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Regular stones of wild hereditary algebras
Journal of Pure and Applied Algebra 93 ( 1994) 15-3 1 North-Holland
15
Regular stones of wild hereditary algebras Otto Kerner and Frank Lukas Mathematisches Germany
Institut, Heinrich-Heine-Universitiit,
Diisseldorf D-40225 Diisseldorf I,
Communicated by C.A. Weibel Received 2 March 1992 Revised 6 January 1993
Abstract Kemer, 0. and F. Lukas, Regular stones of wild hereditary Applied Algebra 93 ( 1994) 15-3 1.
algebras, Journal of Pure and
If A is a finite-dimensional wild hereditary algebra we study the class of regular components containing stones of quasi-length two, where a stone is a brick without self-extensions. We show that there are only finitely many non-sincere components of this type and, provided the algebra has elementary modules with self-extensions, no sincere component with stones of quasi-length two.
Introduction If A is a finite-dimensional tame hereditary algebra over an algebraically closed field, there exist only finitely many inhomogeneous tubes in the Auslander-Reiten quiver r(A) of A, thus there are only finitely many regular components containing stones, that is, bricks without self-extensions. The situation is totally different if A is wild hereditary. If A is connected with at least three simple modules there always exist infinitely many regular components containing (quasi-simple) stones. Thus we restrict to those regular components which contain stones of quasi-length (at least) two.
Correspondence to: 0. Kerner, Mathematisches Institut, Dusseldorf, UniversitltsstraBe 1, D-40225 Dusseldorf 1, Germany. Email: [email protected].
0022-4049/94/$07.00 @ 1994 SSDI 0022-4049 ()E0090-Q
Heinrich-Heine-Universitat,
Elsevier Science B.V. All rights reserved
16
0. Kerner, F. Lukas
In Section 1 we show that there are only finitely many non-sincere components, that is, components containing a non-sincere module, of this type (Theorem 1.1). In Section 2 we study sincere components. If the algebra A has elementary modules with self-extensions then all stones in sincere components are quasisimple, especially there are only finitely many regular components containing stones of quasi-length two in this case (Theorem 2.8). The notation of elementary modules is introduced in [ 8,101. A regular module E is called elementary if it is not in the middle-term of a short exact sequence 0 --+ U -+ E -+ V + 0 with U and I/ regular and non-zero. If A is wild hereditary with y1simple modules, it was first shown by Hoshino [4] that the quasi-length of regular stones is bounded by y1- 2. In Section 3 we give a complete list of all wild hereditary algebras with regular stones of quasi-length n - 2. This result and the subsequent corollary demonstrate that the class of components studied in this paper may help to characterise the algebras.
Notations The word algebra always denotes a finite-dimensional, unitary, basic algebra over some algebraically closed field k. The letter A normally is reserved for wild hereditary, connected algebras. If A is such an algebra, A-mod denotes the category of finitely generated left A-modules and 72 denotes the category of regular A-modules, that is, the full subcategory of A-mod defined by direct sums of indecomposable regular modules. We call a module X a brick if End(X) is isomorphic to k. A brick without self-extensions (and projective dimension at most 1) is called a stone. The number of pairwise non-isomorphic simple A-modules will be denoted by y1(A). By qA we denote the Tits-form which coincides in the hereditary case with the Euler-quadratic form, that is qA(bX)
= dimk End(X)
- dimk Ext’(X,X).
By r (A) we denote the Auslander-Reiten quiver of A. The vertices of r (A) are isomorphism-classes of indecomposable A-modules; if the context is clear, we will not distinguish between an indecomposable module X and its class
[Xl. By J2 (A) we denote the set of regular components of r (A). If C is a regular component-it is of type ZA, by [ 11 ]-and all indecomposable modules in C are sincere, we call C a sincere component. An indecomposable regular module X is called r-sincere, if z’X is sincere for all integers i. If U is a quasi-simple module and r a natural number, we denote by U (Y) ( [r ] U )the indecomposable regular module with quasi-length Y and quasi-socle (quasi-top) U.
Regular stones of wild hereditary algebras
17
If X is an idecomposable regular module, say X = [m] U for U quasisimple, we denote by W (X ), the wing of X (of length m ), the mesh-complete full subquiver of r (A ) defined by the vertices 7’ ( [s] U) with 1 5 s 5 m and 0 5 Y 5 m - s. The wing W(X) is called a standard wing if rada (M, N) = 0 for all modules M, N in W(X). By [ 61 this is the case exactly if X is a brick. From the concept of perpendicular categories we use the following result (see [ 21, [ 15 ] or [ 161): If A is an algebra and X a quasi-simple regular stone, then the right perpendicular category XL, defined by the objects {Y 1 Horn (X, Y) = Ext’ (X, Y) = 0) is an abelian subcategory of A-mod which is equivalent to a module category B-mod. We have B = End(Z ), where 2 is defined by Bongartz’s construction such that X @ 2 is a square-free tilting module, and B is wild, connected and hereditary. Sometimes we write (X, Y) ( ’ (X, Y ), respectively) instead of HomA (X, Y) (Exta (X, Y ), respectively). The Auslander-Reiten translations will be denoted by r andr-; to emphasise the algebra, we sometimes add a subscript, for instance ZA,r;. In general we follow the notations used in [ 12 1. If C is a regular component and X E C is quasi-simple, then we define e(C) as the smallest natural number such that Ext’(X(e(C)),X(e(C))) # 0. By e(A) we understand max{e(C) ] C regular}. Note that e (A ) 5 n - 1, where n is the number of simple A-modules, see [ 41. is a maximal family of Note moreover that for e(C) > 1, (X, . . . , P(c)-lX) pairwise orthogonal quasi-simple stones in C, as Hom(X, &)X) # 0, see [ 51.
1. Non-sincere components containing stones The main result of this part is the following: Theorem 1.1. If A is connected wild hereditary then there exist only finitely
many non-sincere regular components containing stones of quasi-length at least two. If C is a regular component, X a quasi-simple module in C then it was shown in [ 7, Proposition 1.11, that the module X(r) for some r > 1 is a stone if and only if X (r + 1) is a brick and X(r - 1) is a stone. So X(r) is a stone for some r 2 2 if and only if X(2) is a stone. If X is a quasi-simple regular stone in A-mod then there is an isomorphism of vector spaces Exti(X(2),X(2))
g HomA(X,r2X),
see [ 61. We will use this simple fact freely in the sequel and will express the dimension of Ext1(X(2),X(2)) frequently by Hom(X,r2X). Moreover, as X(2) is a brick, we have dimHom(X,r2X) = 1 - q(hX(2)). If C is a non-sincere component then there exists an indecomposable module X E C and a vertex CL)E &o, where Q = (‘20, Q1 ) is the (ordinary) quiver of
0. Kerner, F. Lukus
18
A such that Hom(P (o), X) = 0. Moreover, without loss of generality we may assume that w is a source of Q, that is A is a one-point extension A = B [M] with M = rad P (CO).Further, if X is in B-mod, then the whole wing W(X) is in B-mod. Especially a non-sincere component always has non-sincere quasisimple modules. In order to prove Theorem 1.1 we may restrict to the case where B is representation-infinite. Lemma 1.2. If A = B[M]
is a one-point extension and X E B-ind is not projective, then we have a short exact sequence 0-
ZBX -
7,&X -
with 1 = dimk Hom(M,
I(0)’ rgX)
0,
= dimk Ext(l(w),
r,+X)
= dimk Hom(P(w),rAX). Proof. The existence
of the short exact sequence with
1 = dimk Hom(M,
reX)
= dimk Hom(P(m),
TAX)
was shown in [ 12, 2.51. Applying to the short exact sequence
O-M-P(u)-I(CO)-0 the functor Ext(l(cc,),
HomA (-, Y ) for Y E B-mod Y).
arbitrary
we get Horn (M, Y) G
0
Lemma 1.3. If A = B [M] is wild hereditary, if X is a stone in B-mod is regular and quasi-simple in A-mod, then we have
9A(&[2]X)
which
= 1 + (d&lrBX,d&X) - dimk Horn (M, X ) . dimk Horn (M, zg X ) =
ql3(dim(X CETBX) )
Proof. As A is a one-point extension of B, projective B-modules are projective in A-mod; therefore X is not projective and ZAX, r&C are non-zero and by Lemma 1.2 we have for 1 = dimk Horn (M, TBX),
0Using &[
TBx 21X = &X
z,z,x -
z(m)’ -
0.
+ dim TAX one checks the assertion.
0
If A = B [M] is wild hereditary and B is representation-infinite, it was shown in [ 131 that almost all preprojective indecomposable B-modules are regular in A-mod. Unger additionally proved in [ 171 that almost all indecomposable
Regular stones of wild hereditary algebras
preprojective more is true:
and preinjective
B-modules
19
are quasi-simple
in A-mod.
Even
Proposition 1.4. Let A = B [M] be connected, wild and hereditary and B representation-infinite. If N is a natural number then there exist only jinitely many indecomposable preprojective or preinjective B-modules X such that dimk Hom(X, z:X) 5 N. Proof. By [ 13 ] and [ 17 ] it is enough to consider simple regular in A-mod. Using dimkHom(X,riX)
= -q(dim[2]X)
we can now apply Lemma (dim~~X,dimX)
the case where X is quasi-
+ 1.
1.3. Note that here =
d imk Horn
(7pY,
where rx + 1 is the number of middle-terms
X)
=
rx,
of the Auslander-Reiten
sequence
0-7~x-Y-x--+o.
Especially
rx is constant
on the re-orbit
of X and c = max{rx}
exists. So we
get qA(b[2]X)
= 1 + rx - dimkHom(M,X)
.dimkHom(M,rX).
