The Auslander bijections: how morphisms are determined by modules
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Let be an artin algebra. In his seminal Philadelphia Notes published in 1978, Auslander introduced the concept of morphisms being determined by modules. Auslander was very passionate about these investigations (they also form part of the final chapter of the Auslander–Reiten–Smalø book and could and should be seen as its culmination). The theory presented by Auslander has to be considered as an exciting frame for working with the category of -modules, incorporating all what is known about irreducible maps (the usual Auslander–Reiten theory), but the frame is much wider and allows for example to take into account families of modules—an important feature of module categories. What Auslander has achieved is a clear description of the poset structure of the category of -modules as well as a blueprint for interrelating individual modules and families of modules. Auslander has subsumed his considerations under the heading of “morphisms being determined by modules”. Unfortunately, the wording in itself seems to be somewhat misleading, and the basic definition may look quite technical and unattractive, at least at first sight. This could be the reason that for over 30 years, Auslander’s powerful results did not gain the attention they deserve. The aim of this survey is to outline the general setting for Auslander’s ideas and to show the wealth of these ideas by exhibiting many examples.
KeywordsAuslander bijections Auslander–Reiten theory Right factorization lattice Morphisms determined by modules Finite length categories: global directedness Local symmetries Representation type Brauer–Thrall conjectures Riedtmann–Zwara degenerations Hammocks Kronecker quiver Quiver Grassmannians Auslander varieties Modular lattices Meet semi-lattices
Mathematics Subject Classification (2010)Primary 16G70 18E10 18A25 18A32 16G60 Secondary 16G20 18A20 06C05 14M15 19A49 03C60
There are two basic mathematical structures: groups and lattices, or, more generally, semigroups and posets. A first glance at any category should focus the attention on these two structures: to symmetry groups (for example the automorphism groups of the individual objects), as well as to the posets given by suitable sets of morphisms, for example by looking at inclusion maps (thus dealing with the poset of all subobjects of an object), or at the possible factorizations of morphisms. In this way, one distinguishes between local symmetries and global directedness.
The present survey deals with the category of finite length modules over an artin algebra . Its aim is to report on the work of Auslander in his seminal Philadelphia Notes published in 1978. Auslander was very passionate about these investigations and they also form part of the final chapter of the Auslander–Reiten–Smalø book: there, they could (and should) be seen as a kind of culmination. It seems to be surprising that the feedback until now is quite meager. After all, the theory presented by Auslander has to be considered as an exciting frame for working with the category , incorporating what is called the Auslander–Reiten theory (to deal with the irreducible maps), but this frame is much wider and allows for example to take into account families of modules—an important feature of a module category. Indeed, many of the concepts which are relevant when considering the categories fit into the frame! What Auslander has achieved (but he himself may not have realized it) was a clear description of the poset structure of and of the interplay between families of modules.
Auslander’s considerations are subsumed under the heading of morphisms being determined by modules, but the wording in itself seems to be somewhat misleading, and the basic definition looks quite technical and unattractive, at least at first sight. This could be the reason that for over 30 years, Auslander’s powerful results did not gain the attention they deserve.
It is easy to see that the poset is a lattice, thus we call it the right factorization lattice for .
Looking at maps , we may (and often will) assume that is right minimal, thus that there is no non-zero direct summand of with Note that any right equivalence class contains a right minimal map, and if and are right minimal maps, then any with has to be an isomorphism.
Of course, to analyze the poset is strongly related to a study of the contravariant -functor , however the different nature of these two mathematical structures should be stressed: is an additive functor whereas is a poset, and it is the collection of these posets which demonstrates the global directedness.
What is the relevance? As we have mentioned, usually the lattice itself will not satisfy any chain conditions, but all the lattices are of finite height and often can be displayed very nicely: according to (2) we deal with the submodule lattice of some finite length module over an artin algebra (namely over ) and it is easy to see that any submodule lattice arises in this way. Using the Auslander bijections , one may transfer properties of submodule lattices to the right -factorization lattices this will be one of the aims of this paper. Given a submodule of , let be a right -determined map ending in such that . The composition series of the factor module correspond to certain factorizations of (to the “maximal -factorizations”), and we may define the -type of so that it is equal to the dimension vector of the module (recall that the dimension vector of a module has as coefficients the Jordan–Hölder multiplicities of the various simple modules occurring in ).
