Let . Let k be an algebraically closed field of characteristic , and let be a p-elementary abelian group of rank r. It is well known that the category of finite-dimensional -modules is of wild type, whenever or and . Therefore, subclasses with more restrictive properties have been studied; in , the subclass of modules of constant rank and modules with even more restrictive properties, called equal images property and equal kernels property, were introduced. Let be a complement of in , and set for . We say that has constant Jordan type if the Jordan canonical form of the nilpotent operator is independent of . If the image (kernel) of does not depend on α, we say that M has the equal images (kernels) property.
In , the author defined analogous categories , and in the context of the generalized Kronecker algebra , and in more generality for the generalized Beilinson algebra . Using a natural functor with nice properties, she gave new insights into the categories of equal images and equal kernels modules for of Loewy length . A crucial step is the description of , and in homological terms, involving a family -family of regular “test”-modules.
Building on this approach, we show that the recently introduced modules  of constant socle rank and constant radical rank can be described in the same fashion. For , we introduce modules of constant d-radical rank and constant d-socle rank in . More restrictive – and also easier to handle – are modules with the equal d-radical property and the equal d-socle property . For , we have , and . Studying these classes in the hereditary module category allows us to use tools not available in .
As a first step, we establish a homological characterization of , , and . We denote by the Grassmanian of d-dimensional subspaces of . In generalization of , we define a -family of “test”-modules and show that the modules in this family can be described in purely combinatorial terms by being indecomposable of dimension vector . This allows us to construct many examples of modules of equal socle rank in for by considering pullbacks along natural embeddings .
Since is a wild algebra for and every regular component in the Auslander–Reiten quiver of is of type , it is desirable to find invariants that give more specific information about the regular components. It is shown in  that there are uniquely determined quasi-simple modules and in such that the cone consisting of all modules lying on an oriented path starting in satisfies , and the cone consisting of all modules lying on an oriented path ending in satisfies . Using results on elementary modules, we generalize this statement for and . Our main results may be summarized as follows.
Let and be a regular component of the Auslander–Reiten quiver of .
For each , the category is wild, where .
For each , there exists a unique quasi-simple module in such that .
There exists at most one number such that is non-empty. If such a number exists, is the ray starting in .
If there is no such number as in (c), we set . An immediate consequence of (b) and (c) is that, for , we have or . Moreover, statement (a) shows the existence of a lot of AR-components such that is a ray, and for every such component, we have . With the dual result for , we assign a number to each regular component , giving us the possibility to distinguish different types of regular components.
To prove statement (a), we exploit the fact that every regular module M in has self-extensions with , by applying the process of simplification. This method was introduced in  and produces extension closed subcategories, whose objects may be filtered by pairwise orthogonal bricks. For a p-elementary abelian group of rank r over an algebraically closed field of characteristic , is the only such subcategory. We therefore use the functor , whose essential image (the full subcategory of formed by all modules isomorphic to modules of the form ) consists of all modules of Loewy length , to transfer our results to . We denote by the category of modules in of Loewy length with the equal d-socle property and arrive at the following result.
Let , and . Then has wild representation type, where .
For , we consider the Beilinson algebra . The fact that is a concealed algebra of type allows us to apply the simplification process in . We find a wild subcategory in the category of all modules in with the equal kernels property and conclude the following.
Assume that ; then the full subcategory of modules with the equal kernels property in and Loewy length has wild representation type.
In particular, we generalize results by Benson  and Bondarenko and Lytvynchuk  concerning the wildness of various subcategories of -modules. We also construct examples of regular components such that each module in has constant d-socle rank, but no module in is -stable in the sense of .
Throughout this article, let k be an algebraically closed field and . If not stated otherwise, k is of arbitrary characteristic. We denote by a finite and connected quiver without oriented cycles. For an arrow , we define and . We say that α starts in and ends in . A (finite-dimensional) representation over Q consists of vector spaces and linear maps such that is finite. A morphism between representations is a collection of linear maps such that, for each arrow , there is a commutative diagram
The category of finite-dimensional representations over Q is denoted by , and kQ is the path algebra of Q with idempotents , . The k-algebra kQ is a finite-dimensional, associative, basic and connected k-algebra. Let be the class of finite-dimensional kQ left modules. Given , we set . The categories and are equivalent (see for example [1, Theorem III 1.6]). We will therefore switch freely between representations of Q and modules of kQ if one of the approaches seems more convenient for us. We assume that the reader is familiar with Auslander–Reiten theory and basic results on wild hereditary algebras. For a well written survey on the subjects, we refer to [1, 14, 13].