As M is a projective B-module, say M = eiEe6 P (i )=I with cri 2 0 we have dimk Hom(M, Y) = Ci,_06~iyi where (yi)ieo;, = d&Y for any Bmodule Y. Thus there exist only finitely many indecomposable preinjective or preprojective B-modules X such that dimkHom(M,X) see [ 13, Lemma
.dimkHom(M,rsX)
11, combined
5 c + N,
with [ 1, Proposition
2.21.
0
Corollary 1.5. Zf A = B [M] is a connected, wild hereditary algebra, then there are only finitely many regular components in r(A) containing stones of quasilength two and preprojective or preinjective B-modules simultaneously. 0 If C ia a non-sincere component such that [ r]X is a stone for some r > I and some quasi-simple module X and [s] X is non-sincere for some s 2 1, we may suppose, as mentioned above, that A = B [M] is a one-point extension of a hereditary algebra B and [SIX E B-mod. If s > 1 the whole wing W ( [s] X) is contained as a wing in B-mod. Especially we get 7iX = 7&X for i = 1,. ..,s - 1 and thus by Lemma 1.2 W( [SIX) is not sincere in B-mod, as HomB (M, SBX ) = 0. The wing W ( [SIX) is either contained in the
20
0. Kerner, F. Lukas
preprojective or preinjective component of r (B ), and in this case [SIX is a stone and the quiver Q (B ) of B necessarily has the shape
or W ( [s] X ) is contained in a regular, non-sincere component 23 of r (B ) (provided B is representation-infinite) containing stones of quasi-length at least min{ r, s}. Moreover, we may assume that B has only finitely many non-sincere regular components containing stones of quasi-length bigger than one: This is trivial in case B is tame or is wild with two simple modules. Therefore we can use induction on the number of simple modules. To continue the proof of Theorem 1.1, by Corollary 1.5 we only have to consider now the following case: A = B [M] and C is a regular component containing stones of quasi-length two and additionally a quasi-simple module X such that X E B-mod and X is regular in B-mod. We have to distinguish two cases: Either X is quasi-simple in B-mod, which happens for example if [S ] X for s > 1 is in B-mod, since in that case the whole wing W( [SIX) is in B-mod, or X has quasi-length bigger than one in B-mod. Proposition 1.6. Let A = B [M] be connected wild hereditary and B represen-
tation infinite. If C is a non-sincere component containing stones of quasi-length r > 1 and X is a quasi-simple regular B-module which is contained in C, then C contains modules of quasi-length r which are regular B-modules. Proof. It is straightforward to see that X is quasi-simple in C. By induction on the quasi-length we may suppose that for some s with 1 5 s < r the stone [SIX is in B-mod. This especially implies that for 1 5 i < s - 1 we have r;X = T&X. If Y + [SIX is the irreducible epimorphism in B-mod, then we have CjiIJlY = d&l[s]X
+ dim&r,
whereas in A-mod we get d&[s
By Lemma
+
11X =
dim[slX + dimfJ
1.2 we have
dimT>X
= dim&x
+ 1. dimI(
Regular stones of wild hereditary algebras
21
where 1 = dimk Horn (M, r”,X). As s < Y holds, we have (by Lemma 1 = q(&[s = q(dimY)
+ 11X) = q(dimY
1.2)
+ I.&l(o))
+ I2 + 1 .(dimZ(cc,),dim([s]Xe~$X))
= q(dim Y) - 1. dimk Hom(M,
[SIX).
Thus we have q (dim Y) = 1, that is, Y is a stone, as [SIX is a stone and additionally 1 = 0 or Hom(M, [SIX) = 0. In case 1 = 0 holds, we get r>X = r;X and thus [s + 1 ]X is in B-mod. If Horn (M, [s] X) = 0, then the Auslander-Reiten sequence in B-mod o-
[SIX-
[S-l]X@ST,Y-r-
B [SIX -
0
is an Auslander-Reiten sequence in A-mod and consequently 0 in C with T,Y = [S + l](r-X).
the stone r;Y
is
Notice that this result also means that the components V in B-mod containing X has stones of quasi-length at least Y. Moreover, if the wing W( [r]X) in C is contained in 27, then we have ZBX = r,+X, that is, Hom(M, reX) = 0 and thus D is a non-sincere component in B-mod. By induction there exist only finitely many components 2) of this type in r (B ) . As 2) has only finitely many non-sincere modules for each of these finitely many components we can find only finitely many non-sincere wings W of length r > 1. So there exist only finitely many components C in r(A) as described in Proposition 1.6. Thus there remains only the following case: Lemma 1.7. Zf A = B [M] is wild hereditary and B representation infinite, if C is a non-sincere regular component in r(A) containing stones of quasi-length two, if X is a quasi-simple module in C such that (1) X is a regular B-module. (2) X has quasi-length 1 > 1 considered as B-module. Then we have: In the Auslander-Reiten component 2) c Z(B), containing X there are stones of quasi-length 1 + 1 and the wing W(X) is non-sincere. Proof. In B-mod we have
where TBX is a submodule of U and I’ a submodule of X. For dimk Hom(M, T&Y) where i = 0,l we get in A-mod by Lemma 1.3 1 =
qA(b[2]X)
=
li =
qB(dim(X@TBX))-lc,.l1.
Notice that Ie and 11 both are positive as X is quasi-simple in C but not in 27. Notice further that the wings W ( U) and W ( TB U ) in D are standard
22
0. Kerner, F. Lukas
wings by [5] and thus we have Hom(U,V) Ext( V, Y) = 0. So we get qB(&l(XcBTeX))=
qs(dim(U6?
= Hom(V,U)
V))
= qs(dimU)
= Ext(V,:)
=
+ qs(dimV)
= 1 + %(bU).
Thus 1 = 1 + qB (d& U) - lo . II implies qs (d& U) = 1, that is, U is a stone and rc = Ii = 1. If Y is the quasi-top of X in B-mod, then we have dim X = dim &I; 7kY. Therefore, we get I-1
dimk Hom(M,
X) = 1 = c
dimk Hom(M,
r;Y),
i=O
which shows that the wing W(X) in D is not sincere. The same argument as above then proves that there are only finitely many components C in r (A ) of this type, which finishes the proof of Theorem 1.1. 0
2. Sincere components containing stones Let us remind that a connected wild hereditary algebra with more than two simple modules always has infinitely many non-sincere components containing stones, see for instance [ 13,17 1. Proposition 2.1. Zf A is connected, wild hereditary with at least three simple
modules then there are infinitely many sincere components containing stones. Proof. Let C be a non-sincere regular component containing stones such that the number of non-sincere quasi-simple modules is minimal. Take X E C nonsincere quasi-simple. Now we can choose in Xl an infinite family (Xi)i,, of stones such that (a) Xi is sincere in Xl, (b) dimk Hom(Xi, [21X) > (dimk X)2 + 1, (c) Xi are regular in A-mod (e.g. infinitely many Xl-preprojective modules satisfy these conditions as [21X is regular in Xl). As Hom(Xi, [21X) g Hom(Xi,X), by condition (b) we get Hom(Xi,X) # 0. As each nonzero map f : Xi + X is injective or surjective by [3, 4.11 again by (b) each map has to be surjective. As Hom(Xi, zX) = 0, we get by Unger’s Lemma [ 17, 1.3 ] that the kernel K of any surjective map f : Xi + X is indecomposable with self-extensions, that is K is regular and thus also the maps rrf : Z’Xi --t z”X are surjective for all r E Z. Especially rrXi is sincere if z”X is. If E is one of the indecomposable injective A-modules with Hom(X, E) = 0 then E is injective in Xl and therefore we have, since all Xi
Regular stones of wild hereditary algebras
23
are sincere in Xl, Hom(X,, E) # 0. Together with the surjection f : Xi + X this implies that all Xi are sincere in A-mod and therefore in the r-orbit of Xi there are less non-sincere modules than in the r-orbit of X. Notice that a regular component contains only finitely many vertices, corresponding to indecomposable modules in X l. Notice further that by Theorem 1.1 only finitely many of the Xi can be vertices of quasi-length bigger than one in non-sincere components. Thus by the minimality condition of C we have infinitely many sincere components containing stones. 0 Example.
Let A be the path-algebra of the quiver O=O+O, and let B be the Kronecker-algebra corresponding to the quiver 0:. . By [ 111 an indecomposable A-module M(n) of dimension (n + 1, yt, 0) is regular if and only if n 2 5 holds. Using [ 6, 3.31 we see that r’M( n) is sincere for all i # 0 if n > 5. Consider the tilting module M(n) $ N(n) @ M(n + 1) for n 2 4 and &N(n) = (n(n + l),n2, I), see [14]. Then M(n)@N(n) is a preprojective tilting module in M (n + 1 )I, M (n + 1) J_ E K, - mod, where K, is the path-algebra of the quiver .x0 (n arrows) and thus for example the modules rMcn + , Ji M (n ) with dimension-vector (n 2 5 ) n.d&lN(n)-dimhI
= ((A
l)(n
+ 1)&z-
l)n,n)
are r-sincere quasi-simple stones. Let us now consider a more restricted condition: Again we assume that C is a regular component, X E C is quasi-simple such that X (r + 1) with r > 1 is a brick with self-extensions. Additionally we demand that C is a sincere component. Note that in this case X 1 is contained in R. The next aim is to show that this number r is relatively small. Using again Unger’s lemma [ 17, 1.31 the following was shown in [ lo]: Lemma
2.2. Let X and Y be non-isomorphic regular stones with Horn (X, Y) # 0 but Horn (X, 7Y) = 0. If f : X -+ Y is non-zero and X or Y is T-sincere we have: Either f is injective and the cokernel is indecomposable regular or f is surjective with indecomposable regular kernel.
homomorphismus f :X + Y is injective or surjective if Ext (Y, X) = 0. If 2 denotes the cokernel of f in the first and the kernel in the second case, it was shown in [ 17, 1.31 that Proof. It was shown in [ 3, 4.11 that a non-zero
dimkHom(X,
Y) = dimkExt’(Z,Z)
Assume now Y is r-sincere. 0-X-Y-Z-0 o--+z-x-Y~o
+ 1.
