C l a u s M i c h a e l R i n g e l
L e t k be a ( c o m m u t a t i v e ) field, a n d k < X 1 , . . . , X n >
the free a s s o c i a t i v e a l g e b r a in n (non--commutlng) v a r i a b l e s . D e n o t e by M i the i d e a l of k < X 1 , . . . , X n >
g e n e r a t e d by all m o n o m i a l s of degree i F o r a n y k--algebra A , let A ~ be the c a t e g o r y of all A--modules w h i c h are f i n i t e d i m e n s i o n a l as k--vector spaces. If I is a
t w o s l d e d ideal of k < X 1 , ~ , then f o r A = k ( X 1 , . . , X n > / I , the c a t e g o r y A ~ is just the c a t e g o r y of all (finite d i m e n - s i o n a l ) v e c t o r spaces e n d o w e d w i t h n e n d e m o r p h i s m s w h i c h s a t i s f y the r e l a t i o n s e x p r e s s e d by the e l e m e n t s o f I .
The k--algebra A is c a l l e d local, p r o v i d e d A = k-1 + r a d A , w h e r e r a d A is the J a c c b s o n r a d i c a l of A . If A is a l o c a l k--algebra, we will c o n s i d e r a l s o its c o m p l e t i o n A = l i m A / ( r a d A) n . T h e r e is a c a n o n i c a l r i n g h o m o m o r p h l s m A ~ > A , and A is said to be c o m p l e t e in case this h o m o m o r p h i s m is a n i s o m o r p h i s m . S i n c e ob-- v i o u s l y e v e r y o b j e c t in A N is a n n i h i l a t e d b y some p o w e r
(rad A) n , the c a n o n i c a l h o m o m o r p h i s m A --> A i n d u c e s a n i s o m o r p h i s m of the c a t e g o r i e s A N a n d ~ m . Thus, i n o r d e r to c o n s i d e r the b e h a v i o u r of A ~ for ~ l o c a l a l g e b r a s A , we m a y r e s t r i c t to the case w h e r e A is c o m p l e t e .
The k--algebra A is said to be w i l d (or to be o f w i l d r e p r e s e n t a t i o n type) p r o v i d e d t h e r e is a f u l l and
exact subcategory of A ~ which is representation equivalent to the category k<X,Y>~ " The reason for calling it wild, is that there seems to be no hope to expect a complete classification of the indecomposable objects in k<X,Y>~ ' since for any finitely generated k-algebra B , there is a full and exact embedding of B~ into k<X,Y>~ " On the other hand, the algebra A is said to be tame (or to be of tame representation type), if there exists a complete classification of the indecomposable objects in A~ ' and if there are not only finitely many indecomposables.
In order to distinguish the complete local algebras according to there representation type, we have to find the smallest possible wild algebras (that is, wild alge- bras for which all proper residue algebras are tame or of finite representation type), and the largest possible tame algebras (that is, tame algebras w h i c h do not occur as proper residue algebras of other tame algebras).
(1.1) We will have to consider several algebras which we want to introduce now. First, we mention
(a) k<X,Y,Z>/M2,
the local algebra of dimension 4 wlth radical square zero. Next, we single out certain residue algebras
k<X,Y>/I of k<X,Y> of dimension 5 , namely those with I the twosided ideal generated by the elements
(b) X 2, XY, y2X, y3 (b ~ X 2, YX, XY 2, y3 ;
(c) X 2, XY -- ~YX , y2X, y3 with ~ # 0 ; and
(d) X 2 - y 2 Y XAlso, we are i n t e r e s t e d in a n o t h e r set of local algebras k < X , Y > / I , where the ideal I is g e n e r a t e d by just two elements:
(I) YX , XY
(2) YX -- X n , XY , w i t h n > 27
(3) YX -- X n XY -- y m n > 2, m > 3;
(4) Y X - x 2 x Y - ~ y 2
(5) X 2 - (yx)ny , y2 _ (Xy)nx (6) X 2 -- (yx)ny , y2
(7) X 2 _ (yx)n , y2 _ (xy)n (8) X 2 - ( y x ) n y2
(9) x 2 , y 2
O / ~ # l i n k ~ n >_ ,I~
n > I~
m
n > 2 ; n > 2;
Let us m e n t i o n first w h i c h algebras are ~nown to be tame or wild.
(1.2) The a.lgebras (a), (b), (b~ ( c ) a n d ~d) are wild.
For (a), (b) and (b~ this was proved by H e l l e r and R e i n e r [ 7 ], for (c) this was proved by D r o z d [ ~ ] and B r e n n e r [ 2 ]. In section 3, we will deal w i t h these algebras.
(1.3) The algebras (1.) -- (4) and (7).--.(.9) are tame.
Namely, we have the following theorem:
Le__~t A be a local algebra, and assume there are elements x1' x2' YI' Y2 i_nn rad A such that rad A = A x I + A Y I = A x 2 + A Y 2 an d XlX 2 = yly 2 = o, the___n A is tame.
The case of the a l g e b r a (1) was proved by G e l f a n d and P o n o m a r e v [ G ] and by Szekeres (unpublished, but see [12]). The case (9), w h i c h includes the d e c o m p o s i t i o n
of the m o d u l a r r e p r e s e n t a t i o n s of the dihedral 2--groups, was proved in [11]. A n i n d i c a t i o n of the m e t h o d of tae p r o o f of (1.3) will be given in the last sectlon, we f o l l o w quite closely the ideas d e v e l l o p p e d by G e l f a n d and P o n o m a r e v in the case of a l g e b r a (I). ~)
(1.4) Let k be an a l g e b r a i c a l l y closed field.