Recall the definition of the dimension function
If is an exact sequence, then . The quiver Q defines a (non-symmetric) bilinear form
given by . For the case that are given by dimension vectors of quiver representations, there is another description of known as the Euler–Ringel form 
We denote by the corresponding quadratic form. A vector is called a real root if and an imaginary root if .
Denote by the r-Kronecker quiver, which is given by two vertices and arrows .
We set and , . and are the indecomposable projective modules of , and . As Figure 2 suggests, we write
For example, and . The Coxeter matrix Φ and its inverse are
For M indecomposable, holds if M is not projective and if M is not injective. The quadratic form q is given by .
Figure 3 shows the notation we use for the components in the Auslander–Reiten quiver of which contain the indecomposable projective modules and indecomposable injective modules . The set of all other components is denoted by .
Ringel has proven [18, Theorem 2.3] that every component in is of type if or a homogeneous tube if . A module in such a component is called regular. An irreducible morphism in a regular component is injective if the corresponding arrow is uprising (see Figure 1 for ) and surjective otherwise. A regular module M is called quasi-simple if the AR-sequence terminating in M has an indecomposable middle term. If M is quasi-simple in a regular component , there is an infinite chain (a ray) of irreducible monomorphisms
and an infinite chain (a coray) of irreducible epimorphisms
and for each regular module X, there are unique quasi-simple modules and with . The number l is called the quasi-length of X. We fix the orientation of each regular component in such a way that the quasi-simple modules form the bottom layer of the component (see Figure 1).
The indecomposable modules in are called preprojective modules and the modules in are called preinjective modules. Moreover, we call an arbitrary module preprojective (resp. preinjective, regular) if all its indecomposable direct summands are preprojective (resp. preinjective, regular). It is P in (I in ) if and only if there is with () for . Recall that there are no homomorphisms from right to left [1, Corollary VIII.2.13]. To emphasize this result later on, we just write
Using the canonical equivalence ([1, Theorem III.1.6]) of categories , we introduce the duality by setting and for each , where is the permutation of order 2. Note that for all . We state a simplified version of Kac’s theorem [12, Theorem 1.10] for the Kronecker algebra in combination with results on the quadratic form proven in [17, Lemma 2.3].
Let and .
If for some indecomposable module M , then .
If , then there exists a unique indecomposable module X with . In this case, X is preprojective or preinjective, and X is preprojective if and only if .
If , then there exist infinitely many indecomposable modules Y with and each Y is regular.
Since there is no pair satisfying for , we conclude together with [1, Lemma VIII.2.7] the following.
Let M be an indecomposable -module. Then if and only if M is preprojective or preinjective. If and M is regular, then .
Let be the subcategory of all modules without non-zero projective direct summands and the subcategory of all modules without non-zero injective summands. Since is a hereditary algebra, the Auslander–Reiten translation induces an equivalence from to . In particular, if M and N are indecomposable with not projective, we get . The Auslander–Reiten formula [1, Theorem VI.2.13] simplifies to the following.
Theorem 1.3 ([13, Theorem 2.3]).
For in , there a functorial isomorphisms
2 Modules of constant radical and socle rank
2.1 Elementary modules of small dimension
Let . The homological characterization in  uses an algebraic family of modules of projective dimension 1 for the Beilinson algebra on n vertices. For , we have , and is a hereditary category. Hence every non-projective module is of projective dimension 1. In the following, we study the module family for . We will see later on that each non-zero proper submodule of is isomorphic to a finite number of copies of , and itself is regular. In particular, we do not find a short exact sequence such that A and B are regular and non-zero. In the language of wild hereditary algebras, we therefore deal with elementary modules.
Definition ([16, Definition 1]).
A non-zero regular module E is called elementary if there is no short exact sequence with A and B regular non-zero. In particular, elementary modules are indecomposable and quasi-simple.
Elementary modules are analogues of quasi-simple modules in the tame hereditary case (). If X is a regular module, then X has a filtration such that is elementary for all and the elementary modules are the smallest class with that property. For basic results on elementary modules, used in this section, we refer to .
We are grateful to Otto Kerner for pointing out the following helpful lemma.
Let E be an elementary module and regular with non-zero morphisms and . Then . In particular, .
Since f is non-zero and , is a regular non-zero submodule of E. Consequently, since E is elementary, is preinjective by [16, Proposition 1.3]; hence g cannot factor through . ∎
We use the theory on elementary modules to generalize [21, Corollary 2.7] in the hereditary case.
Let be a non-empty family of elementary modules of bounded dimension, and put
Then the following statements hold.
is closed under extensions, images and τ.
contains all preinjective modules.
For each regular component , the set forms a non-empty cone in , which consists of the predecessors of a uniquely determined quasi-simple module in , i.e. there is quasi-simple such that .