We have to consider the two short exact sequences or
24
0. Kerner, F. Lukas
corresponding and get O=
to f injective
or surjective.
We apply the functor
(Z,Y)--+(Y,Y)-(X,Y)+‘(Z,Y)-0 0+
Assume Z is dimk Hom(X, Y) ular modules are preinjective. The
not
(Y, Y) regular.
Y)
or
(X, Y) --f Then
Hom(-,
Z
is
(Z, Y) -
0.
a
and
stone,
consequently factors of regeither regular or preinjective, in the first case 2 has to be condition dimk Horn (X, Y) = 1 then implies = 1 holds by [ 17, 1.3 1. As indecomposable
0 = Ext’(Z,
Y) g DHom(Y,zZ)
that is, Y is not r-sincere. In the second case Z is preprojective and dimk Hom(X, Y) = 1 implies Hom(Z, Y) = 0 which again would mean Y is not z-sincere. If X is r-sincere, we apply the duality D = Homk (-, k) : A-mod + and consider
mod-A
Df : DY + DX instead of f.
0
The following lemma is almost trivial. As it will be used frequently, formulate it explicitely. The proof is straightforward.
we will
Lemma 2.3. (a) Zf X and Y are regular, f :X + Y is a homomorphism and f(X) is a submodule of some module Y’ such that Y/f(X) is regular, then Y’/f (X) is regular. (b) Zf X is an indecomposable regular module and f : X -+ r”X is an injective homomorphism for some n > 0, then the cokernel z”X/ f (X) is not regular. 0 Theorem 2.4. Let C be a regular sincere component, X E C quasi-simple such that X (r + 1) for r > 1 is a brick with self-extensions. (a) There exists a module S, simple in Xl, with 0 f Horn (S, r-‘+ ‘X ). (b) S is quasi-simple in A-mod and S, T,& . . . , zT-‘S are pairwise orthogonal. Proof. One quickly checks that the modules z --r+ 1x,
. ..)
7-x,
7*x,
. . .)
7’X
are in Xl. Thus there exists a XL-simple module S with Hom(S, 7-‘+‘X) # 0 and we may consider S as submodule of 7P’X. Since 7 is left exact, 7’S then is a submodule of T-‘+~+~X for all i > 0. Of course the simple objects of XL are quasi-simple in A-mod.
Regular stones of wild hereditary algebras
25
Clearly we have Horn (T’S, S) = 0 for all i > 0, as S is a quasi-simple stone, see [ 51. Assume there exists 1 < i 5 2r - 1 with 0 # f : S + 7’s. For i = r - 1 A X. By Lemma 2.2 the map f o I is injective or we then get f o I : S f, P’S surjective. If it is surjective, I is an isomorphism but Hom(r-‘+‘X, X) = 0. Therefore f o I is injective, that is, f is injective. The cokernel Q of f is a submodule of the cokernel Q’ of f 0 I which is regular by Lemma 2.2. So Q also is regular by Lemma 2.3(a), a contradiction to Lemma 2.3 (b). Assume secondly for 1 5 i 5 2r - 1 with i # r - 1, i # r there is a non-zero map f : S + s’S. Now z- r+l+iX is in XI and therefore the composition j- 0 l : s f, z’s A z- r+l+iX is injective as S is simple in Xl, the cokernel of f o I is in X’ and therefore regular. So again f would be injective with regular cokernel. Finally, for i = r a non-zero map f :S + T’S would produce a non-zero map f 0 2 : S + T’S L) TX, which contradicts to the condition Ext(X,S) = 0. 0 If C is a regular component we denote by e (C ) the smallest natural number such that the modules with quasi-length e(C) in C have self-extensions. So C contains stones of quasi-length r 2 1 if and only if e (C ) 2 r + 1. Notice further that e(C) is bounded by n - 1, where n denotes the number of simple A-modules, see [4]. Thus max{e(C) 1C E Q(A)} exists, we denote it by e(A). Theorem 2.4 then says that for a sincere component C with e(C) > 2 the module S is in a component D with e(D) 2 2(e(C) - 1). Corollary 2.5. If C is a sincere regular component, then we have
e(A) e(C) 5 - 2 + 1. Proof. For e (C) > 2 this follows from the above theorem. trivially for e (C ) 5 2. 0
The assertion holds
In [ 8,101, the notion of an elementary module was introduced: A regular module E # 0 is called elementary if it satisfies the two equivalent conditions: (a) If R is regular, if f E Horn (R, E) \ {0}, then the cokernel of f is preinjective. (b) If R is regular, if g E Hom(E, R) \ {0}, then the kernel off is preprojective. Notice that elementary modules always exist for representation-infinite hereditary algebras, that their ring of endomorphisms is trivial and that with E also T’E is elementary for all i E Z. Notice further that elementary modules which are stones in addition are not r-sincere. Notice tinally that for an elementary module E, for regular modules X, Y and non-zero maps f E HomA (X, E), g E HomA (E, Y) the composition f g is non-zero, see [8,10].
0. Kerner,
26
F. Lukas
Proposition 2.6. Let X be an indecomposable r-sincere Hom( X, YX) = 0 for some r 2 2. Let E be an elementary there exists a number n E Z such that for Y = z”E we obtain: (a) Y, . . . ,5 -(r-2)y E XI, (b) Y,...,z -(r-2)Y are regular and simple in Xl.
module module.
with Then
Proof. (a) Choose an integer n such that the elementary module Y = FE satisfies the condition Horn (X, r-‘Y) = 0 for all i 2 0, but Hom(X, 7Y) # 0. So for the assertion (a) it suffices to show that Ext (X, 5-i Y) = 0 for 0 5 i 5 r - 2, which is equivalent to Horn(zY,~‘+~X) = 0 for 0 5 i < r - 2, via the Auslander-Reiten formula. Take f E Hom(X, 7Y) \ (0) and suppose there exists g E Hom(rY, 7 i+2X) \ (0) for some i with 0 5 r 5 r - 2. As the module 7Y is elementary, the composition f g E Hom(X, T~+~X) is non-zero. But, as X is z-sincere, Hom(X, 7’X) = 0 implies by [6, 2.2 and 2.31 that Horn (X, 7jX) = 0 also for 1 5 j < r. Therefore, Ext (X,7-‘Y) = 0, that is, 7-‘Y E Yl for 0 5 i < r - 2. (b) If 7-‘Y E Xl is not simple in Xl, then there exists a short exact sequence 0 -
u
-
7-’
‘Y+V+O
with U, I’ E Xl and non-zero. But, as X is r-sincere, the objects of Xl are all regular in A-mod. Since 7-‘Y is elementary in A-mod, there cannot exist such a short exact sequence. As Y, . . . , T-(‘-~) Y are all in Xl we have 7,‘Y = 7;: Y for 0 5 j 5 r - 2, so it suffices to show that any of the 7PY is regular in Xl. Let 1 > r - 1 be minimal with Horn (7-‘Y, 7X) # 0. Then the modules # 0. As 72x is Y>***>7 -(r-‘)Y are in Xl and we have Horn (7 -(l-‘)Y,z2X) regular in Xl, 7 -crP1)Y is not preinjective in Xl. Also 7-X is regular in XI, indeed .r2X = 7g1 (7-X), and H om( 7-X, Y) # 0 implies Y not preprojective in Xl. Therefore, all the modules 7-‘Y are regular in Xl. 0 If the algebra A has an elementary module E with self-extensions and X is as above, then Y would be a simple module with self-extensions in X1, but Xl is equivalent to C-mod for some finite-dimensional hereditary algebra C, a contradiction. So we obtain the following corollary: Corollary 2.7. Let A be a wild hereditary algebra such that A-mod contains an elementary module Y with Ext (Y, Y) # 0. (i) Let C be a regular component such that X(r) is a stone for some r 2 2 and a quasi-simple module X E C. Then X (r - 1) is not z-sincere. (ii) Especially every stone in a sincere component is quasi-simple.
Regular stones of wild hereditary algebras
27
Let us mention finally that most hereditary algebras have elementary modules with self-extensions. But for example the path-algebras of the quivers E:6, E.7, ,?s only have elementary stones, see [ 81. As main result of this part of the paper we can summarise: Theorem 2.8. Let A be connected wild hereditary with self-extending elementary
modules. Then there are only finitely many regular components containing stones of quasi-length bigger than one. 0
3. Algebras having components with stones of quasi-length n(A) - 2 It is well known and was first tame case also for wild hereditary bounded by n (A) - 2, where n = Straul3 (see [ 161) had remarked 0
-
o-o-
.
.