Submodule lattices have interesting combinatorial features, and it seems to interesting that Auslander himself looked mainly at combinatorial properties (for example at waists in submodule lattices). But we should stress that we really are in the realm of algebraic geometry. Thus, let us assume for a moment that is a -algebra where is an algebraically closed field. If is a finite-dimensional -module, the set of all submodules of is the disjoint union of the sets consisting of all submodules of with fixed dimension vector . It is well-known that is in a natural way a projective variety, called nowadays a quiver Grassmannian. Given -modules and , the Auslander bijections draw the attention on the -module , let be its dimension vector and let be dimension vectors with The quiver Grassmannians corresponds under the Auslander bijection to the set of all right equivalence classes of right -determined maps which end in and have type . We call an Auslander variety. These Auslander varieties have to be considered as an important tool for studying the right equivalence classes of maps ending in a given module.
We end this summary by an outline in which way the Auslander bijections (2) incorporate the existence of minimal right almost split maps: we have to look at the special case where is indecomposable and and to deal with the submodule of . The bijection (2) yields an element in such that ; to say that is right -determined means that is right almost split.
The survey is divided into three parts. Part I presents the general setting, it comprises the Sects. 2 to 10. The Sects. 11 to 15 form Part II, here we show in which way the Auslander bijections deal with families of modules. Finally, in Parts III, we discuss some special cases; these are the Sects. 16 to 18.
I. The setting
2 The right factorization lattice
Let be a -module. Let be the class of all homomorphisms with arbitrary modules (such homomorphisms will be said to be the homomorphisms ending in). We define a preorder on this class as follows: Given and , we write provided there is a homomorphism such that (clearly, this relation is reflexive and transitive). As usual, such a preorder defines an equivalence relation (in our setting, we call it right equivalence) by saying that are right equivalent provided we have both and , and it induces a poset relation on the set of right equivalence classes of homomorphisms ending in . Given a morphism , we denote its right equivalence class by and by definition if and only if As we will see in Proposition 2.2, the poset is a lattice, thus we will call it the right factorization lattice for .
It should be stressed that is a set, not only a class: namely, the isomorphism classes of -modules form a set and for every module , the homomorphisms form a set; we may choose a representative from each isomorphism class of -modules and given a homomorphism , then there is an isomorphism where is such a representative, and is right equivalent to
Recall that a map is said to be right minimal provided any direct summand of with is equal to zero. If is a morphism and such that and is right minimal, then is called a right minimalisation of f. The kernel of a right minimalisation of will be called the intrinsic kernel of f, it is unique up to isomorphism.
Every right equivalence class in contains a right minimal morphism, namely , where is a right minimalisation of . Given right minimal morphisms and , then are right equivalent if and only if there is an isomorphism such that .
Let be a homomorphism ending in . Write such that and is right minimal. Let be the canonical inclusion, the canonical projection. Then and (since ). We see that and , thus and are right equivalent and is right minimal. If the right minimal morphisms and are right equivalent, then there are morphisms and such that and . But implies that is an automorphism, and implies that is an automorphism, thus have to be isomorphisms (see  I.2).
Monomorphisms are always right minimal, and the right equivalence classes of monomorphisms ending in may be identified with the submodules of (here, we identify the right equivalence class of the monomorphism with the image of ).
The poset is a lattice with zero and one. Given and , say with pullback , the meet of and is given by the map , the join of and is given by .
(a trivial verification) Write . We have and , thus and . If is a morphism with and , then and , thus there are morphisms with , for Since , the pullback property yields a morphism such that for . Thus shows that , thus This shows that is the meet of and .
Second, denote the canonical inclusion maps by , for , thus and therefore for Assume that there is given a morphism with for . This means that there are morphisms such that for . Let (with ). Then shows that , thus This shows that is the join of and .
It is easy to check that the map is the zero element of and that the identity map is its unit element.
It should be stressed that ifandare right minimal, say with pullback, then neither the mapnor the direct sum mapwill be right minimal, in general. Thus if one wants to work with right minimal maps, one has to right minimalise the maps in question. Here are corresponding examples:
All path algebras of quivers considered in the paper will have coefficients in an arbitrary field , unless we specify some further conditions. When dealing with the path algebra of a quiver , and is a vertex of , we denote by (or also just by ) the simple module corresponding to , by and the projective cover or injective envelope of , respectively.
Take as maps the canonical projection , this is a right minimal map. The pullback of and is a submodule of which is isomorphic to . Since any map is zero, there is no right minimal map .
Also, the map is not right minimal, since we have
As we have seen, the poset is a lattice. What will be important in the following discussion is the fact that we deal with a meet-semilattice (these are the posets such that any pair of elements has a meet). Note that all the semilattices which we deal with turn out to be lattices, however the poset maps to be considered will preserve meets, but usually not joins, thus we really work in the category of meet-semilattices.
The lattice is modular.
yields a map such that thus the map belongs to .
the map Open image in new window is an element of .