(1) Since , the category is closed under images and extensions. Let ; then the Auslander–Reiten formula yields for all . We first show that M is not preprojective. Assume to the contrary that is preprojective. Let such that is projective; then for an and
This is a contradiction since every non-sincere -module is semi-simple. Hence M is regular or preinjective.
Now we show . In view of the Auslander–Reiten formula, we get
Let be a morphism, and assume that . Since , the module M is not preinjective and therefore regular. As a regular module E has self-extensions (see Corollary 1.2), and therefore . Hence , and we find . Lemma 2.1.1 provides a non-zero morphism
We conclude , a contradiction. Hence
(2) The injective modules are contained in . Now apply (1).
(3) The existence of the cones can be shown as in [21, Theorem 3.3]. We sketch the proof. Let be a quasi-simple module, and denote the upper bound by L. By [13, Lemma 4.6, Proposition 10.5], we find such that for all and each regular representation Y with and for some . In particular, we have and . Now shows that for the uniquely determined quasi-simple module with and . ∎
The next result follows by the Auslander–Reiten formula and duality since is elementary if and only if E is elementary.
Let be a family of elementary modules of bounded dimension, and put
Then the following statements hold.
is closed under extensions, submodules and .
contains all preprojective modules.
For each regular component , the set forms a non-empty cone in , which consists of the successors of a uniquely determined quasi-simple module in .
Note that is a torsion-free class of some torsion pair and is the torsion class of some torsion pair (see for example [1, Proposition VI.1.4]).
Let be indecomposable modules with , . Then the following statements hold.
and are elementary for all . Moreover, every proper factor of M is injective, and every proper submodule of N is projective.
Every proper factor module of is preinjective, and every proper submodule of is preprojective for .
We will give the proofs for M. The statements for N follow by duality.
(a) By [16, Lemma 1.1], M is elementary if and only if all elements in its τ-orbit are elementary. Since is an imaginary root of the quadratic form, M is regular. Now let be a proper submodule with dimension vector . Then since . Hence , and is injective.
(b) Let and be exact with X regular and non-zero. We have to show that . We assume that . Since , we conclude that has no indecomposable preinjective direct summand. Since is right exact [1, Corollary VII.1.9], we get an exact sequence . We conclude with (a) that ; hence . ∎
2.2 An algebraic family of test-modules
Let . Now we take a closer look at the modules . Let us start this section by recalling some definitions from  and the construction of the module family. We use a slightly different notation since we are only interested in the case .
For and , we define and denote by the linear operator associated to .
For , the map , defines an embedding of -modules and is just the left multiplication by . We now set .
These modules are the “test”-modules introduced in . In fact, is a 1-dimensional submodule of contained inside the radical of the local module . From the definition, we get an exact sequence and . Since is local with semi-simple radical , it now seems natural to study embeddings for and the corresponding cokernels. This motivates the next definition. We restrict ourselves to since otherwise the cokernel is the simple injective module.
Let and . For , we define as the -linear map
where denotes the projection onto the i-th coordinate.
The map is injective if and only if T is linearly independent; then we have
and is indecomposable because is local. Moreover, is an imaginary root of q, and therefore is regular indecomposable and by Lemma 2.1.4 elementary. We define .
Let such that ; then if and only if .
If , then the definition of and implies . Hence .
Now let , and assume that is -linear. Since is local with radical and , the map φ is surjective and therefore injective. Recall that has as a basis. Let such that . Since φ is -linear, we get
and hence . Write ; then
The assumption yields . Then and
a contradiction to the injectivity of φ. Hence and . ∎
Let and with basis . We define .
is well defined (up to isomorphism) with dimension vector , and is elementary for and quasi-simple for .
For a module X, we define as the category of summands of finite direct sums of X, and denotes the set of isomorphism classes of indecomposable modules M with dimension vector for .
Let M be indecomposable.
If , then there exists with .
The map , is bijective.
Let and . There is and an epimorphism .
(a) Let be a submodule of M. Then , and X is in . It is
so we find a non-zero map . Since every proper submodule of M is in and , the map is surjective and yields an exact sequence . For , there exist uniquely determined elements such that
where denotes the embedding into the i-th coordinate. Since is an idempotent with () and , we get
Hence . Now define , and . It is , and by the injectivity of ι, we conclude that T is linearly independent, and therefore . Now we conclude
(b) This is an immediate consequence of (a) and Lemma 2.2.1.
(c) By (a), we find U in with basis such that . Let V be the subspace with basis . Then , and we get an epimorphism , with . ∎
As a generalization of , we introduce maps and for and . Note that if and only if .