.
shown by Hoshino [4] that similarly to the algebras the quasi-length of regular stones is n (A) denotes the number of simple modules. that the path-algebra of the quiver -0-o
-
0
always has stones of quasi-length n(A) - 2. The aim of this part is to show that he gave a typical example. It was shown in [ 91 that for a regular (indecomposable) module R there exists a surjective homomorphism zFR + R for some s > 0 and thus R is generated by zPR. Recall that e(C), where C is a regular component, is the smallest number 1 such that indecomposable modules in C of quasi-length 1 have self-extensions and that
e(A) = max{e(C) If C is a regular component
1C E Q(A)} 5 n(A) - 1. with e(C) = e(A), we obtain:
Lemma 3.1. Let A be wild hereditary with e(A) 2 3 and let C be a regular
component with e(C) = e(A). Zf X is quasi-simple in C with z+‘X sincere for all m -> 0, then re(A)X is generated by X. Proof. As the modules re(A)-2X (i) for i = 1,. . . , e (A) - 1 are stones and the wing W(r e(A)-lX (e (A) ) ) is a standard-wing containing the modules ftAjm2X(i), the direct sum T = @f~I’-’ zecAjP2X(i) is a partial tilting module (see [ 161). Denoting by U the trace of T in recA)X, that is U = CIm f and the cokernel of the inclusion U it T~(~)_X by with f E Hom(T,r e(A)X) Q, we get the short exact sequence o-u-7
dA)_,y
-Q-O.
0. Kerner,
28
F. Lukas
The standardness condition of the wing W (r e(A)-lX(e(A))) immediately implies that U also is the trace of X in r e(A)X. Since T is a partial tilting module we have Horn ( T, Q ) = 0; this can be checked for example directly by applying the functor Hom( T, -) to the above short exact sequence. The modules X, . . . , T~(~)-~X are the base of the wing W(recAjp2X(e (A) - 1 )), the indecomposable direct summands of T are the projective vertices in this wing. Thus Hom(T,Q) = 0 implies Hom(r’X,Q) = 0 for i = O,...,e(A) - 2. Note further, as r-‘X are sincere for all i 2 0, that Hom(X, Q) = 0 shows that Q is regular. If we apply 5-l for 2 < i 5 e (A) to the canonical epimorphism recA)X -+ Q we get epimorphisms re(A)-ZX + r-‘Q. Using Hom(?(‘)-‘X, Q) = 0 we immediatly get Hom(r-‘Q, Q) = 0. But this implies that every non-zero direct summand of Q lies in a component c with e (c”) > e (A ). This shows Q = 0, that is, X generates recA)X. q Proposition 3.2. Let A be wild hereditary with e(A) 2 3 and let C be a regular
component with e(C) = e(A). Then C contains a non-sincere stone of quasilength e(A) - 1. Proof. Choosing a quasi-simple module X in C such that r-‘X is sincere for all i > 0, we know from Lemma 3.1 that X generates recA)X. If r = dimk Hom(X, TAX) and {f;}i=l,..,,r is a k-basis of Hom(X, recA)X) we get a short exact sequence (1;) e(A)X _
O-K-Xr--+r
0
where K denotes the kernel of (J;) = f. Note that K is not regular: If K would be regular, we could apply T/ for all 1 E Z to the above short exact sequence to see that C is sincere, which contradicts Corollary 2.5. Therefore K has an indecomposable preprojective direct summand P. Applying for 0 5 i 5 e (A) -2 the functor Hom( riX, -) to the short exact sequence K ---t X’ +f
0 -
@)X
----f 0
we get the exact sequence ... -L
(six,
jyr) O,(Tix,
‘(z’X,K)
-
‘(z’X,X’)
Zen) -....
For i = 0 the map ( z’X, f ) is surjective by construction, that is, 6 is the zeromap. For 1 5 i < e(A) -2 the map 6 also is zero because of Hom(r’X, retA)X) being zero. = 0 As also Ext ( riX, X’) is zero for i = 0,. . . ,e(A) -2, we get Ext(r’X,K) for i = 0,. . . ,e(A) -2 and thus Hom(P,r’+‘X) = 0 for i = O,...,e(A) -2 which proves the assertion. 0
Regular stones of wild hereditary algebras
29
Proposition 3.3. Let A be wild connected and hereditary with n > 3 simple
A-modules. The following are equivalent: (a) There exist regular stones of quasi-length n - 2 in A-mod. (b) The quiver of A has-up to an admissible change of orientation-the following shape: 0
/\
0
T
o-.
. . . . .
--+0-o
:_
for (CY) r # 0, s # 0, r + s > 2, t = 1, (D) (y)
t>l,r#O,s>l, t > 1, s > 1,
r = 0.
Proof. (b) + (a) If the quiver has the shape (a) then consider the full subquiver
A,_1 = 1 --+2+~~~--tn-2~n-l of &(A). As A has a factor-algebra of type An- t,t all simple modules i = I,..., n - 2 are regular in A-mod. Let A
=
S(i),
B p;@p,_, 0
k
where B is the path-algebra of the above quiver A,_I. As Horn (PI @ P;_ t, W(Y)) = 0, where Y = P1/Pn_l = z;Pz, the whole wing W( z;Pz) in the Auslander-Reiten quiver of B is preserved under the canonical embedding B-mod it A-mod. The simple modules S(i), i = 1,. . . , n - 2, are on the base of this wing W (.r;Pz). Therefore, 7iPz is a regular stone of quasi-length n - 2 sitting on the top of the wing W (zg Pz). Similar arguments work in the case (p) and (y ). (a)+ (b) Suppose there exists a regular component C such that there are n - 1 pairwise orthogonal quasi-simple stones X, rX, . . . , znp2X in C, or equivalently [n - 21X is a regular stone of quasi-length n - 2. By Proposition 3.2, up to r-translation there exists a projective module P with Horn (P, W) = 0 where W = W ( [n - 21X). Moreover, we may assume that A is a one-point extension
with A4 = rad P and thus W is contained in B-mod as a wing. Note that this implies Hom(M, T’X) = 0 for i = 1,..., n-2 by [12, 2.51. Since B has only
30
0. Kerner, F. Lukzs
it - 1 simple modules, W cannot be regular in B-mod, that is W is contained as wing in the preprojective or preinjective component of r(B). Especially the quiver of B is of the following type (up to change of orientation): 1 -Tc2 0:o -
An easy computation
n-2 ***
n-1
-0-o
then gives the assertion.
0
Corollary 3.4. Zf A is hereditary with regular stones of quasi-length quiver & of A has a full subquiver of type A,.
m then the
Proof. The assertion is trivial for A tame or A wild and all stones are quasisimple, so assume A is wild and m > 1. We have m 2 n - 2, where II is the number of simple A-modules and we prove Corollary 3.4 by induction on the number n - m: For n - m = 2 Proposition 3.3 works. If r = n - m > 2 we may assume that m + 1 = e(A), that is m is the maximal quasi-length of regular stones. Then by Proposition 3.2 there exists a regular component C containing a stone X of quasi-length m which is not sincere. We may suppose that A = B [M] is a one-point extension of some hereditary algebra B and X and thus the whole wing W(X) is in B-mod. If W(X) is in the preprojective or preinjective component P(B) of r(B) then clearly the quiver Q (B ) of B contains a full subquiver of type A,, as the orbit graph of P (B ) is the underlying graph of Q (B ). If W (X) is regular in B-mod and B is tame we are done. If W (X ) is regular in B-mod and B is wild, we have e(B) - 1 = m’ 2 m and n(B) = n - 1. Thus we have n(B) -m’ < r therefore by induction the quiver Q(B) has a full subquiver of type Amf, but Q(B) is a full subquiver of Q(A). 0 Remark. By Corollary 3.4 all hereditary algebras whose quiver only have multiple arrows have no regular stones of quasi-length bigger than one.
Acknowledgement Both authors thank L. Unger for helpful and stimulating topic.
discussions
on this
References [ 1] D. Baer, Homological properties of wild hereditary Artin algebras, in: Representation Theory I, Proc. ICRA IV, Lecture Notes in Mathematics, Vol. 1177 (1986) l-12.
Regular stones of wild hereditary algebras [2]
[3] [4] [5] [6] [7]
[8] [9]
[ lo] [ 111 [ 121 [13]
[ 141 [ 151 [ 161 [ 171
31
W. Geigle and H. Lenzing, Perpendicular categories with applications to representations and sheaves, J. Algebra 144 ( 199 1) 273-343. D. Happel and C.M. Ringel, Tilted algebras, Trans. Amer. Math. Sot. 274 (1982) 399-443. M. Hoshino, Modules without self-extensions and Nakayama’s conjecture, Arch. Math. 43 (1984) 493-500. 0. Kerner, Stable components of wild tilted algebras, J. Algebra 142 ( 1991) 37-57. 0. Kerner, Exceptional components of wild hereditary algebras, J. Algebra 152 (1992) 184-206. 0. Kerner and F. Lukas, Regular modules over wild hereditary algebras, in: Proceedings of the Tsukuba International Conference, CMS Conference Proceedings, Vol. 11 (American Mathematical Society, Providence, RI, 199 1) 19 l-208. 0. Kerner and F. Lukas, Elementary modules, Preprint, 1992. F. Lukas, Infinite dimensional modules over wild hereditary algebras, J. London Math. Sot. (2) 44 (1991) 401-419. F. Lukas, Elementare Moduln iiber wilden erblichen Algebren, Dissertation, Heinrich-HeineUniversitit Dusseldorf, 1992. C.M. Ringel, Finite dimensional algebras of wild representation type, Math. Z. 161 (1978) 235-255. C.M. Ringel, Tame Algebras and Integral Quadratic Forms, Lecture Notes in Mathematics, Vol. 1099 (Springer, Berlin, 1984). C.M. Ringel, The regular components of the Auslander-Reiten quiver of a tilted algebra, Chinese Ann. Math. Ser. B. 9 (1) (1988) 1-18. C.M. Ringel, Recent advances in the representation theory of finite dimensional algebras, in: Progress in Mathematics, Vol. 95 (Birkiuser, Boston, MA, 199 1) 141-192. A. Schofield, Generic representations of quivers, to appear. H. Strauss, On the perpendicular category of a partial tilting module, J. Algebra 144 ( 1991) 43-66. L. Unger, On wild tilted algebras which are squids, Arch. Math. 55 (1990) 542-550.