Let A be a complete local algebra. Then
e i t h e r (i) A has a residue ping of type (a) -- (d), or (ii) A is a residue ring of the c o m p l e t i o n of one of the algebras (I) -- (9),
o._rr (iii) char k = 2, and A is isomorphic to k < X , Y > / I w i t h I the twosided ideal g e n e r a t e d b~
(5') X 2 -- (YX) nY + y(YX) n+1, (6') X 2 -- (yx)ny + y(YX) n+1, w i t h (T,6) /(o,o).
y2
_(xy)nx + 6(yx)n+1, or
y2 + 8(yx)n+1,
In s e c t i o n 2 we will prove this theorem. The first step in its proof is the c l a s s i f i c a t i o n of the local algebras k<X,Y>/I of d i m e n s i o n 5 given by G a b r i e l (unpublished). C e r t a i n partial results were obtained by Dade [3], Janusz [8] and M U l l e r [1o], w h e n they
considered the p r o b l e m to bring certain algebras (group a l g e b r a s of 2--groups of m a x i m a l rank) into a normal form. Drozd [4] proved the result for commutative A .
W i t h respect to r e p r e s e n t a t i o n theory, the case (iii) in the t h e o r e m is of no real importance. Namely, the algebras (5') and (6') -- as well as (5) and (6) -- are F r o b e n i u s algebras, and modulo the socle, (5') and
(5), as well as (6') and (6), are isomorphic (for fixed n).
S i n c e the only i n d e c o m p o s a b l e m o d u l e w h i c h is n o t a n n i h i l a t e d by the socle, is the a l g e b r a itself, the r e p r e s e n t a t i o n t h e o r y of (5') is i d e n t i c a l to that of
(5), and the r e p r e s e n t a t i o n t h e o r y of (6') is the same a s that of (6).
(1.5) It follows f r o m the p r e c e d i n g p a r a g r a p h s that the only q u e s t i o n w h i c h r e m a i n s is to d e t e r m i n e the r e p r e s e n t a t i o n type of (5) and (6). It is an i n t e r e s t i n g fact that these are "just" the g r o u p a l g e b r a s of the g e n e r a l i s e d q u a t e r n l o n and the semi--dihedral groups.
To be m o r e precise: If k is an a l g e b r a i c a l l y c l o s e d f i e l d of c h a r a c t e r i s t i c 2, and G is a g e n e r a l i s e d q u a t e r n i o n group, t h e n the group a l g e b r a kG is o f type (5'), and if G is seml--dihedral, t h e n kG is of type (6').
It should be noted that for all o t h e r p--groups G, the r e p r e s e n t a t i o n type of kG is known: If c h a r k = p and G is a n o n - c y c l i c p-group, then kG is w i l d
e x c e p t in the case of a two--generator 2--group of m a x i m a l r a n k ~ K r u g l i a k [9] and B r e n n e r [I]), that is e x c e p t in the case of dihedral, semi--dihedral, a n d g e n e r a l i s e d q u a t e r n i o n groups. Namely, in all the o t h e r cases, kG has a residue r i n g of type (a) or (c), and t h e r e f o r e is wild.
~) A t the conference in Ottawa, t h e o r e m (1.3) w a s f o r m u l a t e d by the a u t h o r only w i t h an a d d i t i o n a l h y p o t h e s i s : that k x 1 + k Y I = k x 2 + k Y 2 ~ the g e n e r a l case was c o n J e c t u r e d . A complete proof w i l l appear elsewhere.
2. The c l a s s i f i c a t i o n t h e o r e m
We w a n t to p r o v e t h e o r e m (1.4). T h u s , we a s s u m e t h a t k is a l g e b r a i c a l l y c l o s e d . L e t A be a c o m p l e t e l o c a l a l g e b r a , a n d l e t J = r a d A . We a s s u m e t h a t A h a s no r e s i d u e a l g e b r a of the f o r m (a), (b), (b~ (e) or (d).
A s a c o n s e q u e n c e , d i m k j / j 2 ~ 2 . If d i m k j / 5 2 ~ I , t h e n A is a h o m o m o r p h l c i m a g e of l l m k < X > / ( X n) , a n d t h i s
is a h o m o m o r p h i c i m a g e of the c o m p l e t i o n o f the a l g e b r a (I). Thus, we m a y a s s u m e d i m k j / j 2 = 2 . O f t e n we w i l l d e n o t e by N a ( s u i t a b l e ) k - s u b s p a c e of A w i t h J = N ~ j2.
(2.1) We m a y a s s u m e d l m k j 2 / j 3 = 2.
F i r s t , we s h o w t h a t f o r d i m k j 2 / j 3 ~ 3 , t h e r e is a h o m o m o r p h i c i m a g e of one of the f o r m s (a) -- (d)o T h i s is o b v i o u s for d i m e n s i o n 4 . We m a y a s s u m e j 3 = 0, a n d l e t d i m k j2 = 3. T h e r e is a non--trivlal r e l a t i o n
~ x 2 + ~xy + yyx + 8y 2 = O,
w h e r e x, y is a b a s i s os N . If ~ = 8 = O, t h e n we u s e as a d d i t i o n a l r e l a t i o n x 2 = O , a n d g e t as r e s i d u e a l g e b r a an a l g e b r a of the f o r m (b), (b ~ or (c). Thus, we m a y s u p p o s e ~ = I. U s i n g x' = x + y y i n s t e a d of x, we h a v e a r e l a t i o n of the f o r m
x '2 + ~ ' x ' y + 8'y = Oo
A d d i n g the n e w r e l a t i o n x ' y = O, we g e t as r e s i d u e a l g e b r a one of the f o r m (b) or (d).
