is not the case, we get that F ( M q , χ M q ) ∗ F ( N t , χ N t ) is not isomorphic with F ( M q , χ M q ) ∗ F ( N t ′ , χ N t ′ ) . Similar arguments showthat it is not isomorphic with F ( M q ′ , χ M q ′ ) ∗ F ( N t , χ N t ) as well, for any q ′ = q ∈ C .This shows the assertion. ✷ heorem 4.2. Assume that A is a wild k -algebra over an algebraically closed field k . There exists an independent pair of dense chains of Ξ -pointed modules, for some A -module Ξ . Therefore there exists a super-decomposable pure-injective A -module,if the base field k is countable. Proof.
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