Here, we can assume that all the kernels are equal to , where is a fixed indecomposable module of length . Also, if the ground field is infinite, then there is such a chain of epimorphisms such that all the kernels are pairwise different and of length . In the first case, the kernels of the maps are all indecomposable (namely of the form for ), in the second, they are direct sums of pairwise non-isomorphic modules of length 2.
3 Morphisms determined by modules: Auslander’s First Theorem
The existence of the dashed arrow on the left for all possible maps shall imply the existence of the dashed arrow on the right (of course, the converse implication always holds true: if for some morphism , then for all morphisms ).
Assume that . Then is right -determined if and only if is right -determined.
If is right -determined, then is also right -determined.
We denote by the set of the right equivalence classes of the morphisms ending in which are right -determined. We will see below that also is a lattice, thus we call it the right C-factorization lattice for .
Note that is usually not closed under predecessors or successors inside . But there is the following important property:
The subset of is closed under meets.
Let and be right -determined. As we know, the meet of and is given by forming the pullback of and . Thus assume that is the pullback with maps and and let We want to show that is right -determined. Thus, assume that there is given such that for any , there exists such that . Then we see that for any , we have , thus factors through . Since is right -determined, it follows that factors through , say for some . Similarly, for any , the morphism factors through and therefore for some . Now implies that there is such that and . Thus shows that factors through .
We should stress that usually is not closed under joins, see the examples at the end of the section. One of these examples is chosen in order to convince the reader that this is not at all a drawback, but an important feature if we want to work with lattices of finite height.
where runs through all the -modules (or just through representatives of all multiplicity-free -modules) and this is a filtered union of meet-semilattices.
By definition, the sets are subsets of . By Proposition 3.1(a), we know that only depends on , thus we may restrict to look at representatives of multiplicity-free -modules . Proposition 3.1(b) asserts that both and are contained in , thus we deal with a filtered union. According to Proposition 3.2, we deal with embeddings of meet-semilattices. The essential assertion of Theorem 3.3 is that any morphism is right determined by some module, the usual formulation of Auslander’s First Theorem. A discussion of this assertion and its proof follows.
There is a precise formula which yields for the smallest possible module which right determines . We will call it the minimal right determiner of , any other right determiner of will have as a direct summand.
Since the image of is not contained in the image of , we see that is non-zero, thus is a submodule of
(Determiner formula of Auslander–Reiten–Smalø) Let be a morphism ending in . Let be the direct sum of the indecomposable modules of the form , where is an indecomposable direct summand of the intrinsic kernel of and of the indecomposable projective modules which almost factor through , one from each isomorphism class. Then is right -determined if and only if .
Any morphism is right -determined by some , for example by the module
We have to show that is a direct summand of . The intrinsic kernel of is a direct summand of , thus if is an indecomposable direct summand of the intrinsic kernel of , then is a direct summand of Now assume that is a simple module such that almost factors through . Then is a submodule of , thus is a direct summand of
(Auslander) The module right determines .
Let be a projective module and a right minimal morphism. Then is right -determined if and only if is a monomorphism and the socle of the cokernel of is generated by .
This is an immediate consequence of the determiner formula: First, assume that is right -determined. Then the intrinsic kernel of has to be zero. Since we assume that is right minimal, must be a monomorphism. If is a simple submodule of the cokernel of , then almost factors through , thus is a direct summand of . This shows that the socle of the cokernel of is generated by . Conversely, assume that is a monomorphism and the socle of the cokernel of is generated by . Since is a monomorphism, is the direct sum of all indecomposable projective modules which almost factor through . Such a module is the projective cover of a simple submodule of . Since generates the socle of the cokernel of , it follows that is a direct summand of . Thus is in , therefore is right -determined.
A right minimal morphism is a monomorphism if and only if it is right -determined.
Let be non-zero maps for , these are monomorphisms, thus they are right -determined. The join of and in is given by the map . Clearly, this map is right minimal, but it is not injective. Thus is not right -determined.
Here, the modules are the indecomposable representations of length 2, one from each isomorphism class and all the arrows are inclusion maps.
The join in of two different maps in the height 2 layer is just the identity map , whereas the join of in is the direct sum map More generally, if there are given pairwise different regular modules of length 2 with inclusion maps , then the join in is the direct sum map . Let us stress that all these direct sum maps are right minimal (thus here we deal with a cofork as defined in Sect. 13). Thus, if the base field is infinite, the smallest subposet of closed under meets and joins and containing the inclusion maps with regular of length 2 will have infinite height.