Let and . We denote by and the operators
It is , and since, for and , we have
Let and . Every non-zero quotient Q of is indecomposable. Q is preinjective (injective) if and regular otherwise.
Since is regular, we conclude with that every indecomposable non-zero quotient of is preinjective or regular. Let Q be such a quotient with and . Since is local with radical and , it follows for some . Hence , and Q is an injective module if . Otherwise, Q is also indecomposable since and with imply w.l.o.g. . Hence , which is a contradiction to . ∎
2.3 Modules for the generalized Kronecker algebra
In the following, we will give the definition of -modules with constant radical rank and constant socle rank.
Let M be in and .
M has constant d-radical rank if the dimension of
is independent of the choice of .
M has constant d-socle rank if the dimension of
is independent of the choice of .
M has the equal d-radical property if is independent of the choice of
M has the equal d-socle property if is independent of the choice of
Let . We define
Let M be indecomposable and not simple.
M has the equal d -socle property if and only if .
M has the equal d -radical property if and only if .
(a) Assume that M is in , and let for . Denote by the canonical basis vectors. We find a k-complement of in , say . Then M decomposes into submodules . Since M is not simple, we have and conclude , i.e. (see also [10, Lemma 5.1.1]). Denote by the set of all subsets of of cardinality d. Then
Since and , we get and hence .
(b) Let , and . Since M is not simple, it is [10, Lemma 5.1.1] and hence
For the benefit of the reader, we recall the definitions of the classes , and given in .
Note that , , , and for with basis , we have and . We restrict the definition to since , and therefore every module in is of constant r-socle and r-radical rank.
Let M be in and .
if and only if .
if and only if .
Note that and hence
Hence if and only if . Moreover, M in if and only if , and hence . ∎
For the proof of the following proposition, we use the same methods as in [21, Theorem 2.5].
Let . Then
Let with basis . Consider the short exact sequence
Application of yields
be the natural isomorphisms, where denotes the embedding into the i-th coordinate. Let be the natural projection. The equality holds since both maps are applied to a homomorphism. Hence . Now let . We conclude
This finishes the proof for . Moreover, note that together with Lemma 2.3.1 yields
Since , the next result follows from the Auslander–Reiten formula and Lemma 2.3.2.
Let . Then
For , we have with , , and [21, Proposition 3.1] yields . However, this identity holds if and only if . From Proposition 2.3.3 and Lemma 2.1.4, it follows immediately that, for and , the module is in . If not stated otherwise, we assume from now on that .
Let and a regular component of .
is closed under extensions, submodules and . Moreover, contains all preprojective modules, and forms a non-empty cone in , i.e. there is a quasi-simple module such that
is closed under extensions, images and τ . Moreover, contains all preinjective modules, and forms a non-empty cone in , i.e. there is a quasi-simple module such that
For , we set and , where .
The next result suggests that, for each regular component and , only a small part of vertices in corresponds to modules in . Nonetheless, we will see in Section 4 that, for , the categories and are of wild type.
Let . Let be a regular component and () in the uniquely determined quasi-simple modules such that and .
There exists at most one number such that is non-empty. If such a number exists, then .
There exists at most one number such that is non-empty. If such a number exists, then .
(a) By Proposition 2.3.5, there are such that
We will show that either or that there exists a uniquely determined such that .
Let M be in , and assume . In the following, we show that . There exists with for . Hence we find a non-zero map . Consider an exact sequence . Then , and by Lemma 2.2.3, N is indecomposable. By Proposition 2.2.2, there exists with . Since (see [21, Proposition 3.1]), we get a non-zero morphism and by Lemma 2.1.1 a non-zero morphism
Therefore, by Proposition 2.3.3.
Now assume that for some i and j. Then, in particular, . Hence . By definition, we have , and the above considerations yield . Therefore, since is closed under . Therefore, and . We conclude that there is a uniquely determined such that , and in this case, . Now we set .
(b) This follows by duality. ∎
Let be an almost split sequence such that two modules of the sequence are of constant d-socle rank. Then the third module also has constant d-socle rank.
Let and , and let . Let be the right orthogonal category , and let be the left orthogonal category . Then we set .
Note that every module in is regular by Proposition 2.3.5.
Let and , and let M be quasi-simple regular in a component such that . Then every module in has constant d-socle rank.
Let . We have shown in Proposition 2.3.5 that the set
is closed under (resp. . Since , we have for . The Euler–Ringel form yields
Since is independent of V, has constant d-socle rank. On the other hand, implies that has constant d-socle rank for all . It follows that each quasi-simple module in has constant d-socle rank. Now apply Lemma 2.3.7. ∎
3 Process of simplification and applications
3.1 Representation type
Denote by the path algebra of a connected, wild quiver Q. We use the notation introduced in . Recall that a module M is called brick if , and two modules are called orthogonal if we have .