15
Regular stones of wild hereditary algebras Otto Kerner and Frank Lukas Mathematisches Germany
Institut, Heinrich-Heine-Universitiit,
Diisseldorf D-40225 Diisseldorf I,
Communicated by C.A. Weibel Received 2 March 1992 Revised 6 January 1993
Abstract Kemer, 0. and F. Lukas, Regular stones of wild hereditary Applied Algebra 93 ( 1994) 15-3 1.
algebras, Journal of Pure and
If A is a finite-dimensional wild hereditary algebra we study the class of regular components containing stones of quasi-length two, where a stone is a brick without self-extensions. We show that there are only finitely many non-sincere components of this type and, provided the algebra has elementary modules with self-extensions, no sincere component with stones of quasi-length two.
Introduction If A is a finite-dimensional tame hereditary algebra over an algebraically closed field, there exist only finitely many inhomogeneous tubes in the Auslander-Reiten quiver r(A) of A, thus there are only finitely many regular components containing stones, that is, bricks without self-extensions. The situation is totally different if A is wild hereditary. If A is connected with at least three simple modules there always exist infinitely many regular components containing (quasi-simple) stones. Thus we restrict to those regular components which contain stones of quasi-length (at least) two.
Correspondence to: 0. Kerner, Mathematisches Institut, Dusseldorf, UniversitltsstraBe 1, D-40225 Dusseldorf 1, Germany. Email: [email protected].
0022-4049/94/$07.00 @ 1994 SSDI 0022-4049 ()E0090-Q
Heinrich-Heine-Universitat,
Elsevier Science B.V. All rights reserved
16
0. Kerner, F. Lukas
In Section 1 we show that there are only finitely many non-sincere components, that is, components containing a non-sincere module, of this type (Theorem 1.1). In Section 2 we study sincere components. If the algebra A has elementary modules with self-extensions then all stones in sincere components are quasisimple, especially there are only finitely many regular components containing stones of quasi-length two in this case (Theorem 2.8). The notation of elementary modules is introduced in [ 8,101. A regular module E is called elementary if it is not in the middle-term of a short exact sequence 0 --+ U -+ E -+ V + 0 with U and I/ regular and non-zero. If A is wild hereditary with y1simple modules, it was first shown by Hoshino [4] that the quasi-length of regular stones is bounded by y1- 2. In Section 3 we give a complete list of all wild hereditary algebras with regular stones of quasi-length n - 2. This result and the subsequent corollary demonstrate that the class of components studied in this paper may help to characterise the algebras.
Notations The word algebra always denotes a finite-dimensional, unitary, basic algebra over some algebraically closed field k. The letter A normally is reserved for wild hereditary, connected algebras. If A is such an algebra, A-mod denotes the category of finitely generated left A-modules and 72 denotes the category of regular A-modules, that is, the full subcategory of A-mod defined by direct sums of indecomposable regular modules. We call a module X a brick if End(X) is isomorphic to k. A brick without self-extensions (and projective dimension at most 1) is called a stone. The number of pairwise non-isomorphic simple A-modules will be denoted by y1(A). By qA we denote the Tits-form which coincides in the hereditary case with the Euler-quadratic form, that is qA(bX)
= dimk End(X)
- dimk Ext’(X,X).
By r (A) we denote the Auslander-Reiten quiver of A. The vertices of r (A) are isomorphism-classes of indecomposable A-modules; if the context is clear, we will not distinguish between an indecomposable module X and its class
[Xl. By J2 (A) we denote the set of regular components of r (A). If C is a regular component-it is of type ZA, by [ 11 ]-and all indecomposable modules in C are sincere, we call C a sincere component. An indecomposable regular module X is called r-sincere, if z’X is sincere for all integers i. If U is a quasi-simple module and r a natural number, we denote by U (Y) ( [r ] U )the indecomposable regular module with quasi-length Y and quasi-socle (quasi-top) U.
Regular stones of wild hereditary algebras
17
If X is an idecomposable regular module, say X = [m] U for U quasisimple, we denote by W (X ), the wing of X (of length m ), the mesh-complete full subquiver of r (A ) defined by the vertices 7’ ( [s] U) with 1 5 s 5 m and 0 5 Y 5 m - s. The wing W(X) is called a standard wing if rada (M, N) = 0 for all modules M, N in W(X). By [ 61 this is the case exactly if X is a brick. From the concept of perpendicular categories we use the following result (see [ 21, [ 15 ] or [ 161): If A is an algebra and X a quasi-simple regular stone, then the right perpendicular category XL, defined by the objects {Y 1 Horn (X, Y) = Ext’ (X, Y) = 0) is an abelian subcategory of A-mod which is equivalent to a module category B-mod. We have B = End(Z ), where 2 is defined by Bongartz’s construction such that X @ 2 is a square-free tilting module, and B is wild, connected and hereditary. Sometimes we write (X, Y) ( ’ (X, Y ), respectively) instead of HomA (X, Y) (Exta (X, Y ), respectively). The Auslander-Reiten translations will be denoted by r andr-; to emphasise the algebra, we sometimes add a subscript, for instance ZA,r;. In general we follow the notations used in [ 12 1. If C is a regular component and X E C is quasi-simple, then we define e(C) as the smallest natural number such that Ext’(X(e(C)),X(e(C))) # 0. By e(A) we understand max{e(C) ] C regular}. Note that e (A ) 5 n - 1, where n is the number of simple A-modules, see [ 41. is a maximal family of Note moreover that for e(C) > 1, (X, . . . , P(c)-lX) pairwise orthogonal quasi-simple stones in C, as Hom(X, &)X) # 0, see [ 51.
1. Non-sincere components containing stones The main result of this part is the following: Theorem 1.1. If A is connected wild hereditary then there exist only finitely
many non-sincere regular components containing stones of quasi-length at least two. If C is a regular component, X a quasi-simple module in C then it was shown in [ 7, Proposition 1.11, that the module X(r) for some r > 1 is a stone if and only if X (r + 1) is a brick and X(r - 1) is a stone. So X(r) is a stone for some r 2 2 if and only if X(2) is a stone. If X is a quasi-simple regular stone in A-mod then there is an isomorphism of vector spaces Exti(X(2),X(2))
g HomA(X,r2X),
see [ 61. We will use this simple fact freely in the sequel and will express the dimension of Ext1(X(2),X(2)) frequently by Hom(X,r2X). Moreover, as X(2) is a brick, we have dimHom(X,r2X) = 1 - q(hX(2)). If C is a non-sincere component then there exists an indecomposable module X E C and a vertex CL)E &o, where Q = (‘20, Q1 ) is the (ordinary) quiver of
0. Kerner, F. Lukus
18
A such that Hom(P (o), X) = 0. Moreover, without loss of generality we may assume that w is a source of Q, that is A is a one-point extension A = B [M] with M = rad P (CO).Further, if X is in B-mod, then the whole wing W(X) is in B-mod. Especially a non-sincere component always has non-sincere quasisimple modules. In order to prove Theorem 1.1 we may restrict to the case where B is representation-infinite. Lemma 1.2. If A = B[M]
is a one-point extension and X E B-ind is not projective, then we have a short exact sequence 0-
ZBX -
7,&X -
with 1 = dimk Hom(M,
I(0)’ rgX)
0,
= dimk Ext(l(w),
r,+X)
= dimk Hom(P(w),rAX). Proof. The existence
of the short exact sequence with
1 = dimk Hom(M,
reX)
= dimk Hom(P(m),
TAX)
was shown in [ 12, 2.51. Applying to the short exact sequence
O-M-P(u)-I(CO)-0 the functor Ext(l(cc,),
HomA (-, Y ) for Y E B-mod Y).
arbitrary
we get Horn (M, Y) G
0
Lemma 1.3. If A = B [M] is wild hereditary, if X is a stone in B-mod is regular and quasi-simple in A-mod, then we have
9A(&[2]X)
which
= 1 + (d&lrBX,d&X) - dimk Horn (M, X ) . dimk Horn (M, zg X ) =
ql3(dim(X CETBX) )
Proof. As A is a one-point extension of B, projective B-modules are projective in A-mod; therefore X is not projective and ZAX, r&C are non-zero and by Lemma 1.2 we have for 1 = dimk Horn (M, TBX),
0Using &[
TBx 21X = &X
z,z,x -
z(m)’ -
0.
+ dim TAX one checks the assertion.
0
If A = B [M] is wild hereditary and B is representation-infinite, it was shown in [ 131 that almost all preprojective indecomposable B-modules are regular in A-mod. Unger additionally proved in [ 171 that almost all indecomposable
Regular stones of wild hereditary algebras
preprojective more is true:
and preinjective
B-modules
19
are quasi-simple
in A-mod.
Even
Proposition 1.4. Let A = B [M] be connected, wild and hereditary and B representation-infinite. If N is a natural number then there exist only jinitely many indecomposable preprojective or preinjective B-modules X such that dimk Hom(X, z:X) 5 N. Proof. By [ 13 ] and [ 17 ] it is enough to consider simple regular in A-mod. Using dimkHom(X,riX)
= -q(dim[2]X)
we can now apply Lemma (dim~~X,dimX)
the case where X is quasi-
+ 1.
1.3. Note that here =
d imk Horn
(7pY,
where rx + 1 is the number of middle-terms
X)
=
rx,
of the Auslander-Reiten
sequence
0-7~x-Y-x--+o.