Let be a morphism. If is an indecomposable direct summand of , then
By definition, there are two kinds of indecomposable direct summands of , the non-projective ones are of the form , where is an indecomposable direct summand of the intrinsic kernel of , the remaining ones are the indecomposable projective modules which almost factor through . Of course, if is an indecomposable projective module which almost factors through , then .
If we assume that , then , thus factors through the kernel of , say Consequently, . But is injective, thus and therefore . But this means that is split mono, a contradiction. It follows that , thus
Proposition 3.9 asserts that all the indecomposable direct summands of the minimal right determiner of a map satisfy . Actually, according to , Proposition XI.2.4 (see also ), such a module is equipped with a distinguished non-zero map which is said to “almost factor through” . At the beginning of this section we gave a corresponding definition in the special case when is projective. See also the Remark 3 at the end of Sect. 4.
4 The Auslander bijection. Auslander’s Second Theorem
Let be objects. Let . We always will consider as a -module. For any module , we denote by the set of all submodules (it is a lattice with respect to intersection and sum of submodules).
Here is a reformulation of the definition of .
Let . Then is the set of all which factor through . This subset of is a -submodule.
We have mentioned already, that is a -submodule of . Also, if , then factors through . And conversely, if factors through , then belongs to
If and , then
A trivial verification: First, let us show that is injective. Consider maps and such that Since is right -determined and , we see that . Since is right -determined and , we see that . But this means that , thus
Auslander’s Second Theorem (as established in ) asserts:
of the inclusion map and the map defined by is a lattice isomorphism.
Convention. In the following, several examples of Auslander bijections will be presented. When looking at the submodule lattice of a module , we usually will mark (some of) the elements of by bullets and connect comparable elements by a solid lines. Here, going upwards corresponds to the inclusion relation.
For the corresponding lattices , we often will mark an element (with a right minimal map) by just writing and we will connect neighboring pairs by drawing an (upwards) arrow . On the other hand, sometimes it seems to be more appropriate to refer to the right minimal map with kernel and image by using the short exact sequence notation .
Note that the lattice has two distinguished elements, namely itself as well as its zero submodule. Under the bijection the total submodule corresponds to the identity map of , this is not at all exciting. But of interest seem to be the maps in , we will discuss them in this will be discussed in Proposition 5.5
The special case. It is worthwhile to draw the attention on the special case when .
The special case of the Auslander bijection is the obvious identification of both and with .
First, consider : The determiner formula asserts: a right minimal morphism is right -determined if and only if it is a monomorphism. Thus is just the set of right equivalence classes of monomorphisms ending in , and the map yields an identification between the set of right equivalence classes of monomorphisms ending in and the submodules of .
Next, we deal with . Note that and there is a canonical identification (given by for ), thus (with for a submodule of ).
As a consequence, we see that all possible submodule latticesoccur as images under the Auslander bijections. This assertion can be strengthened considerably, as we want to show now.
By definition, an artin algebra is an artin -algebra for some commutative artinian ring (this means that is a -algebra and that it is finitely generated as a -module). Such an algebra is said to be strictly wild (or better strictly-wild), provided for any artin -algebra , there is a full exact embedding . If is a -module and is a -module, a semilinear isomorphism from to is a pair , where is an algebra isomorphism, and is an isomorphism of abelian groups such that for all and . It is clear that any semilinear isomorphism from to induces a lattice isomorphism
Let be an artin -algebra which is strictly -wild. Let be an artin -algebra and a -module. Then there are -modules such that the -module is semilinearly isomorphic to . Thus there is a lattice isomorphism .
Let be a full embedding (we do not need that it is exact). Let and . Let as well as both be given by applying the functor . Since is a full embedding, is an algebra isomorphism and is an isomorphism of abelian groups. The functoriality of asserts that we also have for all and . This shows that the pair is a semilinear isomorphism.
but even if is exact, such a bijection will not be given by applying directly . Namely, if is right minimal and right -determined, then the kernel of belongs to , thus the kernel of belongs to , whereas the intrinsic kernel of any right -determined map has to belong to and the -modules and may be very different, as the obvious embeddings of the category of -Kronecker modules into the category of -Kronecker modules (using for one arrow the zero map) show.
Note that under a full exact embedding functor , submodule lattices are usually not preserved: given a -module , the functor yields an embedding of into , but usually this is a proper embedding. Actually, for any finite-dimensiona algebra , there are submodule lattices which cannot be realized as the submodule lattice of any -module. Namely, assume that the length of the indecomposable projective -modules is bounded by and take a finite-dimensional algebra with a local -module of length . Then is a modular lattice of height with a unique element of height (the radical of the module ). If is of the form , then has to be a local -module of length , thus a factor module of an indecomposable projective -module. But by assumption, the indecomposable projective -modules have length at most .