If d i m k j 2 / j 3 = I , let A be the c o m p l e t i o n of some l o c a l a l g e b r a k < X , Y > / I , w h e r e I is a t w o s i d e d ideal. We w a n t to c o n s t r u c t a n i d e a l I' ~ I s u c h t h a t
k < X , Y > / I ' a g a i n h a s no r e s i d u e a l g e b r a o f the f o r m (a) -- (d), b u t w i t h d i m k j , 2 / j , 3 = 2 , w h e r e J' = r a d k < X , Y > / I ' . It is f a i r l y e a s y to see t h a t I + M 3 c o n t a i n s e l e m e n t s x 2 x I a n d y 2 y I , w h e r e b o t h x1' Yl as w e l l as x2' Y2 is a b a s i s of a f i x e a ~ w i t h J = N 9 j2. If x ~ x 1 + f a n d y 2 Y 1 + g b e l o n g to I ( w i t h f, g in M 3 ) , t h e n let I' be g e n e r a t e d b y x 2 x 1 + f a n d y 2 Y 1 + g .
(2.2) T h e r e are e l e m e n t s a, b in J \ j2 s u c h t h a t ab b e l o n g s to j3.
A g a i n , we m a y a s s u m e j3 = O. N o w A c a n be w r i t t e n in the f o r m k 9 N 9 N ~ N / U , w h e r e U is a s u b s p a c e o f N ~ N of d i m e n s i o n 2, a n d w h e r e the m u l t i p l i c a t i o n is g i v e n by the t e n s o r p r o d u c t ~. L e t x, y be a b a s i s of N . W e m a y a s s u m e that U i n t e r s e c t s b o t h N ~ x a n d N ~ y t r i v i a l l y , thus U is the g r a p h of an i s o m o r p h i s m N ~ x > N m ~ , a n d t h e r e f o r e t h e r e is a n a u t o m o r p h i s m
~ : N > N w i t h U = { a ~ x + ~ ( a ) ~ y I a e N } . L e t a be an e i g e n v e c t o r of ~ w i t h e i g e n v a l u e a . T h e n 0 ~ a ~ ( x + ~ y ) b e l o n g s to U.
(2.3) T h e r e are e l e m e n t s x1' x2' YI' Y2 w i t h N = k x 1 + k y I = k x 2 + k Y 2 a n d x 2 x I, y2Yl in J.
A g a i n , we m a y a s s u m e j3 = O. F i r s t , a s s u m e t h e r e is x e j k j 2 w i t h x 2 = O. L e t x, y be a b a s i s of N.
T h e r e is a n o t h e r non--trivial r e l a t i o n
~ x y + p y x + yy2 = O.
N o w y # O, s i n c e o t h e r w i s e we h a v e one of the c a s e s (b), (b ~ or (c). Thus, we m a y s u p p o s e T = I, a n d t h e n
(y + ~ x ) ( y + px) =
o,
a n d w e t a k e x I = , x 2 = x a n d Y2 = y + ~x, Yl = y + ~x.
N e x t , a s s u m e x 2 ~ 0 f o r a l l x i n j \ j 2 . B y (2.2), t h e r e is n o w a b a s i s x, y o f ~ w i t h y x = O~ ~ before, w e c o n s i d e r a n o t h e r non--trivial r e l a t i o n , s a y
~ x 2 + ~y2 + Txy = O.
A g a i n , T ~ O, s i n c e o t h e r w i s e we a r e d e a l i n g w i t h the a l g e b r a (d), t h u s a s s u m e y = I. T h e n
(x + ~ y ) ( ~ x + y ) =
o,
w h i c h s h o w s t h a t we m a y t a k e x 2 = y, x I = x, Y 2 = x + ~x a n d Yl = ~ x + y.
(2.4) A / J 3 is r e s i d u e a l g e b r a of o n e o f t h e a l g e b r a s
(1)
- ( 9 ) .P r o o f : A s s u m e first, o n e of the e l e m e n t s x 2 ' Y2' say x2, is l i n e a r l y i n d e p e n d e n t b o t h f r o m x I a n d f r o m Yl . U s i n g a s u i t a b l e m u l t i p l e o f x I f o r x a n d of Yl f o r y, w e m a y a s s u m e x 2 = y-x. If Y 2 is a l s o l i n e a r l y i n d e p e n d e n t b o t h f r o m x a n d y, t h e n a m u l t i p l e o f Y2 is of the f o r m x--~y, w i t h ~ # o,1. T h u s , A / J 3 is of t h e f o r m (4). I f Y 2 is a m u l t i p l e of x, t h e n w e h a v e c a s e (2) w i t h n = 2 , i f Y2 is a m u l t i p l e o f y, t h e n w e h a v e c a s e (8) w i t h n = 2. I n c a s e b o t h x 2 a n d Y 2 are l i n e a r l y d e p e n d e n t o f x I or Yl ' w e g e t the c a s e s (I) a n d (9).
(2.5) It r e m a i n s to be shown: If, f o r P ~ 3 , A / J p is a r e s i d u e r i n g of one os the a l g e b r a s (I)--(9), t h e n the s a m e is t r u e f o r A / J p+I. O b v i o u s l y , w e m a y a s s u m e jp+1 = O. A s a by--product o f o u r c a l c u l a t i o n s , w e a l s o w i l l d e t e r m i n e a b a s i s of the a l g e b r a s (I)--(9).
C a s e ( I ) . T h e r e a r e e l e m e n t s X, Y i n r a d A w i t h Y X a n d X Y in r a d P A . N o w r a d P A is g e n e r a t e d b y X p a n d YP, t h u s t h e r e a r e e l e m e n t s a, ~, T, 8 i n k w i t h
Y~X + aX p + ~ Y P = 0 a n d X Y + TX p + 8Y p = O.
If we r e p l a c e X b y X' = X + ~yp--1 a n d Y b y Y' = Y + ~X p--T, the n e w r e l a t i o n s a r e
Y ' X ' + ~X 'p = 0 a n d X'Y' + 8Y 'p = O.