Especially
rx is constant
on the re-orbit
of X and c = max{rx}
exists. So we
get qA(b[2]X)
= 1 + rx - dimkHom(M,X)
.dimkHom(M,rX).
As M is a projective B-module, say M = eiEe6 P (i )=I with cri 2 0 we have dimk Hom(M, Y) = Ci,_06~iyi where (yi)ieo;, = d&Y for any Bmodule Y. Thus there exist only finitely many indecomposable preinjective or preprojective B-modules X such that dimkHom(M,X) see [ 13, Lemma
.dimkHom(M,rsX)
11, combined
5 c + N,
with [ 1, Proposition
2.21.
0
Corollary 1.5. Zf A = B [M] is a connected, wild hereditary algebra, then there are only finitely many regular components in r(A) containing stones of quasilength two and preprojective or preinjective B-modules simultaneously. 0 If C ia a non-sincere component such that [ r]X is a stone for some r > I and some quasi-simple module X and [s] X is non-sincere for some s 2 1, we may suppose, as mentioned above, that A = B [M] is a one-point extension of a hereditary algebra B and [SIX E B-mod. If s > 1 the whole wing W ( [s] X) is contained as a wing in B-mod. Especially we get 7iX = 7&X for i = 1,. ..,s - 1 and thus by Lemma 1.2 W( [SIX) is not sincere in B-mod, as HomB (M, SBX ) = 0. The wing W ( [SIX) is either contained in the
20
0. Kerner, F. Lukas
preprojective or preinjective component of r (B ), and in this case [SIX is a stone and the quiver Q (B ) of B necessarily has the shape
or W ( [s] X ) is contained in a regular, non-sincere component 23 of r (B ) (provided B is representation-infinite) containing stones of quasi-length at least min{ r, s}. Moreover, we may assume that B has only finitely many non-sincere regular components containing stones of quasi-length bigger than one: This is trivial in case B is tame or is wild with two simple modules. Therefore we can use induction on the number of simple modules. To continue the proof of Theorem 1.1, by Corollary 1.5 we only have to consider now the following case: A = B [M] and C is a regular component containing stones of quasi-length two and additionally a quasi-simple module X such that X E B-mod and X is regular in B-mod. We have to distinguish two cases: Either X is quasi-simple in B-mod, which happens for example if [S ] X for s > 1 is in B-mod, since in that case the whole wing W( [SIX) is in B-mod, or X has quasi-length bigger than one in B-mod. Proposition 1.6. Let A = B [M] be connected wild hereditary and B represen-
tation infinite. If C is a non-sincere component containing stones of quasi-length r > 1 and X is a quasi-simple regular B-module which is contained in C, then C contains modules of quasi-length r which are regular B-modules. Proof. It is straightforward to see that X is quasi-simple in C. By induction on the quasi-length we may suppose that for some s with 1 5 s < r the stone [SIX is in B-mod. This especially implies that for 1 5 i < s - 1 we have r;X = T&X. If Y + [SIX is the irreducible epimorphism in B-mod, then we have CjiIJlY = d&l[s]X
+ dim&r,
whereas in A-mod we get d&[s
By Lemma
+
11X =
dim[slX + dimfJ
1.2 we have
dimT>X
= dim&x
+ 1. dimI(
Regular stones of wild hereditary algebras
21
where 1 = dimk Horn (M, r”,X). As s < Y holds, we have (by Lemma 1 = q(&[s = q(dimY)
+ 11X) = q(dimY
1.2)
+ I.&l(o))
+ I2 + 1 .(dimZ(cc,),dim([s]Xe~$X))
= q(dim Y) - 1. dimk Hom(M,
[SIX).
Thus we have q (dim Y) = 1, that is, Y is a stone, as [SIX is a stone and additionally 1 = 0 or Hom(M, [SIX) = 0. In case 1 = 0 holds, we get r>X = r;X and thus [s + 1 ]X is in B-mod. If Horn (M, [s] X) = 0, then the Auslander-Reiten sequence in B-mod o-
[SIX-
[S-l]X@ST,Y-r-
B [SIX -
0
is an Auslander-Reiten sequence in A-mod and consequently 0 in C with T,Y = [S + l](r-X).
the stone r;Y
is
Notice that this result also means that the components V in B-mod containing X has stones of quasi-length at least Y. Moreover, if the wing W( [r]X) in C is contained in 27, then we have ZBX = r,+X, that is, Hom(M, reX) = 0 and thus D is a non-sincere component in B-mod. By induction there exist only finitely many components 2) of this type in r (B ) . As 2) has only finitely many non-sincere modules for each of these finitely many components we can find only finitely many non-sincere wings W of length r > 1. So there exist only finitely many components C in r(A) as described in Proposition 1.6. Thus there remains only the following case: Lemma 1.7. Zf A = B [M] is wild hereditary and B representation infinite, if C is a non-sincere regular component in r(A) containing stones of quasi-length two, if X is a quasi-simple module in C such that (1) X is a regular B-module. (2) X has quasi-length 1 > 1 considered as B-module. Then we have: In the Auslander-Reiten component 2) c Z(B), containing X there are stones of quasi-length 1 + 1 and the wing W(X) is non-sincere. Proof. In B-mod we have
where TBX is a submodule of U and I’ a submodule of X. For dimk Hom(M, T&Y) where i = 0,l we get in A-mod by Lemma 1.3 1 =
qA(b[2]X)
=
li =
qB(dim(X@TBX))-lc,.l1.
Notice that Ie and 11 both are positive as X is quasi-simple in C but not in 27. Notice further that the wings W ( U) and W ( TB U ) in D are standard
22
0. Kerner, F. Lukas
wings by [5] and thus we have Hom(U,V) Ext( V, Y) = 0. So we get qB(&l(XcBTeX))=
qs(dim(U6?
= Hom(V,U)
V))
= qs(dimU)
= Ext(V,:)
=
+ qs(dimV)
= 1 + %(bU).
Thus 1 = 1 + qB (d& U) - lo . II implies qs (d& U) = 1, that is, U is a stone and rc = Ii = 1. If Y is the quasi-top of X in B-mod, then we have dim X = dim &I; 7kY. Therefore, we get I-1
dimk Hom(M,
X) = 1 = c
dimk Hom(M,
r;Y),
i=O
which shows that the wing W(X) in D is not sincere. The same argument as above then proves that there are only finitely many components C in r (A ) of this type, which finishes the proof of Theorem 1.1. 0
2. Sincere components containing stones Let us remind that a connected wild hereditary algebra with more than two simple modules always has infinitely many non-sincere components containing stones, see for instance [ 13,17 1. Proposition 2.1. Zf A is connected, wild hereditary with at least three simple
modules then there are infinitely many sincere components containing stones. Proof. Let C be a non-sincere regular component containing stones such that the number of non-sincere quasi-simple modules is minimal. Take X E C nonsincere quasi-simple. Now we can choose in Xl an infinite family (Xi)i,, of stones such that (a) Xi is sincere in Xl, (b) dimk Hom(Xi, [21X) > (dimk X)2 + 1, (c) Xi are regular in A-mod (e.g. infinitely many Xl-preprojective modules satisfy these conditions as [21X is regular in Xl). As Hom(Xi, [21X) g Hom(Xi,X), by condition (b) we get Hom(Xi,X) # 0. As each nonzero map f : Xi + X is injective or surjective by [3, 4.11 again by (b) each map has to be surjective. As Hom(Xi, zX) = 0, we get by Unger’s Lemma [ 17, 1.3 ] that the kernel K of any surjective map f : Xi + X is indecomposable with self-extensions, that is K is regular and thus also the maps rrf : Z’Xi --t z”X are surjective for all r E Z. Especially rrXi is sincere if z”X is. If E is one of the indecomposable injective A-modules with Hom(X, E) = 0 then E is injective in Xl and therefore we have, since all Xi
Regular stones of wild hereditary algebras
23
are sincere in Xl, Hom(X,, E) # 0. Together with the surjection f : Xi + X this implies that all Xi are sincere in A-mod and therefore in the r-orbit of Xi there are less non-sincere modules than in the r-orbit of X. Notice that a regular component contains only finitely many vertices, corresponding to indecomposable modules in X l. Notice further that by Theorem 1.1 only finitely many of the Xi can be vertices of quasi-length bigger than one in non-sincere components. Thus by the minimality condition of C we have infinitely many sincere components containing stones. 0 Example.
Let A be the path-algebra of the quiver O=O+O, and let B be the Kronecker-algebra corresponding to the quiver 0:. . By [ 111 an indecomposable A-module M(n) of dimension (n + 1, yt, 0) is regular if and only if n 2 5 holds. Using [ 6, 3.31 we see that r’M( n) is sincere for all i # 0 if n > 5. Consider the tilting module M(n) $ N(n) @ M(n + 1) for n 2 4 and &N(n) = (n(n + l),n2, I), see [14]. Then M(n)@N(n) is a preprojective tilting module in M (n + 1 )I, M (n + 1) J_ E K, - mod, where K, is the path-algebra of the quiver .x0 (n arrows) and thus for example the modules rMcn + , Ji M (n ) with dimension-vector (n 2 5 ) n.d&lN(n)-dimhI
= ((A
l)(n
+ 1)&z-
l)n,n)
are r-sincere quasi-simple stones. Let us now consider a more restricted condition: Again we assume that C is a regular component, X E C is quasi-simple such that X (r + 1) with r > 1 is a brick with self-extensions. Additionally we demand that C is a sincere component. Note that in this case X 1 is contained in R. The next aim is to show that this number r is relatively small. Using again Unger’s lemma [ 17, 1.31 the following was shown in [ lo]: Lemma
2.2. Let X and Y be non-isomorphic regular stones with Horn (X, Y) # 0 but Horn (X, 7Y) = 0. If f : X -+ Y is non-zero and X or Y is T-sincere we have: Either f is injective and the cokernel is indecomposable regular or f is surjective with indecomposable regular kernel.
homomorphismus f :X + Y is injective or surjective if Ext (Y, X) = 0. If 2 denotes the cokernel of f in the first and the kernel in the second case, it was shown in [ 17, 1.31 that Proof. It was shown in [ 3, 4.11 that a non-zero
dimkHom(X,
Y) = dimkExt’(Z,Z)
Assume now Y is r-sincere. 0-X-Y-Z-0 o--+z-x-Y~o
+ 1.