Let then the subsets of are actually -submodules and there is an isomorphism of -modules . Thus maps the lattice bijectively onto the submodule lattice . The lattice contains as a sublattice and both have the same height. However, may be a proper sublattice of , since isomorphisms of subfactors of the various modules yield diagonals in .
When dealing with the Auslander bijections , we always can assume that is multiplicity-free and supporting, here supporting means that for any indecomposable direct summand of . Namely, let be the direct sum of all indecomposable direct summands of with , one from each isomorphism class. Then, on the one hand, (since a map ending in is right -determined if and only if it is right -determined. On the other hand, there is an idempotent such that and , and there is a lattice isomorphism given by , where is a submodule of .
Both objects and related by the Auslander bijection concern morphisms ending in . Of course, in Proposition 3.9 we have seen already that all the indecomposable direct summands of the minimal right determiner of a map satisfy
We use the next two sections in order to transfer well-known properties of the lattice of submodules of a finite length module to the right -factorization lattices, in particular the Jordan–Hölder theorem. In Sect. 5, we introduce the right -length of a right -determined map ending in , it corresponds to the the length of the factor module In Sect. 6 we will define the -type of as the dimension vector of
5 Right -factorizations and right -length
The Auslander bijection asserts that the lattice is a modular lattice of finite height, thus there is a Jordan–Hölder Theorem for ; it can be obtained from the corresponding Jordan–Hölder Theorem for the submodule lattice . In Sects. 5 and 6, we are going to formulate the assertions for explicitly. Here we consider composition series of submodules and factor modules of .
Let be maps, where with composition . The sequence is called a right C-factorization of f of length t provided the maps are non-invertible and the compositions are right minimal and right -determined, for It sometimes may be helpful to deal also with right -factorizations of length ; by definition these are just the identity maps (or, if you prefer, the isomorphisms).
If is a right -factorization of a map , then any integer sequence defines a sequence of maps with for . We use the following lemma inductively, in order to show that is again a right -factorization of and we say that is a refinement of . In particular, any right -factorization of is a refinement of .
We only have to check that cannot be invertible. Assume is invertible. Then is a split epimorphism. Since is right minimal, it follows that is invertible, a contradiction.
We say that a right -factorization is maximal provided it does not have a refinement of length .
This is a direct consequence of Auslander’s Second Theorem.
Any right -factorization has a refinement which is a maximal right -factorization and all maximal right -factorizations of have the same length.
This follows from Proposition 5.2 and the Jordan–Hölder theorem.
In particular, any right minimal right -determined map has a refinement which is a maximal right -factorization, say and its length will be called the right-length of , we write for the right -length of . There is the following formula:
where denotes the length of the -module and the length of its -submodule .
The right equivalence class As we have mentioned in Sect. 4, it is of interest to determine the maps in the right equivalence class
Let be modules. Up to right equivalence, there is a unique right -determined map ending in with maximal. The submodule of is the zero module. If is any right -determined map ending in , then for some .
The lattice has a unique zero element, namely . Let for some right minimal map . Then, the right -length of has to be maximal and for any right -determined map ending in .
In general it seems to be quite difficult to describe the maps such that But one should be aware that such a map always does exist: any pair of -modules determines uniquely up to right equivalence a map ending in , namely the right minimal, right -determined map with
Let be modules. The set is the right equivalence class of the zero map if and only if belongs to .
This is a direct consequence of Corollary 3.7.
The special case ofbeing projective. For an arbitrary projective module , there is the following description of the right -length of a right minimal, right -determined morphism . Here, we denote by the Jordan–Hölder multiplicity of the simple module in the module , this is the number of factors in a composition series of which are isomorphic to .
Let be projective. The right minimal, right -determined maps are up to right equivalence just the inclusion maps of submodules of such that the socle of is generated by .
The minimal element of is the inclusion map , where is the intersection of the kernels of all maps , where is a simple module with a direct summand of .
Let be the set of modules , where is a simple module with a direct summand of . Let be the intersection of the kernels of all maps with Since is of finite length, there are finitely many maps with , say , such that Then embeds into , thus its socle is generated by . It follows that the inclusion map is right -determined. On the other hand, if is right minimal and right -determined, then it is a monomorphism, thus we can assume that it is an inclusion map. In addition, we know that the socle of is generated by , thus embeds into a finite direct sum of modules in . It follows that is the intersection of some maps , where , thus
There is the following consequence: The-length of any inclusion map (such a map is obviously right minimal and right -determined) is precisely the length of
For further results concerning the right -length of maps, see Sect. 9.