We s h o w h o w to g e t r i d of ~ a n d 8 . I f a = 8 = O, w e are a g a i n in c a s e (I). If ~ / O, a n d 8 = O, w e r e p l a c e X' b y X" = P - - ~ X', a n d are in c a s e (2). I f ~ = O a n d 8 # O, t h e n w e i n t e r c h a n g e X' a n d Y' , a n d a r e i n the p r e v i o u s s i t u a t i o n . F i n a l l y , a s s u m e ~ # 0 # 8.
C o n s i d e r X' = Z X" a n d Y' = ~ Y " w h e r e ~ , ~ a r e e l e m e n t s of k w h i c h we w a n t to d e t e r m i n e m o w , in o r d e r to h a v e X" a n d Y" s a t i s f y i n g the r e l a t i o n s (3). T h e o l d r e l a t i o n s b e c o m e
~ Y " X " + ~ P x "p = 0 a n d ~ X " Y " + 8 ~ P Y " p = O.
T h i s m e a n s t h a t w e h a v e to f i n d ~ , ~ s u c h t h a t
~ I > - 1 ~ - 1
= -1 a n d 8 ~ P - I ~ -1 = - 1 , i n o r d e r to h a v e
Y~'X" -- X " p = O a n d X " Y " -- Y " P = O.
O f c o u r s e it is e a s y to w r i t e d o w n ~ a n d ~ e x p l i c i t l y , a n d Lf X ~ a n d Y ~ a r e g e n e r a t o r s of r a d A , s i n c e X' a n d Y' h a d t h i s p r o p e r t y . S u c h a c h a n g e o f X' a n d Y' w i l l be c a l l e d a s c a l a r t r a n s f o r m a t i o n i n the l a t e r p a r t
of the p r o o f , a n d u s u a l l y w i l l be l e f t to the r e a d e r .
C a s e (2 2 . We c a n a s s u m e n < p. N o w the e l e m e n t s X Y a n d YX--X n b o t h b e l o n g to JP, t h e r e f o r e X n+1 = X Y X = O, a n d J P is g e n e r a t e d by the s i n g l e e l e m e n t YP.
A s s u m e t h e r e is a r e l a t i o n Y X -- X n + a Y P = O,
t h e n we r e p l a c e X by X' = X + ~yp--1, a n d g e t t h a t A is e i t h e r r e s i d u e r i n g of a n a l g e b r a of t y p e (2) or of one of type (3); in the l a t t e r case we u s e a n o b v i o u s s c a l a r t r a n s f o r m a t i o n .
C a s e (3). W e c o n s i d e r the case n ~ m = p--l, a n d we w a n t to p r o v e that J P = O. T h i s t h e n i m p l i e s that the a l g e b r a of type (3) h a s d i m e n s i o n n + m + 1 . B y a s s u m p t i o n , t h e r e a r e e l e m e n t s X, Y in J w i t h Y X -- X n a n d X Y -- y m in JP. A s in case (2), J P is g e n e r a t e d by YP, b u t
Y P = Y X Y = X n Y = xn--IY m = xn--2Y 2m--I = O, s i n c e n + 2 m - 3 ~ p+1.
Case(4). We a s s u m e j4 = 0 a n d s h o w j 5 = O.
T h e r e are e q u a l i t i e s
X 3 = X Y X = a y 2 x = a Y X 2 = ~X 3 a n d X 2 y = a X y 2 = ~ 2 y 3 = ~ Y X Y = ~X2y.
S i n c e a / I, the m o n o m i a l s X 3, X Y X a n d X 2 Y a r e zero.
S i n c e a / 0, a l s o all the o t h e r m o n o m i a l s v a n i s h . C a s e (9). A s s u m e A / J p is a r ~ s i d u e r i n g o f the a l g e b r a of type (9). We d i s t i n g u i s h two cases. F i r s t , l e t p be even, p = 2q. T h e n J P is g e n e r a t e d b y the two e l e m e n t s (YX) q a n d (XY) q, t h u s t h e r e a r e r e l a t i o n s
X 2 + a ( Y X ) q + ~(XY) q = O, y 2 + T ( y x ) q + 6 ( x y ) q . If we r e p l a c e X by X' = X + 8(YX)q--IY a n d Y' = Y + T ( x Y ) q - I x , t h e n the r e l a t i o n s in X' a n d Y' ( a f t e r s o m e s c a l a r t r a n s f o r m a t i o n ) h a v e the f o r m (7), (8) or (9). If p is odd, say p = 2q+I, t h e n J P is g e n e r a t e d b y the e l e m e n t s ( x Y ) q X and (YX)qY, a n d we h a v e r e l a t i o n s
X 2 + ~ ( x Y ) q X + ~ ( Y X ) q Y : O, y 2 + y ( x y ) q x + 6 ( Y X ) q y = O.
T h i s time, we r e p l a c e X by X + ~ ( X Y ) q a n d Y b y Y + 6 ( Y X ) q, and, a g a i n a f t e r some s c a l a r t r a n s f o r m a t i o n , the n e w e r e - l a t i o n s are of the f o r m (5), (6) or (9).
C a s e (8) Now, let p = 2n+I, a n d a s s u m e A / J p is g e n e r a t e d by two e l e m e n t s X and Y w h i c h s a t i s f y the r e l a t i o n s X 2 -- (YX) n = 0 a n d y2 = 0. N o w JP is g e n e r a t e d by the e l e m e n t s ( X Y ) n X a n d ( y x ) n T , but
(+) (xy)nx = X ~ = X ( X Y ) n = X 2 y ( x Y ) n-1 = ( X Y ) n Y ( X Y ) n-1 = O, t h e r e f o r e JP is g e n e r a t e d by the s i n g l e e l e m e n t (yx)ny.
T h e r e are r e l a t i o n s
X 2 -- (YX) n + ~ ( y x ) n y = O, y 2 + ~ ( y x ) n y = O.
W e r e p l a c e X by X' = X -- ~YX ~ ~ X Y -- ~ 2 y x Y , a n d Y by Y' = Y + ~(YX) n. T h e n we get
X '2 -- (Y'X') n = 0 and y , 2 = O.
To see the first, we n o t e that
X ,2 = X 2 ~ ~ X 2 y = X 2 + ~ ( y x ) n y ,
w h e r e the f i r s t e q u a l i t y s t e m s f r o m the f a c t that all the o t h e r s u m m a n d s c a n c e l e a c h other, a n d the s e c o n d f o l l o w s f r o m the fact that X 2_ ( y ~ ) n b e l o n g s to JP.
Thus, X' and Y' s a t i s f y r e l a t i o n s of t h e f o r m (8).
Next, let p = 2n+2, and A/J p be of type (8).
Then, as we have seen above, (xy)nx b e l o n g s to JP.
But then JP = O, and therefore the a l g e b r a of type (8) has d i m e n s i o n 4n+2.
Case (7). We assume p = 2n+I. We w a n t to show that JP = 0 in case A/J p is residue a l g e b r a of the a l g e b r a (7). U s i n g the c a l c u l a t i o n (+) of the previous case, we see that (XY)~X = O. Similarly, we have n o w also (yx)ny = O. This proves the assertion. As a consequence, we see that the a l g e b r a of type (7) has d i m e n s i o n 4n+I.
Cases (5),(6). Finally, we have to c o n s i d e r the s i t u a t i o n where A/J p is residue a l g e b r a of an a l g e b r a of type (5) or (6). We first leek at the case p = 2n+2.
Since X 2- (yx)ny belongs to JP, it follows that (yx) n+1 = X 3 = (Xy) n+l .
Thus, if JP J O, then A is a F r o b e n i u s algebra, with socle g e n e r a t e d by the element (YX) n+l. This shows that A is of the form (5') of (6'). But if the c h a r a c t e r i s t i c
of k is different of 2, then it is easy to b r i n g (5') into the form (5),and (6') into the form (6).
If p = 2m+3, we k n o w from the p r e v i o u s c o n s i d e r a - tion that (yx)n+I-(xY) n+l belongs to JP, and therefore
(xy)n+Ix = (yx)n+Ix = (YX)~yx2 = (yx)ny(Yx)ny = O, and then also (yx)n+Iy = O. As a consequence, the a l g e b r a s of type (5), (5'), (6), (6') all are of d i m e n s i o n 4n+4.
3. The w i l d a l g e b r a s
In o r d e r to s h o w that a g i v e n a l g e b r a A is wild, we w i l l use the f o l l o w i n g p r o c e d u r e . We w i l l start w i t h a c a t e g o r y ~ w h i c h we k n o w is wild, w i t h a full sub-- c a t e g o r y ~ of A ~ ' and w i t h f u n c t o r e
U: ~ -- > ~ , and P: ~ > ~ ,
s u c h t h a t the c o m p o s i t i o n P U is the i d e n t i t y f u n c t o r on ~ . Then, obviously, ~ is r e p r e s e n t a t i o n e q u i v a l e n t to the full s u b c a t e g o r y of A ~ of all m o d u l e s w h i c h are i m a g e s u n d e r U .
(3.1) The a l g e b r a A = k < X , Y , Z > / M 2 is wild.
F o l l o w i n g H e l l e r and R e i n e r [ 7 ], we e m b e d the c a t e g o r y
= k < x , y > ~ into A ~ " Let ~ be the f u l l s u b c a t e g o r y of A m c o n s i s t i n g of all A M w i t h Z--tO = Z M (that is, all A - m o d u l e s w h i c h are free w h e n c o n s i d e r e d as K < Z > / ( Z 2 ) - modules). The f u n c t o r U a s s o c i a t e s w i t h k < x , y > V the m o d u l e A M g i v e n by the d i a g r a m
x V ~ - - ~ V ,
thus, as v e c t o r s p a c e , k M = V ~ V , a n d X o p e r a t e s o n V ~ V by I~ ~ , and so on . C o n v e r s e l y , g i v e n A M in B , t h e n P(A M) is the v e c t o r space ZM t o g e t h e r w i t h the two e n d o m o r p h i s m s x = X Z -I and y = Y Z -I . Note that, for example, X Z -I is well--defined, since XZ--Io = X Z M = O a c c o r d i n g to the c o n d i t i o n Z--tO = ZM , a n d that its image lies in ZM , u s i n g a g a i n the same c o n d i t i o n .
(3.2) The a l g e b r a A = K < X , Y > / ( X 2 , y x , x Y Z , Y 3) i__ss wild. Again, we f o l l o w Heller--Reiner [ q ] . As ~ , we use the category
thus, an object of ~ is given by a t r s (W,V,~) w i t h W a v e c t o r space, V ~ W a subspace, and ~ a n
e n d o m o r p h i s m of W . Let ~ be the full s u b c a t e g o r y of all A M in A ~ w i t h Xy--Io = O and y - - I o q Y M . For (W,V,~) in ~ , define A M = U ( W , u by the d i a g r a m
y X=~
V > W ~ ~ W ,
Y=I
thus M = V 8 W 8 W , and X and Y o p e r a t e on M as indicated.
Conversely, for A M in ~ , let P(A M) = (y--IO,y2M, Xy--I).
Obviously, y 2 M is a subspace of Y--tO, a n d XY -I is well-defined, since we assume Xy--Io = O . Also, the image of X Y -I lies in y--tO, since Y X = O.
(3.5) The a l g e b r a A = E < X , Y > / ( X R , X y , y 2 x , y 3) is wild, since it is just the opposite a l g e b r a to the p r e v i o u s l y discussed one.