We have to consider the two short exact sequences or
24
0. Kerner, F. Lukas
corresponding and get O=
to f injective
or surjective.
We apply the functor
(Z,Y)--+(Y,Y)-(X,Y)+‘(Z,Y)-0 0+
Assume Z is dimk Hom(X, Y) ular modules are preinjective. The
not
(Y, Y) regular.
Y)
or
(X, Y) --f Then
Hom(-,
Z
is
(Z, Y) -
0.
a
and
stone,
consequently factors of regeither regular or preinjective, in the first case 2 has to be condition dimk Horn (X, Y) = 1 then implies = 1 holds by [ 17, 1.3 1. As indecomposable
0 = Ext’(Z,
Y) g DHom(Y,zZ)
that is, Y is not r-sincere. In the second case Z is preprojective and dimk Hom(X, Y) = 1 implies Hom(Z, Y) = 0 which again would mean Y is not z-sincere. If X is r-sincere, we apply the duality D = Homk (-, k) : A-mod + and consider
mod-A
Df : DY + DX instead of f.
0
The following lemma is almost trivial. As it will be used frequently, formulate it explicitely. The proof is straightforward.
we will
Lemma 2.3. (a) Zf X and Y are regular, f :X + Y is a homomorphism and f(X) is a submodule of some module Y’ such that Y/f(X) is regular, then Y’/f (X) is regular. (b) Zf X is an indecomposable regular module and f : X -+ r”X is an injective homomorphism for some n > 0, then the cokernel z”X/ f (X) is not regular. 0 Theorem 2.4. Let C be a regular sincere component, X E C quasi-simple such that X (r + 1) for r > 1 is a brick with self-extensions. (a) There exists a module S, simple in Xl, with 0 f Horn (S, r-‘+ ‘X ). (b) S is quasi-simple in A-mod and S, T,& . . . , zT-‘S are pairwise orthogonal. Proof. One quickly checks that the modules z --r+ 1x,
. ..)
7-x,
7*x,
. . .)
7’X
are in Xl. Thus there exists a XL-simple module S with Hom(S, 7-‘+‘X) # 0 and we may consider S as submodule of 7P’X. Since 7 is left exact, 7’S then is a submodule of T-‘+~+~X for all i > 0. Of course the simple objects of XL are quasi-simple in A-mod.
Regular stones of wild hereditary algebras
25
Clearly we have Horn (T’S, S) = 0 for all i > 0, as S is a quasi-simple stone, see [ 51. Assume there exists 1 < i 5 2r - 1 with 0 # f : S + 7’s. For i = r - 1 A X. By Lemma 2.2 the map f o I is injective or we then get f o I : S f, P’S surjective. If it is surjective, I is an isomorphism but Hom(r-‘+‘X, X) = 0. Therefore f o I is injective, that is, f is injective. The cokernel Q of f is a submodule of the cokernel Q’ of f 0 I which is regular by Lemma 2.2. So Q also is regular by Lemma 2.3(a), a contradiction to Lemma 2.3 (b). Assume secondly for 1 5 i 5 2r - 1 with i # r - 1, i # r there is a non-zero map f : S + s’S. Now z- r+l+iX is in XI and therefore the composition j- 0 l : s f, z’s A z- r+l+iX is injective as S is simple in Xl, the cokernel of f o I is in X’ and therefore regular. So again f would be injective with regular cokernel. Finally, for i = r a non-zero map f :S + T’S would produce a non-zero map f 0 2 : S + T’S L) TX, which contradicts to the condition Ext(X,S) = 0. 0 If C is a regular component we denote by e (C ) the smallest natural number such that the modules with quasi-length e(C) in C have self-extensions. So C contains stones of quasi-length r 2 1 if and only if e (C ) 2 r + 1. Notice further that e(C) is bounded by n - 1, where n denotes the number of simple A-modules, see [4]. Thus max{e(C) 1C E Q(A)} exists, we denote it by e(A). Theorem 2.4 then says that for a sincere component C with e(C) > 2 the module S is in a component D with e(D) 2 2(e(C) - 1). Corollary 2.5. If C is a sincere regular component, then we have
e(A) e(C) 5 - 2 + 1. Proof. For e (C) > 2 this follows from the above theorem. trivially for e (C ) 5 2. 0
The assertion holds
In [ 8,101, the notion of an elementary module was introduced: A regular module E # 0 is called elementary if it satisfies the two equivalent conditions: (a) If R is regular, if f E Horn (R, E) \ {0}, then the cokernel of f is preinjective. (b) If R is regular, if g E Hom(E, R) \ {0}, then the kernel off is preprojective. Notice that elementary modules always exist for representation-infinite hereditary algebras, that their ring of endomorphisms is trivial and that with E also T’E is elementary for all i E Z. Notice further that elementary modules which are stones in addition are not r-sincere. Notice tinally that for an elementary module E, for regular modules X, Y and non-zero maps f E HomA (X, E), g E HomA (E, Y) the composition f g is non-zero, see [8,10].
0. Kerner,
26
F. Lukas
Proposition 2.6. Let X be an indecomposable r-sincere Hom( X, YX) = 0 for some r 2 2. Let E be an elementary there exists a number n E Z such that for Y = z”E we obtain: (a) Y, . . . ,5 -(r-2)y E XI, (b) Y,...,z -(r-2)Y are regular and simple in Xl.
module module.
with Then
Proof. (a) Choose an integer n such that the elementary module Y = FE satisfies the condition Horn (X, r-‘Y) = 0 for all i 2 0, but Hom(X, 7Y) # 0. So for the assertion (a) it suffices to show that Ext (X, 5-i Y) = 0 for 0 5 i 5 r - 2, which is equivalent to Horn(zY,~‘+~X) = 0 for 0 5 i < r - 2, via the Auslander-Reiten formula. Take f E Hom(X, 7Y) \ (0) and suppose there exists g E Hom(rY, 7 i+2X) \ (0) for some i with 0 5 r 5 r - 2. As the module 7Y is elementary, the composition f g E Hom(X, T~+~X) is non-zero. But, as X is z-sincere, Hom(X, 7’X) = 0 implies by [6, 2.2 and 2.31 that Horn (X, 7jX) = 0 also for 1 5 j < r. Therefore, Ext (X,7-‘Y) = 0, that is, 7-‘Y E Yl for 0 5 i < r - 2. (b) If 7-‘Y E Xl is not simple in Xl, then there exists a short exact sequence 0 -
u
-
7-’
‘Y+V+O
with U, I’ E Xl and non-zero. But, as X is r-sincere, the objects of Xl are all regular in A-mod. Since 7-‘Y is elementary in A-mod, there cannot exist such a short exact sequence. As Y, . . . , T-(‘-~) Y are all in Xl we have 7,‘Y = 7;: Y for 0 5 j 5 r - 2, so it suffices to show that any of the 7PY is regular in Xl. Let 1 > r - 1 be minimal with Horn (7-‘Y, 7X) # 0. Then the modules # 0. As 72x is Y>***>7 -(r-‘)Y are in Xl and we have Horn (7 -(l-‘)Y,z2X) regular in Xl, 7 -crP1)Y is not preinjective in Xl. Also 7-X is regular in XI, indeed .r2X = 7g1 (7-X), and H om( 7-X, Y) # 0 implies Y not preprojective in Xl. Therefore, all the modules 7-‘Y are regular in Xl. 0 If the algebra A has an elementary module E with self-extensions and X is as above, then Y would be a simple module with self-extensions in X1, but Xl is equivalent to C-mod for some finite-dimensional hereditary algebra C, a contradiction. So we obtain the following corollary: Corollary 2.7. Let A be a wild hereditary algebra such that A-mod contains an elementary module Y with Ext (Y, Y) # 0. (i) Let C be a regular component such that X(r) is a stone for some r 2 2 and a quasi-simple module X E C. Then X (r - 1) is not z-sincere. (ii) Especially every stone in a sincere component is quasi-simple.
Regular stones of wild hereditary algebras
27
Let us mention finally that most hereditary algebras have elementary modules with self-extensions. But for example the path-algebras of the quivers E:6, E.7, ,?s only have elementary stones, see [ 81. As main result of this part of the paper we can summarise: Theorem 2.8. Let A be connected wild hereditary with self-extending elementary
modules. Then there are only finitely many regular components containing stones of quasi-length bigger than one. 0
3. Algebras having components with stones of quasi-length n(A) - 2 It is well known and was first tame case also for wild hereditary bounded by n (A) - 2, where n = Straul3 (see [ 161) had remarked 0
-
o-o-
.
.
.
shown by Hoshino [4] that similarly to the algebras the quasi-length of regular stones is n (A) denotes the number of simple modules. that the path-algebra of the quiver -0-o
-
0
always has stones of quasi-length n(A) - 2. The aim of this part is to show that he gave a typical example. It was shown in [ 91 that for a regular (indecomposable) module R there exists a surjective homomorphism zFR + R for some s > 0 and thus R is generated by zPR. Recall that e(C), where C is a regular component, is the smallest number 1 such that indecomposable modules in C of quasi-length 1 have self-extensions and that
e(A) = max{e(C) If C is a regular component
1C E Q(A)} 5 n(A) - 1. with e(C) = e(A), we obtain:
Lemma 3.1. Let A be wild hereditary with e(A) 2 3 and let C be a regular
component with e(C) = e(A). Zf X is quasi-simple in C with z+‘X sincere for all m -> 0, then re(A)X is generated by X. Proof. As the modules re(A)-2X (i) for i = 1,. . . , e (A) - 1 are stones and the wing W(r e(A)-lX (e (A) ) ) is a standard-wing containing the modules ftAjm2X(i), the direct sum T = @f~I’-’ zecAjP2X(i) is a partial tilting module (see [ 161). Denoting by U the trace of T in recA)X, that is U = CIm f and the cokernel of the inclusion U it T~(~)_X by with f E Hom(T,r e(A)X) Q, we get the short exact sequence o-u-7
dA)_,y
-Q-O.
0. Kerner,
28
F. Lukas
The standardness condition of the wing W (r e(A)-lX(e(A))) immediately implies that U also is the trace of X in r e(A)X. Since T is a partial tilting module we have Horn ( T, Q ) = 0; this can be checked for example directly by applying the functor Hom( T, -) to the above short exact sequence. The modules X, . . . , T~(~)-~X are the base of the wing W(recAjp2X(e (A) - 1 )), the indecomposable direct summands of T are the projective vertices in this wing. Thus Hom(T,Q) = 0 implies Hom(r’X,Q) = 0 for i = O,...,e(A) - 2. Note further, as r-‘X are sincere for all i 2 0, that Hom(X, Q) = 0 shows that Q is regular. If we apply 5-l for 2 < i 5 e (A) to the canonical epimorphism recA)X -+ Q we get epimorphisms re(A)-ZX + r-‘Q. Using Hom(?(‘)-‘X, Q) = 0 we immediatly get Hom(r-‘Q, Q) = 0. But this implies that every non-zero direct summand of Q lies in a component c with e (c”) > e (A ). This shows Q = 0, that is, X generates recA)X. q Proposition 3.2. Let A be wild hereditary with e(A) 2 3 and let C be a regular
component with e(C) = e(A). Then C contains a non-sincere stone of quasilength e(A) - 1. Proof. Choosing a quasi-simple module X in C such that r-‘X is sincere for all i > 0, we know from Lemma 3.1 that X generates recA)X. If r = dimk Hom(X, TAX) and {f;}i=l,..,,r is a k-basis of Hom(X, recA)X) we get a short exact sequence (1;) e(A)X _
O-K-Xr--+r
0
where K denotes the kernel of (J;) = f. Note that K is not regular: If K would be regular, we could apply T/ for all 1 E Z to the above short exact sequence to see that C is sincere, which contradicts Corollary 2.5. Therefore K has an indecomposable preprojective direct summand P. Applying for 0 5 i 5 e (A) -2 the functor Hom( riX, -) to the short exact sequence K ---t X’ +f
0 -
@)X
----f 0
we get the exact sequence ... -L
(six,
jyr) O,(Tix,
‘(z’X,K)
-
‘(z’X,X’)
Zen) -....
For i = 0 the map ( z’X, f ) is surjective by construction, that is, 6 is the zeromap. For 1 5 i < e(A) -2 the map 6 also is zero because of Hom(r’X, retA)X) being zero. = 0 As also Ext ( riX, X’) is zero for i = 0,. . . ,e(A) -2, we get Ext(r’X,K) for i = 0,. . . ,e(A) -2 and thus Hom(P,r’+‘X) = 0 for i = O,...,e(A) -2 which proves the assertion. 0
Regular stones of wild hereditary algebras
29
Proposition 3.3. Let A be wild connected and hereditary with n > 3 simple
A-modules. The following are equivalent: (a) There exist regular stones of quasi-length n - 2 in A-mod. (b) The quiver of A has-up to an admissible change of orientation-the following shape: 0
/\
0
T
o-.
. . . . .
--+0-o
:_
for (CY) r # 0, s # 0, r + s > 2, t = 1, (D) (y)
t>l,r#O,s>l, t > 1, s > 1,
r = 0.
Proof. (b) + (a) If the quiver has the shape (a) then consider the full subquiver
A,_1 = 1 --+2+~~~--tn-2~n-l of &(A). As A has a factor-algebra of type An- t,t all simple modules i = I,..., n - 2 are regular in A-mod. Let A
=
S(i),
B p;@p,_, 0
k
where B is the path-algebra of the above quiver A,_I. As Horn (PI @ P;_ t, W(Y)) = 0, where Y = P1/Pn_l = z;Pz, the whole wing W( z;Pz) in the Auslander-Reiten quiver of B is preserved under the canonical embedding B-mod it A-mod. The simple modules S(i), i = 1,. . . , n - 2, are on the base of this wing W (.r;Pz). Therefore, 7iPz is a regular stone of quasi-length n - 2 sitting on the top of the wing W (zg Pz). Similar arguments work in the case (p) and (y ). (a)+ (b) Suppose there exists a regular component C such that there are n - 1 pairwise orthogonal quasi-simple stones X, rX, . . . , znp2X in C, or equivalently [n - 21X is a regular stone of quasi-length n - 2. By Proposition 3.2, up to r-translation there exists a projective module P with Horn (P, W) = 0 where W = W ( [n - 21X). Moreover, we may assume that A is a one-point extension
with A4 = rad P and thus W is contained in B-mod as a wing. Note that this implies Hom(M, T’X) = 0 for i = 1,..., n-2 by [12, 2.51. Since B has only
30
0. Kerner, F. Lukzs
it - 1 simple modules, W cannot be regular in B-mod, that is W is contained as wing in the preprojective or preinjective component of r(B). Especially the quiver of B is of the following type (up to change of orientation): 1 -Tc2 0:o -
An easy computation
n-2 ***
n-1
-0-o
then gives the assertion.
0
Corollary 3.4. Zf A is hereditary with regular stones of quasi-length quiver & of A has a full subquiver of type A,.
m then the
Proof. The assertion is trivial for A tame or A wild and all stones are quasisimple, so assume A is wild and m > 1. We have m 2 n - 2, where II is the number of simple A-modules and we prove Corollary 3.4 by induction on the number n - m: For n - m = 2 Proposition 3.3 works. If r = n - m > 2 we may assume that m + 1 = e(A), that is m is the maximal quasi-length of regular stones. Then by Proposition 3.2 there exists a regular component C containing a stone X of quasi-length m which is not sincere. We may suppose that A = B [M] is a one-point extension of some hereditary algebra B and X and thus the whole wing W(X) is in B-mod. If W(X) is in the preprojective or preinjective component P(B) of r(B) then clearly the quiver Q (B ) of B contains a full subquiver of type A,, as the orbit graph of P (B ) is the underlying graph of Q (B ). If W (X) is regular in B-mod and B is tame we are done. If W (X ) is regular in B-mod and B is wild, we have e(B) - 1 = m’ 2 m and n(B) = n - 1. Thus we have n(B) -m’ < r therefore by induction the quiver Q(B) has a full subquiver of type Amf, but Q(B) is a full subquiver of Q(A). 0 Remark. By Corollary 3.4 all hereditary algebras whose quiver only have multiple arrows have no regular stones of quasi-length bigger than one.
Acknowledgement Both authors thank L. Unger for helpful and stimulating topic.
discussions
on this
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Regular stones of wild hereditary algebras [2]
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[8] [9]
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31
W. Geigle and H. Lenzing, Perpendicular categories with applications to representations and sheaves, J. Algebra 144 ( 199 1) 273-343. D. Happel and C.M. Ringel, Tilted algebras, Trans. Amer. Math. Sot. 274 (1982) 399-443. M. Hoshino, Modules without self-extensions and Nakayama’s conjecture, Arch. Math. 43 (1984) 493-500. 0. Kerner, Stable components of wild tilted algebras, J. Algebra 142 ( 1991) 37-57. 0. Kerner, Exceptional components of wild hereditary algebras, J. Algebra 152 (1992) 184-206. 0. Kerner and F. Lukas, Regular modules over wild hereditary algebras, in: Proceedings of the Tsukuba International Conference, CMS Conference Proceedings, Vol. 11 (American Mathematical Society, Providence, RI, 199 1) 19 l-208. 0. Kerner and F. Lukas, Elementary modules, Preprint, 1992. F. Lukas, Infinite dimensional modules over wild hereditary algebras, J. London Math. Sot. (2) 44 (1991) 401-419. F. Lukas, Elementare Moduln iiber wilden erblichen Algebren, Dissertation, Heinrich-HeineUniversitit Dusseldorf, 1992. C.M. Ringel, Finite dimensional algebras of wild representation type, Math. Z. 161 (1978) 235-255. C.M. Ringel, Tame Algebras and Integral Quadratic Forms, Lecture Notes in Mathematics, Vol. 1099 (Springer, Berlin, 1984). C.M. Ringel, The regular components of the Auslander-Reiten quiver of a tilted algebra, Chinese Ann. Math. Ser. B. 9 (1) (1988) 1-18. C.M. Ringel, Recent advances in the representation theory of finite dimensional algebras, in: Progress in Mathematics, Vol. 95 (Birkiuser, Boston, MA, 199 1) 141-192. A. Schofield, Generic representations of quivers, to appear. H. Strauss, On the perpendicular category of a partial tilting module, J. Algebra 144 ( 1991) 43-66. L. Unger, On wild tilted algebras which are squids, Arch. Math. 55 (1990) 542-550.