6 The right -type of a right -determined map
Recall that we consider as a -module, where The indecomposable projective -modules are of the form , where is an indecomposable direct summand of , thus the simple -modules are of the form
Given an artin algebra , we denote by its Grothendieck group (of all -modules modulo all exact sequences), it is the free abelian group with basis the set of isomorphism classes of the simple -modules . Given a -module , we denote by the corresponding element in , called the dimension vector of . Of course, can be written as an integral linear combination where the coefficient of is just the Jordan–Hölder multiplicity of in . The elements of with non-negative coefficients will be said to be the -dimension vectors. If is a -dimension vector and is a -module, we denote by the subset of consisting of all submodules of with dimension vector
Let us return to the artin algebra , where is a -module. Thhe Grothendieck group is the free abelian group with basis the set of modules , where runs through a set of representatives of the isomorphism classes of the indecomposable direct summands of . We are interested here in the dimension vectors of and of its factor modules. Actually, we want to attach to each right -determined map ending in its right -type so that We start with pairs of neighbors in the right -factorization lattice , since they correspond under to the composition factors of .
Let and be right minimal, right -determined maps. We say that the pair is a pair of C-neighbors provided Note that the pair in is a pair of -neighbors provided and there is no with (of course, it is the condition which implies that there is a map with ).
Let us consider a composition , where both are right minimal and right -determined. It can happen that is also right minimal and right -determined, but is not right -determined. Also it can happen that both maps and are right minimal and right -determined, whereas is not right -determined. Here are corresponding examples.
First, let , thus and both and are right -determined, whereas is not right -determined.
Second, let thus . Then both and are right -determined, whereas is not right -determined.
Let and such that is a pair of neighbors. We say that is of type (or better of type ) where is an indecomposable direct summand of , provided there is a map such that does not factor through . Such a summand must exist, since otherwise would factor through , due to the fact that is right -determined. The following proposition shows that is uniquely determined.
If is a pair of -neighbors of type , then is isomorphic to the simple -module Thus, the type of a pair of -neighbors is well-defined.
Thus, if is a pair of -neighbors of type , we may write
We note the following: Ifis a pair of-neighbors and, thenmay be neither injective nor surjective. Let us exhibit examples with .
As in the examples 5, let be the linearly directed quiver of type and take now as the path algebra of modulo the zero relation .
Now consider a right minimal right -determined map ending in . As we have mentioned, we want to attach to an element
into a finite number of disjoint subsets.
for every -dimension vector .
Assume now that is an algebraically closed field and that and are -algebra. If is a -module and a dimension vector for , we write instead of . Note that is in a natural way an algebraic variety, it is called a quiver Grassmannian. Namely, all the -modules with dimension vector have the same -dimension, say (if , then ). Denote by the usual Grassmannian of all -dimensional subspaces of the vector space Using Plücker coordinates one knowns that is a closed subset of a projective space, thus is a projective variety. Now is a subset of defined by the vanishing of some polynomials (which express the fact that we consider submodules with a fixed dimension vector), thus also is an algebraic variety and indeed a projective variety (but usually not even connected).
Proposition 6.3 can be reformulated as follows:
for every -dimension vector .
In particular, we see that the set is a projective variety: these Auslander varieties (as they should be called) furnish an important tool for studying the right equivalence classes of maps ending in a given module. As we have mentioned at the end of Sect. 2, the study of the set of right minimal maps ending in a fixed module can be separated nicely into that of the local symmetries described by the right automorphism groups and that of the global directedness given by the right factorization lattice. Auslander’s first theorem describes the right factorization lattice as the filtered union of the right -factorization lattices, and, as we now see, these right -factorization lattices are finite disjoint unions of (transversal) subsets which are projective varieties, the Auslander varieties.
It seems that quiver Grassmannians first have been studied by Schofield  and Crawley-Boevey  in order to deal with generic properties of quiver representations. In 2006, Caldero and Chapoton  observed that quiver Grassmannians can be used effectively in order to analyse the structure of cluster algebras as introduced by Fomin and Zelevinsky. Namely, it turns out that cluster variables can be described using the Euler characteristic of quiver Grassmannians. In this way quiver Grassmannians are now an indispensable tool for studying cluster algebras and quantum cluster algebras. We should add that quiver Grassmannians were also used (at least implicitly) in the study of quantum groups, see for example the calculation of Hall polynomials in . A large number of papers is presently devoted to special properties of quiver Grassmannians.