Let be a non-empty class of pairwise orthogonal bricks in . The full subcategory is by definition the class of all modules Y in with an -filtration, that is, a chain
with for all .
In [17, Theorem 1.2], the author shows that is an exact abelian subcategory of , closed under extensions, and is the class of all simple modules in . In particular, a module M in is indecomposable if and only if it is indecomposable in .
Let , and let be a non-empty class of pairwise orthogonal bricks with self-extensions (and therefore regular).
Every module in is regular.
Every regular component contains at most one module of .
Every indecomposable module is quasi-simple in .
is a wild subcategory of .
(a) and (b) are proven in [15, Lemma 1.1, Proposition 1.4] for any wild hereditary algebra, and (c) follows by [15, Proposition 1.4] and the fact that every regular brick in is quasi-simple [13, Proposition 9.2]. Let . Then we have by Corollary 1.2. Due to [11, Section 7] and [15, Remark 1.4], the category is equivalent to the category of finite-dimensional modules over the power-series ring in non-commuting variables . Since , the category is wild, and also . ∎
We will use the above result to prove the existence of numerous components such that all of its vertices correspond to modules of constant d-socle rank. By duality, all results also follow for constant radical rank. As a by-product, we verify the wildness of and . Using the functor , we show the wildness of the corresponding full subcategories in of -modules of Loewy length .
3.2 Passage between and
Let . Denote by the functor that assigns to a -module M the module with the same underlying vector space so that the action of on stays unchanged and all other arrows act trivially on . Moreover, let be the natural k-algebra monomorphism given by for and . Then each -module N becomes a -module via pullback along ι. Denote the corresponding functor by . In the following, will be fixed, so we suppress the index and write just and .
Let . The functor is fully faithful and exact. The essential image of is a subcategory of closed under factors and submodules. Moreover, is indecomposable if and only if M is indecomposable in .
Clearly, is fully faithful and exact. Now let , and let be a submodule. Then () acts trivially on U, and hence the pullback is a -module with . Now let , and let be an epimorphism. Let and such that . It follows for all . This shows that .
Since is fully faithful, we have . Hence is local if and only if is local. ∎
Statement (a) of the following lemma is stated in [7, Proposition 3.1] without proof.
Let , and let M be an indecomposable -module that is not simple. The following statements hold.
is regular and quasi-simple.
for all .
(a) Write . Since M is not simple, and . It follows
Hence , and is regular.
Assume that is not quasi-simple; then for U quasi-simple with . By Lemma 3.2.1, we have and for some indecomposable in . Fix an irreducible monomorphism . Since is full, we find with . The faithfulness of implies that g is an irreducible monomorphism . By the same token, there exists an irreducible epimorphism . As all irreducible morphisms in are injective and all irreducible morphisms in are surjective, M is located in a component. It follows that in . Let ; then the Coxeter matrices for and yield
This is a contradiction since .
(b) Denote by the canonical basis of . Let , and set . Then . Let such that acts non-trivially on M. Let such that . Then , and M does not have constant m-socle rank. ∎
Let and , and let M be an indecomposable and non-simple -module. Then the following statements hold.
If , then .
If , then .
If , then is contained in a regular component with .
By definition, it is . Now fix , and note that
which is the dimension vector of every -module for .
(a) Assume that , and let . By Lemmata 3.2.1 and 2.1.4, the -module is regular and every proper submodule of is preprojective. Hence f is surjective onto . Again, Lemma 3.2.1 yields indecomposable with such that . By Proposition 2.2.2, there exists with . Since is fully faithful, it follows , a contradiction.
(b) Assume that , and let be non-zero. Since is regular indecomposable, the module is not injective, and Lemma 2.2.3 yields that is indecomposable and regular. As is a submodule of , there exists an indecomposable module with . Since is not simple, we have for . Hence for , and by Proposition 2.2.2 (d), there exists and an epimorphism . We conclude with and the surjectivity of that , a contradiction to the assumption.
The following two examples will be helpful later on.
Let . Ringel has shown that the representation with the linear maps , and is elementary. Let E be the corresponding -module. Then , and it is easy to see that every indecomposable submodule of E has dimension vector or . In particular, for each indecomposable module with dimension vector . Assume now that is non-zero; then f is surjective since every proper submodule of W is projective. Since E is elementary, is a preprojective module with dimension vector , a contradiction. Hence .
Recall that and . Given a regular component , there are unique quasi-simple modules and in such that and . The width is defined [21, Theorem 3.3] as the unique integer satisfying . In fact, it is shown that , and an example of a regular component with and is given. Since for (see [21, Theorem 3.1]), we conclude for an arbitrary regular component that
Let and . Then there exists a regular module with the following properties.
is a (quasi-simple) brick in .
There exist with .
We start by considering and . Pick the elementary module from the preceding example. E is a brick, and . Set , and , . By the definition of E, we have
Now let . If , consider . In view of Proposition 3.2.3, we have Moreover, is a brick in and for the canonical basis vectors and , we get as before
Now let . Set , consider a regular component for with such that is a brick and set . Then , and Proposition 3.2.3 yields . Since M is a brick, is a brick in . Recall that for all implies that, viewing M as a representation, the linear map corresponding to is injective. Since the map is not affected by , is also injective. Therefore, we conclude for the first basis vector and that . By Lemma 3.2.2, we find with . ∎
3.3 Numerous components lying in
In this section, we use the simplification method to construct a family of regular components such that every vertex in such a regular component corresponds to a module in . By the next result, it follows that implies .
Lemma 3.3.1 ([13, Lemma 1.9]).
Let be modules with non-zero. If X and Y have filtrations
then there are with .
For a regular module , denote by the regular component that contains M.
Let , and let be a family of pairwise orthogonal bricks in . Then
is an injective map such that, for each component in , we have . Here denotes the category of a chosen set of representatives of non-isomorphic indecomposable objects of in .
Since each module in is regular, Proposition 3.1.1 implies that every module is contained in a regular component and is quasi-simple. By Lemma 3.3.1, the module N satisfies for all . But now Lemma 2.3.8 implies that every module in has constant d-socle rank. The injectivity of φ follows immediately from Proposition 3.1.1. ∎
There exists an infinite set Ω of regular components such that, for all ,
, in particular, every module in has constant rank,
does not contain any bricks.
Let be a regular component that contains a brick and (such a component exists by the example above). Let ; then . Apply Proposition 3.3.2 with , and set . Let be indecomposable. N is quasi-simple in and has a -filtration with and . Hence is a non-zero homomorphism that is not injective. Therefore, N is not a brick. This finishes the proof since every regular brick in is quasi-simple [13, Proposition 9.2] and for all . ∎
Now we apply our results on the simplification method to modules constructed in Lemma 3.2.4.
Definition ([6, Proposition 3.6]).
Denote with the group of invertible -matrices which acts on via for , . For , let be the algebra homomorphism with , and , . For a -module M, denote the pullback of M along by . The module M is called -stable if for all .
Let ; then there exists a wild full subcategory and an injection
such that, for each component in , we have and no module in is -stable.
such that each component in satisfies .
Moreover, is a wild full subcategory of by Proposition 3.1.1. Let be indecomposable. Then M has a filtration with for all . By Lemma 3.3.1, we have , and since , we conclude . This proves that M does not have constant 1-socle rank. Therefore, contains a module that is not of constant 1-socle rank. By [6, Proposition 3.6], the module M is not -stable. Assume that contains an -stable module N. Since acts as an auto-equivalence on (see also [9, Lemma 2.2]), we conclude that g sends the Auslander–Reiten sequence to the Auslander–Reiten sequence . Hence and for all , and therefore X and E are -stable. If E is not indecomposable, we write with indecomposable such that the quasi-lengths satisfy . We get and therefore and . Hence every direct summand in the Auslander–Reiten sequence is -stable. Now one can easily conclude that every module in is -stable, a contradiction since M is not -stable. ∎
3.4 Components lying almost completely in
Let M be an indecomposable -module, and . M is called U-trivial if
Note that the sequence and left-exactness of imply that
Let M be a regular U-trivial module. If M is not elementary, then
Assume that ; then we find an epimorphism and an exact sequence . Note that . We apply and conclude that is non-zero. In particular, we have and therefore . Let such that . Since is a brick, we conclude that f is an isomorphism and is elementary.
Assume that . Consider together with an epimorphism (see Proposition 2.2.2). We conclude with that M is W-trivial. Now the equation (which follows from the proof of Proposition 2.3.3) shows that acts as zero on M. Hence acts as zero on , and is W-trivial with . Hence we find such that . Since (see [21, Proposition 3.1]) and , we conclude that . Now the above arguments show that and therefore . ∎
Let M be regular quasi-simple in a regular component such that
If M does not have constant d-socle rank, then a module X in has constant d-socle rank if and only if X is in .
Let and , and let and . Let M be an indecomposable -module in that is not elementary. Denote by the regular component that contains .
Every module in has constant b -socle rank.
has constant l -socle rank if and only if .
(a) is an immediate consequence of Proposition 3.2.3.
(b) Consider the indecomposable projective module in . We get
we find with . Now let . By Proposition 2.2.2, there is and an epimorphism . Let be a basis of . Since π is surjective, the set is linearly independent. Hence
holds, and is U-trivial.
Since M is not elementary, is not elementary, and therefore Lemma 3.4.1 yields that
Let , and let be a regular component with such that is not a brick (see Corollary 3.3.3) and in particular not elementary. Then , and we can apply Corollary 3.4.3. Figure 4 shows the regular component of containing . Every module in has constant socle rank. But for , a module in this component has constant q-socle rank if and only if it lies in the shaded region.
4 Wild representation type
4.1 Wildness of strata
As another application of the simplification method and the inflation functor , we get the following result.
Let and . Then is a wild subcategory, where .
For , consider a regular component for that contains a brick F. By Proposition 2.3.5, we find a module E in the τ-orbit of F that is in and set . Then E is brick since and by Corollary 1.2. Therefore, is wild category (see Proposition 3.1.1). As is closed under extensions, it follows . Note that this case does only require the application of Proposition 3.1.1.
Now let and . Consider the projective indecomposable -module with . By Lemma 3.2.2, is a regular quasi-simple module in with
Since P is in , we have for all . Hence Proposition 3.2.3 implies
so that is in .
Let ; then is a wild category, and since is extension closed, it follows . Since , we find (see Proposition 2.2.2) with
That means . Since is closed under submodules, we have . Hence . ∎
Let us collect the following observations.
Note that all indecomposable modules in the wild category are quasi-simple in and .
For , we define . One can show that if and only if is injective for all linearly independent tuples T in . From the definitions, we get a chain of proper inclusions . By adapting the preceding proof, it follows that is wild. Moreover, it can be shown that, for each regular component , the set is empty or forms a ray.
We will use the following result later on to prove the wildness of the subcategory in consisting of modules of Loewy length 3 and the equal kernels property. We denote by the Beilinson algebra with 3 vertices and 2 arrows.
Let be the full subcategory of modules in with the equal kernels property (see [21, Definition 2.1, Theorem 2.5]). The category is of wild representation type.
Consider the path algebra A of the extended Kronecker quiver . Since the underlying graph of Q is not a Dynkin or Euclidean diagram, the algebra A is of wild representation type by . It is known that there exists a preprojective tilting module T in with ; see for example  or [23, Section 4]. We sketch the construction. The start of the preprojective component of A is illustrated in Figure 5, and the direct summands of T are marked with a dot.
One can check that is a tilting module. Since preprojective components are standard [19, Proposition 2.4.11], one can show that is given by the quiver in Figure 6, bound by the relation . Moreover, it follows from the description as a quiver with relations that .
Since A is hereditary, it follows that the algebra is a concealed algebra [1, Definition 4.6]. By [2, Theorem XVIII.5.1], the functor induces an equivalence G between the regular categories and , and we have an isomorphism between the two Grothendieck groups with for all . Now we make use of a homological characterization of the class given in [21, Theorem 2.5]: for each , there exist certain indecomposable -modules , such that
The modules , arise as cokernels of embeddings similar to the embeddings studied in Section 2.2. We do not need the exact definition of . Let us show that each is regular. Clearly, for , so Z is not in . Moreover, the equality (see [22, Proposition 3.14])
holds (D denotes a certain duality on ). Since is not in , we conclude that is not in . Since is closed under , we conclude that is not in . The assumption yields that is in . Since is closed under , we get that is in , a contradiction. Therefore, are not in and by [21, Corollary 2.7] regular, so is a regular module as well.
Moreover, is independent of α. Hence we find for each a regular indecomposable module in with and for all . Now let M be in a regular brick with (see [15, Proposition 5.1]). By the dual version of [13, Lemma 4.6], we find with for all . Set and . N is a regular brick with , and therefore is a wild category in (see [15, Proposition 1.4]). By Lemma 3.3.1, we have for all L in and all . Hence for all . This shows that the essential image of under G is a wild subcategory contained in . ∎
4.2 The module category of
Throughout this section, we assume that and . Moreover, let be a p-elementary abelian group of rank r with generating set . For , we get an isomorphism
of k-algebras by sending to for all i. We recall the definition of the functor introduced in . Given a module M, is by definition the vector space M and
where . Moreover, is the identity map on morphisms, that is, , for all .
Definition ([6, Definition 2.1]).
Let . For U in with basis and a -module M, we set
Definition ([6, Definition 3.1]).
Let and .
M has constant d- rank ( d- rank, respectively) if the dimension of (, respectively) is independent of the choice of .
M has the equal d- property ( d- property, respectively) if (, respectively) is independent of the choice of .
Let M be a non-simple indecomposable -module, and let .
M is in if and only if has constant d - rank.
M is in if and only if has the equal d - property.
We fix . In the following, we denote for with the induced linear map on M. Let with basis , write for all , and set . Then is linearly independent, and
Hence implies that has constant d- rank.
Now assume that is linearly independent, and set . Then
We have shown that M is in if and only if has constant d- rank. The other equivalence follows in the same fashion. ∎
Let , and . Then has wild representation type.
that reflects isomorphisms and with essential image . Let be a wild subcategory. Since and reflect isomorphisms, we have for all . Hence the essential image of under is a wild category. ∎
Assume that ; then the full subcategory of modules with the equal kernels property in and Loewy length is of wild representation type.
By [21, Proposition 2.3] (, ), the functor is a representation embedding with essential image in . ∎
The results of this article are part of my doctoral thesis, which I have written at the University of Kiel. I would like to thank my advisor Rolf Farnsteiner for fruitful discussions, his continuous support and helpful comments on an earlier version of this paper. I also would like to thank the whole research team for the very pleasant working atmosphere and the encouragement throughout my studies. Furthermore, I thank Otto Kerner for answering my questions on hereditary algebras and giving helpful comments, and Claus Michael Ringel for sharing his insights on elementary modules for the Kronecker algebra. I would like to thank the anonymous referee for the detailed comments.
I. Assem, D. Simson and A. Skowroński, Elements of the Representation Theory of Associative Algebras. I, London Math. Soc. Stud. Texts 72, Cambridge University, Cambridge, 2006. Google Scholar
I. Assem, D. Simson and A. Skowroński, Elements of the Representation Theory of Associative Algebras. III, London Math. Soc. Stud. Texts 72, Cambridge University Press, Cambridge, 2007. Google Scholar
D. J. Benson, Representations of Elementary Abelian p-groups and Vector Bundles, Cambridge Tracts in Math. 208, Cambridge University, Cambridge, 2017. Google Scholar
V. M. Bondarenko and I. V. Lytvynchuk, The representation type of elementary abelian p-groups with respect to the modules of constant Jordan type, Algebra Discrete Math. 14 (2012), no. 1, 29–36. Google Scholar
J. F. Carlson, E. M. Friedlander and J. Pevtsova, Representations of elementary abelian p-groups and bundles on Grassmannians, Adv. Math. 229 (2012), no. 5, 2985–3051. Web of ScienceCrossrefGoogle Scholar
P. Donovan and M. R. Freislich, The Representation Theory of Finite Graphs and Associated Algebras, Carleton Math. Lecture Notes 5, Carleton University, Ottawa, 1973. Google Scholar
R. Farnsteiner, Categories of modules given by varieties of p-nilpotent operators, preprint (2011), https://arxiv.org/abs/1110.2706.
R. Farnsteiner, Nilpotent operators, categories of modules, and auslander-reiten theory, Lectures notes (2012), http://www.math.uni-kiel.de/algebra/de/farnsteiner/material/Shanghai-2012-Lectures.pdf.
P. Gabriel, Indecomposable representations. II, Symposia Mathematica Vol. XI (Rome 1971), Academic Press, London (1973), 81–104. Google Scholar
V. G. Kac, Root systems, representations of quivers and invariant theory, Invariant Theory (Montecatini 1982), Lecture Notes in Math. 996, Springer, Berlin (1983), 74–108. Google Scholar
O. Kerner, Representations of wild quivers, Representation Theory of Algebras and Related Topics (Mexico City 1994), CMS Conf. Proc. 19, American Mathematical Society, Providence (1996), 65–107. Google Scholar
O. Kerner, More representations of wild quivers, Expository Lectures on Representation Theory, Contemp. Math. 607, American Mathematical Society, Providence (2014), 35–55. Google Scholar
O. Kerner and F. Lukas, Regular modules over wild hereditary algebras, Representations of Finite-dimensional Algebras (Tsukuba 1990), CMS Conf. Proc. 11, American Mathematical Society, Providence (1991), 191–208. Google Scholar
C. M. Ringel, Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math. 1099, Springer, Berlin, 1984. Google Scholar
L. Unger, The concealed algebras of the minimal wild, hereditary algebras, Bayreuth. Math. Schr. (1990), no. 31, 145–154. Google Scholar
J. Worch, Module categories and Auslander–Reiten theory for generalized Beilinson algebras, PhD thesis, Christian-Albrechts-Universität zu Kiel, 2013. Google Scholar
About the article
Published Online: 2019-09-11
Published in Print: 2020-01-01
Funding Source: Deutsche Forschungsgemeinschaft
Award identifier / Grant number: DFG priority program SPP 1388
Partly supported by the DFG priority program SPP 1388 “Darstellungstheorie”.