(3.4) The a l g e b r a A = k < X , Y > / ( X 2 , X y - m Y X , y 2 x , y 3) i_~s wild. We may assume a / O, and give a c o n s t r u c t i o n due
to D r o z d [ 4 ]. Again, ~ is the c a t e g o r y , ~ e ~ @ Let ~ be the full subcategory of all A M in A ~ w i t h YXy--2Xy--20 = 0 and Y X M Q y 2 M , X y - - 2 ~ C y 2 M . For
(W,V,~) in ~ , define A M = U(W,V,~) b y the d i a g r a m
inclusion= X
Thus, A M is the direct sum of six copies of W and one copy of V , and X and Y operate on it as i n d i c a t e d (where all but three maps are i d e n t i t y maps, one is g i v e n by ~ , one is m u l t i p l i c a t i o n by ~ and one is the i n c l u s i o n V ~ W ). It r e m a i n s to d e f i n e P . G i v e n A M in ~ , let P(A M) = (YXM, XY--SO YXM, YXy--2Xy--2).
By the a s s u m p t i o n s on B , YXy--2Xy -2 is r e a l l y an endo-- m o r p h i s m of Y X M , and it is easy to check that PU is
the identity on ~ .
(3.5) The algebraA=k<X,Y>/(XY,X2--Y 2) is wild.
(Note that the ideal (k~,X2-y 2) contains M 3
We start w i t h the category ~ with objects V g i v e n as V a < ~ - - V b ~ ~ V c ~ - - V d * - ~ V e - - * > V f * - ~ V g - * > V h
V i
that is, we consider the category of r e p r e s e n t a t i o n s of the c o r r e s p o n d i n g quiver such that the maps are monomorph-- isms or e p i m o r p h i s m s as indicated. This is a well--known wild category. The functor U: ~ - - > A ~ maps the
r e p r e s e n t a t i o n V onto the A - m o d u l e A M g i v e n as
~ V d ~ ~ V e ~ ~ V g ~
V~ V ~ - - V ~ V ~ - V h
Va V - Vc Y
where (besides two identity maps) all maps are the ones g i v e n by V .
We define a functor P: A~ ~ > w'= , where w'= is the category of all r e p r e s e n t a t i o n s of the q u i v e r
I 2 3
4 5 6 7 8
9 1o 11
for w h i c h the square is c o m m u t a t i v e . The c a t e g o r y w is ( e q u i v a l e n t to) the full s u b c a t e g o r y of _w_' of all
r e p r e s e n t a t i o n s for w h i c h the maps w i t h ~ are isomorph-- isms, those w i t h + are m o n o m o ~ p h i s m s , a n d the r e m a i n i n g ones are e p i m o r p h i s m s . We w i l l u s e as u the full
s u b c a t e g o r y of a l l m o d u l e s A M in A m w i t h P(A M) in =w . In o r d e r to define P , we n o t e t h a t there is a c h a i n of s u b f u n c t o r s F i (o < i < 11 ) of t h e f o r g e t f u n c t o r F o f r o m the c a t e g o r y A m into the c a t e g o r y of k--vector
spaces, n a m e l y Fo~AM) = M , FI(AM) = X--IyM ,
~ 2 ( A M) -- X--Iyx--IyM ,
F3(AM) = X-1.YXM + Y M + X M , F4(AM) = Y M + X M ,
~ 5 ( A M) -- Y M , P 6 ( A M) = y x - I Y M , F7(AM) = yx-Iyx--IyM , F S ( A M) = YX--I~XM + Y X ~ , F9(AM) -- Y X M + X 2 M ,
F o(A M) : X2 ,
F 1 1(X M) = 0 9
M o s t of the i n c l u s i o n s F i _ I ~ F i are t r i v i a l , o t h e r w i s e we use the r e l a t i o n s X M ~ X--IyM , Y M C X--Io a n d
X M ~ X--IY](--Io .
The s P: A m - - > ~' is n o w d e f i n e d component-- w i s e by Pi = Fi/Fi--1 ' and those n a t u r a l t r a n s f o r m a t l e n s Pi --> PJ w h i c h we need, are the ones i n d u c e d b y
m u l t i p l i c a t i o n by X or Y, r e s p e c t i v e l y :
y.P2 y X / P 3 y
# NPs
P5 P6
y
Again, in order to s h o w that these m a p s are defined, we n e e d only the r e l a t i o n X Y = 0 . Of couzse, the s q u a r e is commutative, since we a s s u m e X 2 = y2.
It is e a s y to check that the c o m p o s i t i o n P U is the i d e n t i t y f u n c t o r on ~ .
4. Tame a l g e b r a s
We w a n t to give some i n d i c a t i o n s a b o u t the p r o o f of t h e o r e m (1.3). In order to s h o w that a g i v e n a l g e b r a A is tame, it is r e a s o n a b l e to de two things: f i r s t to w r i t e d o w n a list of c e r t a i n i n d e c o m p o s a b l e modules, and t h e n to prove that e v e r y o b j e c t of A ~ can be d e c o m p o s e d as a d i r e c t sum of copies of these m o d u l e s . In our case, the d e c o m p o s i t i o n w i l l be a c h i e v e d by u s i n g several f u n c t o r s and n a t u r a l t r a n s f o r m a t i o n s .
We will start w i t h an i n d e x set W on w h i c h a f u n c t i o n W u > ~ is d e f i n e d w h i c h a s s o c i a t e s t o , v e r y D in ~ a n a t u r a l n u m b e r IDI ~ I, t h e " l e n g t h " o f D.
To e v e r y D in W we w i l l d e f i n e e i t h e r one indecom-- p o s a b l e m o d u l e M(D), or a whole set of i n d e c o m p o s a b l e m o d u l e s M ( D , ~ ) i n d e x e d by the set of ( e q u i v a l e n c e classes of) i n d e c o m p o s a b l e a u t o m o r p h i s m s of k--vector spaces (thus, if k is a l g e b r a i c a l l y closed, we m a y Zake as i n d e x set the set of J o r d a n m a t r i c e s ) .
Then, we w i l l c o n s i d e r the f o r g e t f u n c t o r A ~ u > k ~ ' w h i c h a s s o c i a t e s to e v e r y A--module the u n d e r l y i n g v e c t o r
space. F o r e v e r y D in W , we w i l l c o n s t r u c t 2-1D I s u b f u n c t o r s of it, d e n o t e d by F(D,i) + a n d F(D,i)-- , w h e r e I ~ i ~ IDI, s u c h that F(D,i)-- ~ F(D,i) +. We
w i l l denote by F(D,i) the q u o t i e n t f u n c t o r F ( D , i ) + / F ( D , i ) -.
Then, we w i l l c o n s t r u c t n a t u r a l t r a n s f o r m a t i o n s
F(D,i) --> F(D,i+I) or F(D,I) <-- F(D,I+I),
for I ~ i < IDI, and for c e r t a i n e l e m e n t s D in W also for i = IDI, c a l c u l a t i n g m o d u l o IDI.In this way, we w i l l d e t e r m i n e for e v e r y A M in A m a s u b m o d u l e of "type D" (that is, one w h i c h is a d i r e c t sum of copies e i t h e r of M(D), or of some of the M ( D , ~ ) . ) , such that A M is the d i r e c t s u m of these
s u b m o d u l e s .
Obviously, the i n d e x set ~ w i l l d e p e n d on the p a r t i c u l a r a l g e b r a A . The m e t h o d w i l l be e a s i e r to visualise, if we use a s p e c i f i c example. We have c h o s e n the ease of the a l g e b r a (4), that is k(X,Y~/(YX--X2,Xy--~Y 2) w i t h a # o,1, since, on the one hand, the a l g e b r a is
r a t h e r small, and, on the other hand, the b e h a v i o u r of the r e m a i n i n g a l g e b r a s is s o m e w h a t i n t e r m e d i a t e b e t w e e n that of the a l g e b r a (4) and of the well--known cases (I) and (9).
Thus, let A = k<X,Y>/(YX--X2,Xy--~Y 2) , and ~ #o,1.
We w i l l denote the e l e m e n t s X, Y, X--~Y a n d X--Y by a , b , c , d , in order to point out that these are just f o u r e l e m e n t s of k X + k Y w h i c h are p a i r w i s e l i n e a r l y i n d e p e n d e n t . The set ~ w i l l be the d i s j o i n t u n i o n of two subsets
~I a n d ~2 " Now, ~I is the set of all finite w o r d s in the letters a,b--l,c,d -I ( i n c l u d i n g the empty w o r d 1), subject to the f o l l o w i n g rules: a f t e r c or b -1 f o l l o w s e i t h e r a or d -I, a f t e r a f o l l o w s o n l y b -I, a n d a f t e r d -I follows only c . Thus, an example is the w o r d
D = ab--ld--lcd -I,
and its l e n g t h is d e f i n e d to be 6 (= n u m b e r of l e t t e r s +I).
If D and E are words, and DE is a l s o a word, t h e n we call DE the product of D and E . Of course, D 2
s t a n d s for DD, and so on. W e call a w o r d D non--periodic p r o v i d e d D 2 is also a word, and D is not of the f o r m D = E n for some w o r d E and n>1. Two n o n - p e r i o d i c w o r d s D and E are c a l l e d e q u i v a l e n t , p r o v i d e d one is a c y c l i c
p e r m u t a t i o n of the other, and ~ 2 w i l l be the set of all e q u i v a l e n c e classes of non--periodic words. N o t e that for e l e m e n t s of ~ 2 the l e n g t h is d e f i n e d different: it is the p r e c i s e n u m b e r of l e t t e r s of the c o r r e s p o n d i n g word. (The w o r d D above does n o t give rise to an e l e m e n t of ~2 ' since D 2 is not a n a d m i s s i b l e word. A n example of a n e l e m e n t of ~ 2 is the set of c y c l i c p e r m u t a t i o n s of ab--ld--lcd--lc .)
Next, we s h o w h o w to d e f i n e for D in ~I a m o d u l e M(D). Namely, let M(D) be a IDI--dimensional v e c t o r s p a c e w i t h base v e c t o r s e l , . . . , e i D l , s u c h t h a t X and Y
o p e r a t e on the base v e c t o r s a c c o r d i n g to the w o r d D.
Thus, for D = ab-ld--lcd -I, we h a v e the f o l l o w i n g s c h e m a j e 2 ~
e 1 e 3 e 5
e 4 e 6
w h i c h m e a n s that ae 2 = e I, be 2 = e 3, de 3 (= (a-b)e 3) = e 4, and so on. Note that in all but the t e r m i n a l p o i n t s e I and e 6 the a c t i o n of a and b is u n i q u e l y defined.
By d e f i n i t i o n this is true for e 2 . It is obvious for e5, since the e l e m e n t s c and d are l i n e a r l y independent.
S i n c e e 3 is image u n d e r b, we m u s t h a v e ce 3 = 0 , thus also on e 3 the m u l t i p l i c a t i o n b y two l i n e a r l y i n d e p e n d e n t e l e m e n t s (namely c and d) is given. Also,
e 4 is i m a g e b o t h u n d e r a a n d b , t h u s w e m u s t h a v e ce 4 = de 4 = 0 . F o r the t e r m i n a l p o i n t s , w e m a k e the f o l l o w i n g c o n v e n t i o n . If, as i n o u r c a s e , a l s o c D i s a w o r d , t h e n w e l e t ce I = 0, if a D is a w o r d , w e l e t ae I = O, a n d i f D s t a r t s w i t h c, t h e n w e l e t a e I = he I = 0. C o n s e q u e n t l y , w e d e f i n e i n o u r c a s e a l s o ae 6 = be 6 = 0.
I n a s i m i l a r w a y , w e d e f i n e f o r a w o r d D i n ~ 2 a n d a n a u t o m o r p h i s m ~ of a v e c t o r s p a c e V , t h e m o d u l e M ( D , ~ ) . N a m e l y , w e t a k e as u n d e r l y i n g v e c t o r s p a c e the d i r e c t s u m of IDI c o p i e s o f V , a n d d e f i n e a g a i n t h e a c t i o n o f X a n d Y a c c o r d i n g to t h e w o r d D , w h e r e a l l a r r o w s b u t the l a s t a r e t a k e n as t h e i d e n t i t y m a p b e t w e e n t h e c o r r e s p o n d i n g c o p i e s (as i n d u c e d b y t h e e l e m e n t o f k X + k Y w h i c h c o r r e s p o n d to t h e l e t t e r ) , a n d w h e r e the l a s t l e t t e r g i v e s j u s t t h e m a p ~ b e t w e e n t h e l a s t a n d t h e f i r s t c o p y o f V .
I n o r d e r to d e f i n e the s u b f u n c t o r s o f the f o r g e t f u n c t o r A ~ - > k ~ w h i c h a r e o f i n t e r e s t to us, w e n o t e t h a t the f o r g e t f u n c t o r h a s t w o c a n o n i c a l f i l t r a - t i o n s , g i v e n b y t h e e q u a t i o n s d a = 0 a n d c b = 0.
C o n s i d e r f i r s t t h e e q u a t i o n d a = 0. W e f o r m f i n i t e a n d i n f i n i t e w o r d s i n the l e t t e r s a a n d d -I, a n d d e n o t e b y ~ a t h e s e t of a l l f i n i t e w o r d s t o g e t h e r w i t h t h o s e i n f i n i t e w o r d s w h i c h a r e o f t h e f o r m D E ~ = D E E E - . - , w h e r e D a n d E a r e f i n i t e w o r d s . F o r e v e r y w o r d D i n ~ a ' t h e r e a r e t w o o b v i o u s f u n c t o r s A ~ t ~ k ~ ' one d e f i n e d b y M ~--> D ( 0 M ) , the o t h e r b y
M ~ ~ D(M). Here, we use the definition E~(0M) = ~ J E n ( 0 M ), and E~(M) = ~ E n ( M ) . It is easy to see that the set of all such functors is linearly ordered by inclusion, and we call this set the a--filtration. In a similar way, the equation cb = 0 gives rise to a set ~b of finite and infinite words in the letters b, c -I , and then to the b-filtration.
If F Z ~ F I are two subfunctors of the forget functor, we call [~!] an intervall. The intersection of the two intervalls [ ~ ] and [GI] is defined to be
G 2 the intervall
FInG I
For any word D in W, the functors F(D,i) + and F(D,i)-- are defined by intersecting suitable intervalls of the a--filtration with those of the b--filtration. We indicate the choice of the intervalls in the case of the word D = ab--ld--lcd -I :
F D , I +
[FID, II__ ] _ _ rad--1(d--la) M ~0--I0.
= [ad--1(d--la) 0 ] ~ [ b M ]
IF(D,2) [ d [ l ( d - - l a ) M .c--20 . F(D,2)_ ] = 4~1(d..1~) O] m [c_lbMJ
F(D,3) ] (d--la) M .be--20
F(D,3)--~ = [(d--la) 0 ] m [bc--lbM J
IF(D,4) (ad -I) M .b2c--20
~(D,4)-] = [(ad-~)0 ] ~ [b2c-lbMJ
+
F(D,5)_ ] = [(d_la ) O] R b2c_IbMJ F(D,6) (ad -I) M rbc--lb2c--20 .
F(D,6)-] = [(ad-1) 0 ] ~ Lbc--lb2c--~b~J
We n o w u s e the m u l t i p l i c a t i o n m a p s i n o r d e r to d e f i n e n a t u r a l t r a n s f o r m a t i o n s b e t w e e n the q u o t i e n t f u n c t o r s F(D,i). Again, we u s e the w o r d D as g u i d e line. In our case, for example, we w a n t to h a v e the f o l l o w i n g t r a n s f o r m a t i o n s :
F ( D , 2 )
a J
F(D,I) F(D,3) F(D,5) / c F(D,4)
w h e r e the l e t t e r i n d i c a t e s the m u l t i p l y i n g e l e m e n t . O f course, it has to be c h e c k e d that the m u l t i p l i c a t i o n m a p s are w e l l ~ e f i n e d and act as i n d i c a t e d , and t h a t t h e y i n d u c e e v e n i s o m o r p h i s m s of the c o r r e s p o n d i n g c o m p o n e n t s.
It then only r e m a i n s to be s h o w n t h a t the F D , i ( ~ +
i n t e r v a l l s [F(D,il- ] cover the forget f u n c t o r (that means, for e v e r y M and e v e r y o / x ~ M, there is such a n i n t e r v a l l w i t h x E F ( D , i ) ~ \ F ( D , i ) - - ( M ) . )
A n outline of the b a c k g r o u n d of the p r o o f , m a y be f o u n d i n G a b r i e l ' s p a p e r [ 5] w h e r e he d i s c u s s e s the v a l u e of f u n c t o r c a t e g o r i e s in o r d e r to d e t e r m i n e all i n d e c o m p o s a b l e objects of a g i v e n c a t e g o r y .
A k n o w l e d g e m e n t : The a u t h o r is i n d e p t e d to P . G a b r i e l for m a n y f r u i t f u l d i s c u s s i o n s and h e l p f u l c o m m e n t s , and he w o u l d like to t h a n k him.