There is the famous assertion that any projective variety is a quiver Grassmannian, see the paper  by Reineke (answering in this way a question by Keller) as well as blogs by Le Brujn  (with a contribution by Van den Bergh) and by Baez . Actually, the construction as proposed by Van den Bergh in Le Bruyn’s blog is much older, it has been mentioned explicitly already in 1996 by Hille  dealing with moduli spaces of thin representations (see the example at the end of that paper), and it can be traced back to earlier considerations of Huisgen-Zimmermann dealing with moduli spaces of serial modules, even if they were published only later (see , Theorem G, but also , Corollary B, and , Example 5.4). It follows from Proposition 4.6 above that given a strictly wild algebra and any projective variety , there are -modules and a dimension vector such that is isomorphic to . We will show in  that this holds true for all controlled wild algebras.
7 Maps of right -length 1
By definition, a right minimal right -determined map has right -length 1 provided is not invertible and given any factorization with right minimal right -determined, then one of the maps is invertible. Let us denote by the set of right equivalence classes of the maps ending in which have right -length 1.
Warning: an irreducible map is of course right minimal, but if is irreducible and right -determined, we may have For example, consider the Kronecker quiver, take . The irreducible map has (note the factorizazion ).
Here is an immediate consequence of Proposition 5.4.
Let be right minimal and right -determined. Then if and only if is a maximal -submodule of .
In order to analyze maps of right -length 1, we will need the following lemma.
If is right minimal and is a split epimorphism, then also is right minimal.
Observe that it is not enough to assume that is an epimorphism. As an example, take the indecomposable injective Kronecker module of length , let be a submodule of length 2, and Then is right minimal. But the induced sequence is just the short exact sequence which splits.
Denote the kernel of by , thus we can assume that such that is the canonical projection with kernel . Assume that , where is contained in the kernel of , thus Since , there are submodules of both containing such that and , with and
Consider , this is a submodule of the kernel of . Also, (using the modular law). Thus we have and . This shows that is a direct summand of which is contained in the kernel of . Since is right minimal, we see that Since , it follows that
Let be a module. Let be a right minimal right -determined epimorphism with . Then the kernel of is indecomposable.
Since is right minimal and is a split epimorphism, lemma 7.2 asserts that is right minimal. Since is also right -determined, we see that a contradiction.
with the upper row being the sequence . Since the lower sequence does not split, the map is right minimal. The kernel shows that is also -determined. Since is not invertible, the factorization shows that is a -factorization of of length at least , thus a contradiction. This shows that belongs to the -socle of
where again the upper row is . Write with endomorphisms . If all belong to the radical of , then all the sequences induced from by the maps split, thus also the lower sequence splits, since it is induced from by Thus, at least one of the maps has to be invertible and therefore is a split monomorphism.
If , then the lower sequence splits off a sequence , but this means that is not right minimal. Thus But then is an automorphism, thus is invertible, a contradiction. This shows that
We say that an epimorphism is epi-irreducible, provided for any factorization with a proper epimorphism, the map is a split epimorphism (the dual concept of mono-irreducible maps has been considered in ).
Let be indecomposable, non-projective and let . If is an epi-irreducible epimorphism with kernel , then belongs to .
Since is a proper epimorphism, also is a proper epimorphism, thus is a split epimorphism. But this implies that also the exact sequence induced from by splits.
We will need some basic facts concerning the Gabriel–Roiter measure of finite length modules, see . The Gabriel–Roiter measure of a module will be denoted by We recall that any indecomposable module which is not simple has a Gabriel–Roiter submodule , this is a certain indecomposable submodule of and the embedding is called a Gabriel–Roiter inclusion. Recall that a Gabriel–Roiter inclusion is mono-irreducible: this means that for any proper submodule of with , the inclusion splits. As a consequence, given any nilpotent endomorphism of , the sequence induced from using splits. Also, it follows that the cokernel of a Gabriel–Roiter inclusion is indecomposable (and not projective).
Of course, we may use duality and consider a Gabriel–Roiter submodule of , the corresponding projection will be called a co-Gabriel–Roiter projection. By duality, a co-Gabriel–Roiter projection is an epi-irreducible epimorphism with (non-injective) indecomposable kernel.
Let be an indecomposable module which is not simple and let be a co-Gabriel–Roiter projection, say with kernel . Let Then is right minimal, right -determined and thus belongs to .
If is indecomposable and not simple, we also may consider a Gabriel–Roiter submodule of , say with projection and consider Then is right minimal and right -determined, however in general there is not a fixed number such that belongs to or to . A typical example is example 8 presented in the next section. The two modules and both have as a Gabriel–Roiter submodule with factor module . Let The projection belongs to whereas the projection belongs to
8 Epimorphisms in
The set of right equivalence classes where is an epimorphism is obviously a coideal of the lattice we denote it by . Since the pullback of an epimorphism is again an epimorphism, we see that is closed under meets. Also, for any module , the subset of consisting of the right equivalence classes of all right -determined epimorphisms ending in is a coideal which is closed under meets. Since is a lattice of finite height, we see that has a unique minimal element, say and our first aim will be to describe
Before we deal with this question, let us point out in which way the projectivity or non-projectivity of indecomposable direct summands of are related to the fact that right minimal right -determined morphisms are mono or epi. If is a monomorphism, then is right minimal and right -determined (see Corollary 3.8), thus right -determined for some projective module . Conversely, if is projective, then any right minimal, right -determined morphism is a monomorphism (see Corollary 3.7). Namely, if is an indecomposable direct summand of the kernel of , where is right minimal, then is not injective and is a direct summand of any module such that is right -determined. Of course, since is not injective, is an indecomposable non-projective module. One should be aware that a morphism may be right -determined for some module without any indecomposable projective direct summand, without being surjective.
As in example 6, we take as the path algebra of the linearly directed quiver of type , modulo the zero relation . Again, let and . As we have mentioned already, the non-zero maps are not surjective, but right -determined, and, of course, is not projective. (As we will see below, it is essential for this feature that the kernel of has injective dimension at least 2; for a general discussion of maps which are not surjective, but right -determined by a module without any indecomposable projective direct summands, we refer to ).
The submoduleof We denote by the set of morphisms which factor through a projective module. Note that is a -submodule of .
Assume that is right -determined. Then is surjective if and only if .
One direction is a trivial verification: First assume that is surjective. Let belong to , thus where and with projective. Since is surjective and is projective, there is such that Thus shows that belongs to
The converse is more interesting, here we have to use that is right -determined. We assume that . Let be a projective cover of . Consider an arbitrary morphism . The composition belongs to , thus to . Since is right -determined, it follows that itself factors through , say for some . Now the composition is surjective, there has to be surjective.
Let us denote by the subset of given by all elements with an epimorphism.
Here, the vertical maps are the canonical inclusions.
It is well-known that given a module and a submodule , then the lattice of submodules of the factor module is canonically isomorphic to the lattice of the submodules of satisfying This is the vertical map on the right.
More generally, dealing with a morphism which is right -determined, we can recover the image of as follows:
Let be right -determined. Then one recovers the image of as the largest submodule of (with inclusion map ) such that
Let be the image of with inclusion map and (with surjective). First of all, we show that Let and (the maps obtained in this way generate additively). We want to show that factors through . Since is surjective, there is such that (since is projective). Thus Thus factors through
On the other hand, let be a submodule of such that . Let be a projective cover. Consider the map It has the property that for all maps the composition factors through (namely belongs to ). But is right -determined, thus we conclude that factors through say for some Thus the image of is contained in the image of . This is what we wanted to prove.
We recover in this way Proposition 8.1. Namely, if is surjective, then is the image of , thus is one of the submodule with , thus
Conversely, if then is one of the submodules with and therefore the image of contains , thus is equal to . This shows: If f is right C-determined, then f is surjective if and only if
All maps in are epimorphisms if and only of
Kernels with injective dimension at most.
The injective dimension of is at most .
If is any module, then all maps in are epimorphisms.
We have for all modules .
Recall from , 2.4 that has injective dimension at most if and only if Thus, if has injective dimension at most and is an arbitrary module, then this shows that (i) implies (iii). Conversely, assume the condition (iii), thus for all modules . If the injective dimension of would be at least , then But . This contradiction shows that (iii) implies (i). For the equivalence of (ii) and (iii) see Corollary 8.4(a).
Let be hereditary and a module without any indecomposable projective direct summand. Then any right -determined morphism is an epimorphism.
Let . Since has no indecomposable projective direct summand, it follows that Since is hereditary, the injective dimension of any module is at most . Since the injective dimension of is at most , it follows from the proposition that all right -determined maps are epimorphisms.
Of course, we also can show directly that Namely, let be in . Then with , where is a projective module. The image of is a submodule of , thus, since is hereditary, the module is also projective. Thus, we have a surjective map with projective. Such a map splits. This shows that is isomorphic to a direct summand of . It follows that and therefore
In terms of the Auslander bijection, we may deal with these data in several different ways: namely, we may look at the right equivalence classes of both and in as well as at the right equivalence class in In case we deal with , one should be aware that this map is the join of the two maps and in .
In addition, we also may concentrate on the possible maps and (sometimes called steering maps).
When dealing with epimorphisms in , Riedtmann–Zwara degenerations play a decisive role, as the following proposition shows: