V l a s t i m i l D l a b a n d C l a u s M i c h a e l R i n g e l
T h e p r e s e n t p a p e r g e n e r a l i z e s a r e c e n t r e s u l t o f I.M. G e l f a n d a n d V . A . P o n o m a r e v [4] r e p o r t e d a t t h e C o n f e r e n c e b y V . A . R o j t e r .
A m o d u l a t e d g r a p h ~ = (F , M , ST). is g i v e n b y
i i 3 l l,j C I
d i v i s i o n r i n g s F. f o r a l l i E I b y b i m o d u l e s (iM4)F•j f o r a l l
i ' F i
J i @ j i n I f i n i t e l y g e n e r a t e d o n b o t h s i d e s a n d b y n o n - d e g e n e r a t e b i l i n e a r f o r m s E? : .M. ® .M. + F. ; h e r e , I is a f i n i t e i n d e x set.
l ± J 3 i ±
N o t e t h a t t h e f o r m s f~ g i v e r i s e to c a n o n i c a l e l e m e n t s c. c .M. ® i
i 3 j 1
.M• N a m e l y , if X l , . . , , x d is a b a s i s o f ( j M i ) F i a n d Y l .... ' Y d i J
t h e c o r r e s p o n d i n g d u a l b a s i s o f (.M.) w i t h r e s p e c t t o ST , t h e n
F. 1 3 i
1 i
c. : E x ~ y p ; s e e s e c t i o n i.
J p P
D e f i n e t h e r i n g H~r~ a s f o l l o w s . L e t T~/~ b e t h e t e n s o r r i n g o f ~ : T ~ = ~ T , w h e r e T : ~ F i , T 1 : ~ iM~ a n d
t s ~ t o i i,j J
T t + 1 = T 1 ~ T t w i t h t h e m u l t i p l i c a t i o n g i v e n b y t h e t e n s o r p r o d u c t . o
T h e n , b y d e f i n i t i o n , H ~ = T ~ / < c > , w h e r e < c > is t h e p r i n c i p a l i d e a l o f T ~ g e n e r a t e d b y t h e e l e m e n t c = Z c~ .
i,j i
L e t Q b e a n ( a d m i s s i b l e ) o r i e n t a t i o n o f ~ ; t h u s , f o r e v e r y p a i r i , j w i t h .M. ~ 0 , w e p r e s c r i b e a n o r d e r i n d i c a t e d b y a n a r r o w
1 3
i - - > j , o r i < - - j i n s u c h a w a y t h a t n o o r i e n t e d c y c l e s o c c u r . L e t R ~ , ~ ) b e t h e c o r r e s p o n d i n g t e n s o r r i n g o f ~ , ~ ) : R ~ , ~ ) =
R w i t h R = K F. , R 1 = • . M a n d = R 1 ~ R t •
t ~ ~ t o i i i÷j I 3 R t + l R o
F o r t h e r e p r e s e n t a t i o n t h e o r y o f R ~ ) w e r e f e r t o [3].
T H E O R E M . For each orientation ~ of 2 ~ , R ( ~ Q ) is a sub- ring o f ~ and, as a (right) R ~ ) -module, H ~ is the direct sum o f all indecomposable preprojective R ~ , ~ ) - m o d u l e s (each occurring with multiplicity one).
T h i s t h e o r e m s u g g e s t s t o c a l l H ~ t h e p r e p r o j e c t i v e a l g e b r a o f 22~. R e c a l l t h a t a n i n d e c o m p o s a b l e R ~ Q ) - m o d u l e P is p r e p r o - j e c t i v e i f a n d o n l y i f t h e r e i s o n l y a f i n i t e n u m b e r o f i n d e c o m p o s a b l e m o d u l e s X w i t h H o m (X,P) ~ 0 .
C O R O L L A R Y . The ring H ~ is artinian i f and only if the modulated graph is a disjoint union o f Dynkin graphs.
O b s e r v e t h a t i f ~ i s a K - m o d u l a t i o n ( w h e r e K is a c o m m u t - a t i v e f i e l d ) , t h e n K ~ is a K - a l g e b r a . I n t h i s c a s e , t h e c o r o l l a r y m a y b e r e f o r m u l a t e d a s f o l l o w s : T h e a l g e b r a ~ ( ~ is f i n i t e -
d i m e n s i o n a l i f a n d o n l y i f ~ i s a d i s j o i n t u n i o n o f D y n k i n g r a p h s . C o n s i d e r , i n p a r t i c u l a r , t h e c a s e w h e n (~,~) i s g i v e n b y a q u i v e r ; t h u s , F. = K f o r a l l i a n d .M. is a d i r e c t s u m o f a
1 1 3
f i n i t e n u m b e r o f c o p i e s o f K F o r e v e r y a r r o w x o f t h e q u i v e r , K K
d e f i n e a n " i n v e r s e " a r r o w x * w h o s e e n d is t h e o r i g i n o f x a n d w h o s e o r i g i n i s t h e e n d o f x . T h e n T ~ is t h e p a t h a l g e b r a g e n e r a t e d b y a l l a r r o w s x a n d x* , a n d H ~ is t h e q u o t i e n t o f T ~ b y t h e i d e a l g e n e r a t e d b y t h e e l e m e n t ~ (xx* + x ' x ) .
a l l x
C O R O L L A R Y . If ~ , ~ ) is given by a quiver, then H ~ ) is finite-dimensional i f and only i f the quiver is of finite type.
F o r a q u i v e r w h i c h is a t r e e , t h e l a s t r e s u l t h a s b e e n a n n o u n c e d b y A . V . R o j t e r [6] i n h i s r e p o r t o n t h e p a p e r [4]. I n c o n t r a s t t o t h e p r o o f s i n [4], o u r a p p r o a c h a v o i d s u s e o f r e f l e c t i o n f u n c t o r s a n d i s b a s e d o n t h e e x p l i c i t e d e s c r i p t i o n o f t h e c a t e g o r y
P ~ , ~ ) o f a l l p r e p r o j e c t i v e R ~ , 9 ) - m o d u l e s . T h e a u t h o r s a r e i n d e b t e d to P. G a b r i e l f o r p o i n t i n g o u t t h a t t h e t h e o r e m is, i n t h e c a s e w h e n ( 9 9 ~ , ~ ) is g i v e n b y a q u i v e r , a l s o d u e t o Jh. R i e d t m a n n [7].
i. P r e l i m i n a r i e s o n d u a l i z a t i o n
G i v e n a f i n i t e - d i m e n s i o n a l v e c t o r s p a c e M , d e n o t e b y * M F
i t s (left) d u a l s p a c e H O m ( F M , F F F ) . I f F M G is a b i m o d u l e a n d X , Y v e c t o r s p a c e s , t h e a d j o i n t m a p f : X + * M (9 Y to a m a p
G F F
d
f : M 0 X ÷ Y is g i v e n b y f(x) = ~ ~ (9 f ( m (9 x) , w h e r e x s X ,
G p : l P P
{ m l , m 2 ... m d } is a b a s i s o f F M a n d { ~ 1 , ~ 2 ... ~ d ) is t h e r e s p e c - t i v e d u a l b a s i s o f ( M ) F In p a r t i c u l a r , if M is a n E n d Y - E n d X - s u b m o d u l e o f t h e h i m o d u l e H o m ( X , Y ) a n d X M : M (9 X + Y t h e
e v a l u a t i o n m a p X M ( m (9 x) : m(x) , t h e n "m~"(x) : Z _~p (9 m (x)
p P
N o t e t h a t ~ M is a (left) G - h o m o m o r p h i s m .
N o w , g i v e n b i m o d u l e s FMG , GNF s u c h t h a t F M a n d N F a r e f i n i t e d i m e n s i o n a l , l e t C : M (9 N + F b e a n o n - d e g e n e r a t e b i l i n e a r
G
f o r m . T h u s , t h e a d j o i n t c is a n i s o m o r p h i s m c : N + * M ; l e t { n l , n 2 ... n d ) b e a b a s i s o f N F a n d { ~ i , ~ 2 ... ~ d } t h e b a s i s o f
(*M) F s u c h t h a t _~p : ~ ( n p ) f o r a l l 1 < p < d. F u r t h e r m o r e , l e t { m l , m 2 . . . . , m d } b e t h e d u a l b a s i s o f F M . T h u s ,
6 ( m (9 n ) = ( m ) [~(n )] : ( m ) } q : d
P q P q P P q
D e f i n e t h e c a n o n i c a l e l e m e n t c o f N (9 M ( w i t h r e s p e c t t o S) b y F
d
C = > n (9 m
e ~ p p
p = l
L e m m a 1.1. T h e e l e m e n t c
C d o e s n o t d e p e n d o n t h e G h o i c e o f a b a s i s .
P r o o f . L e t { n l , n 2 , . .,n~} . a n d { m l , m ~ , . . . ,m~} b e a n o t h e r b a s e s o f N F a n d F M , r e s p e c t i v e l y , s o t h a t
C(m' (9 n') =
P q P q
T h e n n' = Z n . b . a n d m' = Z a .m. w i t h b.
q j 3 3 q P i p l i 3 q
S i n c e 6 P q
a n d a . f r o m F . p ±
= E ( m ' (9 n ' ) = E a . C ( m . (9 n . ) b . = E a .b. ,
P q i,j p l l 3 3 q i p l l q
w e h a v e a l s o Z b. a = 6..
3P p i 3 x
P T h u s ,
E n' O m' = Z n . b . @ a . m.
P P • . 3 3P p l 1
P 1 , 3 , P
n. (p~ b. a p i ] (9 m i = [ n. ~ m . . ,
i,j 3 3P i l ±
If w e take, i n p a r t i c u l a r , G N F = * ( F M G ) X : M Q N ÷ F d e f i n e d b y
G
x ( m ~ ~) : (m)~ ,
a n d t h e e v a l u a t i o n m a p
w e o b t a i n , f o r e v e r y b i m o d u l e M , the c a n o n i c a l e l e m e n t c(M) = c
× G i v e n a b i m o d u l e F M G , d e f i n e t h e h i g h e r d u a l s p a c e s (t) F M G i n d u c t i v e l y b y
(t+l) M = *((t) .
F G F M G ) I
T h u s , (t)M is a n F - G - b i m o d u l e f o r t e v e n a n d a G - F - b i m o d u l e f o r odd.
a n d t
T h e n
L e m m a 1.2. L e t F M G a n d G N F b e b i m o d u l e s a n d £ : M Q N F + [ F F G : G N F O M G + ~ G G n o n - d e g e n e r a t e b i l i n e a r forms. D e f i n e the m a p s i n d u c t i v e l y as f o l l o w s :
0 (0)
= 1 M : F M G ÷ M = M ; 1 = ~ : N ÷ ( 1 ) M : * M ;
G F
.- ( 2 r ) M a n d 2 r n = @ [ ( 2 r - 1 )-i O IM] F M G ÷
: (2r+l) M
2 r + l n = ~ [ ( 2 r n ) - i 0 IN] G N F +
[ 2 r + l O 2 r + 2 n ] (ct) = c ( ( 2 r ) M ) a n d [2r n ~ 2 r + l ] (c6) = c ( ( 2 r + l ) M ) .
: [ n 0 m , w h e r e { m l , m 2 ... m d}
P r o o f . R e c a l l t h a t c s P P P
is a b a s i s o f F M a n d { n l , n 2 , .... .nd} the d u a l b a s i s o f N F w i t h r e s p e c t to s . H e n c e , in o r d e r to p r o v e t h e f i r s t e q u a l i t y , i t is s u f f i c i e n t t o s h o w that, f o r m s M a n d n c N ,
(n 0 m) = (2r+ir](n)) [ 2 r + 2 (m) ] .
But, ( 2 r + i D ( n ) ) [ 2 r + 2 D ( m ) ] : ( 2 r + l q ( n ~ [ @ [ ( 2 r + l ~ ) - i ~ i M ] ( m ) ] : : @ E ( 2 r + l )-i ~ I M ] (2r+l (n)) : 6 [ ( 2 r + i )-i 2 r + i n ( n ) @ m ] =
= @ (n ~ m ) .
S i m i l a r l y , s i n c e
( 2 r ~ ( m ) ) [ 2 r + l u ( n ) ] : ( 2 r ~ ( m ) ) [ s [ ( 2 r ~ ) - i ® i N ] ( n ) ] :
: E E ( 2 r )-i @ i N ] ( 2 r (m)) = £ E ( 2 r )-i 2r (m) @ n] =
: S (m • n) ,
w e c a n d e r i v e t h e s e c o n d e q u a l i t y f o r c ( ( 2 r + l ) M ) .
2. I r r e d u c i b l e m a p s
R e c a l l t h e d e f i n i t i o n o f a n i r r e d u c i b l e m a p [2]: a m a p
f : X ÷ Y is c a l l e d i r r e d u c i b l e if f is n e i t h e r a s p l i t m o n o m o r p h i s m n o r a s p l i t e p i m o r p h i s m a n d if, f o r e v e r y f a c t o r i z a t i o n f = f ' f " , e i t h e r f" is a s p l i t m o n o m o r p h i s m o r f' is a s p l i t e p i m o r p h i s m . A l s o , r e c a l l t h e d e f i n i t i o n o f t h e r a d i c a l o f a m o d u l e c a t e g o r y . If X a n d Y a r e i n d e c o m p o s a b l e m o d u l e s , l e t r a d (X,Y) b e the s e t o f a l l n o n - i n v e r t i b l e h o m o m o r p h i s m s . If X : • X a n d Y = • Y
p P q q
w i t h i n d e c o m p o s a b l e m o d u l e s X a n d Y , d e f i n e r a d (X,Y) :
P q p , q
t a d (X ,Y ) , u s i n g the i d e n t i f i c a t i o n H o m ( X , Y ) = • H o m (X ,Y ).
P q p , q P q
T h e s q u a r e r a d 2 ( x , Y ) o f t h e r a d i c a l is t h u s the s e t o f a l l h o m o - m o r p h i s m s f : X ÷ Y s u c h t h a t f : f ' f " , w h e r e f" s r a d ( X , Z ) a n d f' s r a d ( Z , Y ) f o r s o m e m o d u l e Z . N o t e t h a t b o t h r a d a n d r a d 2 a r e i d e a l s in o u r m o d u l e c a t e g o r y ; i n p a r t i c u l a r , r a d (X,Y) a n d r a d 2 ( X , Y ) a r e E n d Y - E n d X - s u b m o d u l e s o f t h e b i m o d u l e
E n d y H ° m ( X ' Y ) E n d X" F o r i n d e c o m p o s a b l e X a n d Y, t h e e l e m e n t s in r a d ( X , Y ) ~ r a d 2 ( X , Y ) a r e j u s t t h e i r r e d u c i b l e m a p s . In t h i s case, w e w r i t e I r r ( X , Y ) : r a d ( X , Y ) / r a d 2 ( X , Y ) , a n d c a l l I r r ( X , Y ) t h e
b i m o d u l e o f i r r e d u c i b l e m a p s (see [5]). In w h a t f o l l o w s , o u r m a i n o b j e c t i v e is to s e l e c t a d i r e c t c o m p l e m e n t o f r a d 2 ( X , Y ) in r a d ( X , Y ) w h i c h is a n E n d Y - E n d X - s u b m o d u l e , a n d r e a l i z e i n t h i s w a y
I r r ( X , Y ) a s a s u b s e t o f H o m ( X , Y ) r a t h e r t h a n j u s t a s a f a c t o r g r o u p . W e s h a l l s e l e c t s u c h c o m p l e m e n t s i n d u c t i v e l y , u s i n g A u s l a n d e r - R e i t e n s e q u e n c e s .
R e c a l l t h a t a n e x a c t s e q u e n c e 0 ÷ X ~ Y ~ Z + 0 is c a l l e d a n A u s l a n d e r - R e i t e n s e q u e n c e i f b o t h m a p s f a n d g a r e i r r e d u c i b l e . T h i s i m p l i e s t h a t b o t h m o d u l e s X a n d Z a r e i n d e c o m p o s a b l e , X i s n o t i n j e c t i v e a n d Z i s n o t p r o j e c t i v e . C o n v e r s e l y , g i v e n a n i n d e c o m p o s a b l e n o n - i n j e c t i v e m o d u l e X , t h e r e e x i s t s a n A u s l a n d e r - R e i t e n s e q u e n c e s t a r t i n g w i t h X , a n d a l s o d u a l l y , g i v e n a n i n d e c o m - p o s a b l e n o n - p r o j e c t i v e Z , t h e r e i s a n A u s l a n d e r - R e i t e n s e q u e n c e e n d i n g w i t h Z. M o r e o v e r , i f 0 ~ X + Y + Z + 0 f is a n A u s l a n d e r - R e i t e n s e q u e n c e a n d h : X ÷ X' i s a m a p w h i c h i s n o t a s p l i t m o n o - m o r p h i s m , t h e n t h e r e e x i s t s ~ : Y + X' s u c h t h a t h : ~ f . (For a l l t h e s e p r o p e r t i e s , w e r e f e r to [2]).
I n t h e s e q u e l , w e w i l l c o n s i d e r d i r e c t s u m s o f t h e f o r m U(Y) , w h e r e U(Y) is a n a b e l i a n g r o u p d e p e n d i n g o n Y , w i t h Y Y
r a n g i n g o v e r " a l l " i n d e c o m p o s a b l e m o d u l e s . H e r e , o f c o u r s e , w e c h o o s e f i r s t f i x e d r e p r e s e n t a t i v e s Y o f a l l i s o m o r p h i s m c l a s s e s o f i n d e c o m p o s a b l e m o d u l e s a n d t h e n i n d e x t h e d i r e c t s u m b y t h e s e
r e p r e s e n t a t i v e s . I n f a c t , a l l d i r e c t s u m w h i c h w i l l o c c u r i n t h i s w a y w i l l h a v e e v e n o n l y a f i n i t e n u m b e r o f n o n - z e r o s u m m a n d s .
P R O P O S I T I O N 2.1. L e t X b e a n i n d e c o m p o s a b l e n o n - i n j e c t i v e m o d u l e a n d G b e a d i v i s i o n r i n g w i t h
E n d X = G • r a d E n d X .
A s s u m e t h a t , f o r e v e r y i n d e c o m p o s a b l e m o d u l e y , t h e r e i s g i v e n a d i r e c t c o m p l e m e n t M ( X , Y ) o f r a d 2 ( x , y ) i n E n d Y r a d ( X ' Y ) G L e t
0 - - > X ( X M ( X ' Y ) ) Y > • * M ( X , Y ) ~ Y ~--> Z - - > 0
Y E n d Y
b e e x a c t . T h e n , t h i s i s a n A u s l a n d e r - R e i t e n s e q u e n c e . M o r e o v e r , G e m b e d s i n t o t h e e n d o m o r p h i s m r i n g E n d Z o f Z a s a r a d i c a l c o m p l e m e n t , a n d f o r e v e r y Y , t h e r e i s a n e m b e d d i n g O o f * M ( X , Y ) o n t o a c o m p l e m e n t o f r a d 2 ( y , z ) i n G r a d ( Y , Z ) E n d y s u c h t h a t
X@*M(X,y) : z I *M(X,Y) 0 Y . Proof. Let
(f'y,p)y,p> dy
0 - - > X • ~ Y - - > Z' - - > 0 Y p=l
be an A u s l a n d e r - R e i t e n s e q u e n c e s t a r t i n g w i t h X , w h e r e f' : X ÷ Y Y,p
for 1 < p < dy T h e n the r e s i d u e c l a s s e s o f the e l e m e n t s fY,l' f' Y,2''''' Y , d y f' f o r m a b a s i s o f the G - v e c t o r space rad(X,Y)
G/rad2 (X,y)
G (see L e m m a 2.5 of [5]). Let fY,l' fY,2''''' fY,dy be a G - b a s i s of M(X,Y). By the f a c t o r i z a t i o n p r o p e r t y of A u s l a n d e r - R e i t e n sequences, there is a m a pdy dy
~ : ~ @ Y - - > @ @ Y Y p:l Y p = l
such t h a t ~o(f'x,p).x,p = (fy,p)y,p It follows that ~ is an a u t o - d
m o r p h i s m . For, let E = E n d ( @ ~ Y Y ) and c o n s i d e r the r e s i d u e Y p=l
class ~ of ~ in E / r a d E. Also, c o n s i d e r the factor g r o u p
dy dy
M = rad(X, ~ ~ Y ) / r a d 2 ( X , ~
Y p = I Y p : I
Y ) ,
and let f a n d f' b e the r e s i d u e c l a s s e s of f = (fy,p)y,p and
i t
f = (fy,p)y,p , r e s p e c t i v e l y . T h e n r a d E a n n i h i l a t e s M , a n d the e q u a l i t y <~ f' = f shows that ~ i n d u c e s b a s e c h a n g e s b e t w e e n the
-- f'y,p)p o f Irr(X,Y). This i m p l i e s t h a t b a s e s (fy,p)p a n d (
is invertible. S i n c e rad E is n i l p o t e n t , ~ is i n v e r t i b l e , as well. Thus, we c a n f o r m the f o l l o w i n g c o m m u t a t i v e d i a g r a m
f, ~ y d ,
0 - - > X - - > (~ Y - - > Z - - > 0 Y p = l
Is B
d
0 - - > X f--> • ~ Y Y - - > Z - - > 0 , Y p = l
w h e r e b o t h ~ and ~ are i s o m o r p h i s m s . As a c o n s e q u e n c e , a l s o the l o w e r s e q u e n c e is an A u s l a n d e r - R e i t e n sequence.
d
N o t e t h a t w e can r e w r i t e ~ Y Y as *M(X,Y) O Y , a n d
p=l E n d Y
then (fy,p)p b e c o m e s X M ( X , y ) . For, if ~Y,l' ~Y,2 ... ~ Y , d y is the dual b a s i s of * M ( X ' Y ) E n d Y / r a d E n d Y w i t h r e s p e c t to the b a s i s fY,l' fY,2 ... fY,dy o f End Y/Fad End Y N(X,Y] , then we
i d e n t i £ y d d
*M(X,Y) 0 Y = ~ Y % y , p (9 Y z ~ Y Y ,
EndY p = l p = l
a n d
d
XL(X'Y) (x) = p:l~Y ~ Y ' p ~ fy,p(X)
is i d e n t i f i e d w i t h
.
Now, M(X,Y)
(fy,p(X))p
is a l e f t G-module, a n d X M ( X , y ) : X - - > M(X,Y) @ Y
End Y
m
is a G - m o d u l e h o m o m o r p h i s m . Hence, u n d e r (XM(X,y))y , the m o d u l e X b e c o m e s a G - s u b m o d u l e of • *M(X,Y) ~ Y , a n d t h e r e f o r e a l s o the
Y E n d Y
f a c t o r m o d u l e Z has a left G - m o d u l e structure. Thus, G e m b e d s c a n o n i c a l l y into E n d Z a n d in this way, G b e c o m e s a r a d i c a l complement. T h i s f o l l o w s f r o m the c a n o n i c a l i s o m o r p h i s m
E n d X / r a d E n d X ~ E n d Z / r a d E n d Z ,
w h i c h is a l w a y s v a l i d for the o u t e r terms o f an A u s l a n d e r - R e i t e n sequence.
T h e r e s t r i c t i o n o f z to *M(X,Y) ~ Y d e f i n e s a m a p ~ o f
*M(X,Y) into Hom(Y,Z) w h i c h is a G - E n d Y - h o m o m o r p h i s m . If w e d e n o t e a g a i n b y
~ Y , I ' ~ Y , 2 ' ' ' ' ' ~Y,dy an End Y / r a d End Y - b a s i s
of *M(X,Y) , t h e n w I*M(X,Y) ~ Y - - > Z can b e i d e n t i f i e d w i t h E n d Y
d d
(~y,p)p : ~ Y Y ~ ~ Y ~ y , p ® g - - > Z . p = l p = l
A g a i n , u s i n g L e m m a 2.5 o f [5], w e see t h a t t h e r e s i d u e c l a s s e s o f
~ Y , I ' ~ Y , 2 ' ' ' ' ' ~Y,dy in I r r ( Y , Z ) form an End Y/rad End Y - b a s i s
a n d t h a t M ( X , Y ) i s t h e r e f o r e m a p p e d i n j e c t i v e l y o n t o a c o m p l e m e n t o f r a d - ( Y , Z ) i n G r a d ( Y , Z ) E n d y. T h i s c o m p l e t e s t h e p r o o f .
Now, a s s u m e t h a t X is a n i n d e c o m p o s a b l e , n o n - i n j e c t i v e m o d u l e a n d t h a t G is a r a d i c a l c o m p l e m e n t i n E n d X. If t h e r e a r e g i v e n d i r e c t c o m p l e m e n t s M ( X , Y ) o f r a d 2 ( X , Y ) in E n d
y r a d ( X ' Y ) G '
,
t h e n t h e U M ( X , Y ) a r e d i r e c t c o m p l e m e n t s o f r a d 2 ( y , z ) in G r a d ( Y , Z ) E n d y , a n d t h e A u s l a n d e r - R e i t e n s e q u e n c e s t a r t i n g w i t h X is o f t h e f o r m
( ~ M ( X ' Y ) ) Y > * M ( X U * M ( X ' Y ) ) Y >
0 - - > X - • (X,Y) ~ Y - Z ----> 0 .
D e n o t e b y c ( M ( X , Y ) ) t h e c a n o n i c a l e l e m e n t i n * M ( X , Y ) ~ M ( X , Y ) . N o w 1 : M ( X , Y ) ¢--> H o m ( X , Y ) a n d ~ : * M ( X , Y ) ~----> H o m ( Y , Z ) , a n d t h u s w e h a v e a c a n o n i c a l m a p
* M ( X , Y ) ~ M ( X , Y ) - - > H o m ( X , Z ) , n a m e l y ~ 0 I f o l l o w e d b y t h e c o m p o s i t i o n m a p H •
P R O P O S I T I O N 2.2. U n d e r t h e m a p
* M ( X , Y ) ~ M ( X , Y ) ~ ( ~ @ I~ ~ ~ H o m ( Y , Z ) @ H o m ( X , Y )
Y Y
t h e e l e m e n t [ c ( M ( X , Y ) ) g o e s t o z e r o . Y
(H)> H o m ( X , Z ) ,
O b s e r v e t h a t , f o r a f i x e d m o d u l e X , t h e r e is o n l y a f i n i t e n u m b e r o f m o d u l e s Y s u c h t h a t M ( X , Y ) = I r r ( X , Y ) ~ 0 ; t h e r e f o r e , w e m a y f o r m t h e s u m ~ c ( M ( X , Y ) ) .
Y
P r o o f o f P r o p o s i t i o n 2.2. F i r s t , w e a r e g o i n g to s h o w t h a t c ( M ( X , Y ) ) m a p s o n t o X U , M ( X , y ) 0 X M ( X , y ) . L e t f l ' f 2 ' ' ' ' ' f d b e an E n d Y / r a d E n d Y - b a s i s o f E n d Y / r a d E n d y M = M ( X , Y ) , a n d
~ i ' ~ 2 ' .... ~ d t h e c o r r e s p o n d i n g d u a l b a s i s i n * M E n d Y / t a d E n d Y"
T h e n , f o r x E X , w e h a v e
= E ~ p (9 fp(X) ,
XM(x) p
a n d f o r ~ s M, y E Y ,
~ j * M ( 4 ~ y) : 0(4 ) (y) Thus,
X~, M X M ( X ) = X U , M (~ ~p ~ fp(X)) = PZ ~(~p) (fp(X))
This shows t h a t XO, M X M is e q u a l to ~ O ( ¢ p ) f , a n d this is the
p P
image of ~ 4 O f = c(M(X,Y)) u n d e r ~ ( ~ (9 I) A s a c o n s e q u e n c e ,
p P P
we c o n c l u d e that u n d e r the m a p ~ *M(X,Y) O M(X,Y) ~ ( O (9 I)>
Y
Hom(Y,Z) (9 Hom(X,Y) (~)> Hom(X,Z) , the e l e m e n t ~ e(M(X,Y)) goes
Y Y
to Y~ X ( j . M ( X , y ) XM(X,y) , w h i c h i s t h e c o m p o s i t e o f t h e two m a p s i n t h e c o r r e s p o n d i n g A u s l a n d e r - R e i t e n s e q u e n c e a n d t h u s z e r o . T h e p r o o f is c o m p l e t e d .
L e t us p o i n t o u t that, in w h a t follows, w e shall n o t s p e c i f y a n y l o n g e r the e m b e d d i n g o of *M(X,Y) into Hom(Y,Z) , b u t shall s i m p l y c o n s i d e r *M(X,Y) to b e a s u b s e t of H o m ( Y , Z ) .
REMARK. L e t us u n d e r l i n e the use of the two d i s t i n c t t e n s o r p r o d u c t s M(X,Y) (9 *M(X,Y) a n d *M(X,Y) (9 M(X,Y) . W h e r e a s the f i r s t one is u s e d for the o r d i n a r y e v a l u a t i o n m a p
k : M(X,Y) (9 *M(X,Y) ----> E n d Y / r a d E n d Y
g i v e n b y X (f (9 4) = f(4) , it is the s e c o n d o n e w h i c h h a s to b e u s e d for the c o m p o s i t i o n m a p p . Namely, u s i n g the a b o v e e m b e d d i n g
*M(X,Y) ~---> Hom(Y,Z) , w e can c o n s i d e r
*M(X,Y) (9 M(X,Y) ~--> Hom(Y,Z) (9 Hom(X,Y) ~ > Hom(X,Z) , a n d ~ ( 4 (9 f) = 4 0 f .
3. The p r e p r o j e c t i v e m o d u l e s
Now, l e t us c o n s i d e r the p a r t i c u l a r case o f the i r r e d u c i b l e m a p s b e t w e e n i n d e c o m p o s a b l e p r e p r o j e e t i v e R ~ Q ) - m o d u l e s . First, r e c a l l the w a y in w h i c h these m o d u l e s can b e i n d u c t i v e l y o b t a i n e d f r o m the i n d e c o m p o s a b l e p r o j e c t i v e ones.
F o r e a c h i s I , there is an i n d e c o m p o s a b l e p r o j e c t i v e R ~ ) - m o d u l e P(i). Indeed, d e n o t i n g b y e. the p r i m i t i v e idem-
1
p o t e n t o f R ~ , ~ ) c o r r e s p o n d i n g to the i d e n t i t y e l e m e n t of the i th
f a c t o r F. i n R = ~ F i , P(i) = eiR~,~).__ N o t e t h a t
i o i
P ( i ) / r a d P(i) i s t h e s i m p l e R ( ~ ) - m o d u l e c o r r e s p o n d i n g t o t h e v e r t e x i w h i c h d e f i n e s P(i) u n i q u e l y u p t o a n i s o m o r p h i s m . M o r e o v e r , n d t e t h a t E n d P(i) = F , , a n d t h u s i t is a d i v i s i o n r i n g .
1
T h e i r r e d u c i b l e m a p s b e t w e e n p r o j e c t i v e m o d u l e s a r e a l w a y s r a t h e r e a s y t o d e t e r m i n e . H e r e , f o r R ~ , ~ ) , t h e r e a r e i r r e d u c i b l e m a p s f r o m P(j) t o P(i) i f a n d o n l y i f i ÷ j i n ~ . In f a c t , .M.
i 3 c a n b e e a s i l y e m b e d d e d i n H o m (P(j), P ( i ) ) i n s u c h a w a y t h a t
. M ~ r a d 2 ( p ( j ) , P ( i ) ) = r a d (P(j), P ( i ) ) i 3
a s F . - F . - b i m o d u l e s . T h i s f o l l o w s e i t h e r f r o m t h e e x p l i c i t
1 ]
d e s c r i p t i o n o f t h e m o d u l e s P(i) g i v e n i n [3], o r f r o m t h e f a c t t h a t
• .M. is a d i r e c t c o m p l e m e n t o f r a d 2 R ( ~ , ~ ) in r a d R ~ , ~ ) . A s a i 3
r e s u l t , g i v e n t w o i n d e c o m p o s a b l e p r o j e c t i v e R ( ~ ) - m o d u l e s P a n d
p W i
, w e c a n a l w a y s c h o o s e a d i r e c t c o m p l e m e n t M ( P , P ) o f r a d 2 ( p , P ') i n E n d P' r a d ( P ' P ' ) E n d P ' a n d w e c a n i d e n t i f y t h e s e M ( P , P ' ) w i t h t h e g i v e n b i m o d u l e s .M., w h e r e i ÷ j .
i 3
Now, t h e i n d e c o m p o s a b l e p r e p r o j e c t i v e m o d u l e s c a n b e d e r i v e d f r o m t h e p r o j e c t i v e o n e s b y u s i n g p o w e r s o f t h e C o x e t e r f u n c t o r C -
(as d e f i n e d i n [3]) o r o f t h e A u s l a n d e r - R e i t e n t r a n s l a t i o n A - = T r D ( " t r a n s p o s e o f d u a l " o f [2], a n d a l s o [i]). T h u s , w e d e n o t e b y
t h
P ( i , r ) t h e m o d u l e o b t a i n e d f r o m P(i) b y a p p l y i n g t h e r p o w e r o f o n e o f t h e m e n t i o n e d c o n s t r u c t i o n s . (It is c l e a r f r o m t h e u n i q u e n e s s r e s u l t i n [3] t h a t C - r P i) ~ A - r P ( i ) . )
L E M M A 3.1. A s s u m e t h a t x a n d Y a r e i n d e c o m p o s a b l e m o d u l e s a n d t h a t t h e r e e x i s t s a n i r r e d u c i b l e m a p X + Y . I f o n e o f t h e m o d u l e s X, Y i s p r e p r o j e c t i v e , t h e n b o t h a r e . F u r t h e r m o r e , i f X = P ( i , r ) a n d Y = P ( j , s ) , t h e n e i t h e r s = r a n d i ÷ j , o r s = r + l a n d i ÷ j .
P r o o f . T h i s l e m m a is w e l l - k n o w n , s o l e t u s j u s t o u t l i n e a +
p r o o f . U s i n g s h i f t s b y p o w e r s o f t h e C o x e t e r f u n c t o r s C a n d C - (see [3]) o r o f t h e A u s l a n d e r - R e i t e n t r a n s l a t i o n s A = D T r a n d A = T r m (see [2] a n d [i]), w e c a n a s s u m e t h a t X i s p r o j e c t i v e . If Y is n o t p r o j e c t i v e , t h e n w e g e t f r o m t h e A u s l a n d e r - R e i t e n s e q u e n c e e n d i n g w i t h Y , a n i r r e d u c i b l e m a p f r o m A Y t o X .
S i n c e X is p r o j e c t i v e , this m a p c a n n o t be a n e p i m o r p h i s m a n d thus it has to be a m o n o m o r p h i s m . C o n s e q u e n t l y , A Y is p r o j e c t i v e .
Now, in v i e w of P r o p o s i t i o n 2.1, we o b t a i n b y i n d u c t i o n on the
"layer" r o f the i n d e c o m p o s a b l e p r e p r o j e c t i v e R ~ , ~ ) - m o d u l e s P(i,r) the f o l l o w i n g result.
P R O P O S I T I O N 3.2. a) I f w e c h o o s e , f o r a n y t w o i n d e c o m p o s a b l e p r o j e c t i v e m o d u l e s P a n d P' , a d i r e c t c o m p l e m e n t M(P,P') o f r a d 2 ( p , P ') i n E n d p ' r a d ( P ' P ' ) E n d P ' t h e n t h i s d e t e r m i n e s a d i r e c t c o m p l e m e n t M(P,P') o f r a d 2 ( p , P ') i n r a d (P,P') f o r a n y i n d e -
l c o m p o s a b l e p r e p r o j e c t i v e m o d u l e s P, P .
b) I f w e i d e n t i f y , f o r a n y a r r o w i ÷ j t h e b i m o d u l e M(P(j), P(i)) w i t h .M. , t h e n t h i s y i e l d s a n
I 3 (2r)M.
i d e n t i f i c a t i o n o f a n y M(P(j,r) , P(i,r)) w i t h . a n d a n y
± 3 M ( P ( i , r ) , P(j,r+l) w i t h ( 2 r + l ! M f o r i + j .
i 3
P R O P O S I T I O N 3.3. E v e r y m a p b e t w e e n t w o i n d e c o m p o s a b l e p r e p r o - j e c t i v e m o d u l e s i s a s u m o f c o m p o s i t e s o f m a p s f r o m t h e v a r i o u s M(P,P') .
Proof. L e t Y b e an i n d e c o m p o s a b l e p r e p r o j e c t i v e module, say Y = P(i,r). T h e n the r a d i c a l of the e n d o m o r p h i s m r i n g E of
P(j,s) is g e n e r a t e d (by u s i n g the a d d i t i o n a n d m u l t i p l i c a t i o n ) j s I
0 < s < r
b y an a r b i t r a r y c o m p l e m e n t of R a d 2 E in Rad E. So we m a y c h o o s e as a c o m p l e m e n t the d i r e c t sum of M ( P ( j , s ) , P ( j ' , s ' ) ) .
4. A b s t r a c t d e f i n i t i o n of the full s u b c a t e g o r y o f the p r e p r o j e c t i v e m o d u l e s
First, let us i n t r o d u c e the f o l l o w i n g n o t a t i o n i n d i c a t i n g the o p e r a t i o n of the d i v i s i o n rings F. a n d F. : For i ÷ j , p u t
l 3
2~M. : (2r) (.M.) a n d 2r+l.M. = (2r+l) (.M.) .
1 3 l ] ] l 1 3
NOW, d e f i n e the c a t e g o r y ~ ( ~ ) as follows: The o b j e c t s of
~ , ~ ) are p a i r s (i,r) , i c I , r > 0 w i t h the e n d o m o r p h i s m r i n g s F. . F o r i ÷ j ,
1
2r
M((j,r) , (i,r)) = . M
1 3
a n d
2 r + l M ( ( i , r ) , (j,r+l)) : M .
3 l
D e n o t e b y ~(~,~) t h e f r e e c a t e g o r y g e n e r a t e d b y t h e s e m o r p h i s m s u s i n g t h e t e n s o r p r o d u c t s o v e r F. . F u r t h e r m o r e , f o r e v e r y (j,r),
1 t a k e
c ( j , r ) = [ c( 2r M ) + Z c(2r+~-Mj)n E
]
i ÷ j i 3 j + k
2 r + l 2r M 2 r + 2 2 r + l •
( .M. @ .) • • ( j ~ ® kMj} ,
i+j 3 i i 3 j+k
a n d d e n o t e b y J t h e c a t e g o r y i d e a l g e n e r a t e d b y a l l e l e m e n t s c ( j , r ) . T h e c a t e g o r y P(~,Q) is t h e n d e f i n e d as t h e f a c t o r c a t e g o r y o f ~ ( ~ ) b y t h e i d e a l J .
O b s e r v e t h a t t h e d e f i n i t i o n o f P ~ , ~ ) r e q u i r e s o n l y t h e k n o w l e d g e o f t h e b i m o d u l e s M . f o r i ÷ j (and n e i t h e r the
1 3 cj i
c o r r e s p o n d i n g b i m o d u l e s M , n o r t h e b i l i n e a r f o r m s a n d £.).
] l i 3
P R O P O S I T I O N 4.1. The full subcategory of the preprojective modules o f the category of all T ~ , ~ ) - m o d u l e s is equivalent to
p(~)
P r o o f . U s i n g P r o p o s i t i o n 3.2, t h e r e is a c a n o n i c a l f u n c t o r F f r o m ~(~,Q) to t h e s u b c a t e g o r y o f p r e p r o j e c t i v e T ( ~ , Q ) - m o d u l e s g i v e n b y t h e c h o i c e o f M ( P ( i ) , P ( j ) ) : .M. f o r p r o j e c t i v e m o d u l e s
3 ±
P ( i ) , P ( j ) w h e r e j + i . A l s o b y P r o p o s i t i o n 3.3, F is s u r j e c t i v e . M o r e o v e r , a c c o r d i n g to P r o p o s i t i o n 2.2, t h e e l e m e n t s c ( j , r ) a r e m a p p e d to zero.
C o n v e r s e l y , l e t a m o r p h i s m f : (j,r) + (j',r') f r o m
~ , ~ ) b e m a p p e d u n d e r F to zero. W e a r e g o i n g to s h o w t h a t f m u s t l i e i n t h e i d e a l J . T h i s is c l e a r if r = r' ; for, t h e n f = 0 . T h u s , a s s u m e t h a t f ~ 0 a n d p r o c e e d b y i n d u c t i o n o n r' - r . N o w j a n d r a r e f i x e d ; l e t { . . . g p . . . } b e t h e u n i o n o f
2 r M
b a s e s o f a l l v e c t o r s p a c e s ( . .) f o r a l l i w i t h i ÷ j a n d
F. i 3
1
] I
F k ( 2 r + k M j ) f o r a l l k w i t h j + k , a n d l e t {... g p ...] b e t h e
2 r + l • (2r+2
u n i o n o f t h e c o r r e s p o n d i n g d u a l b a s e s o f ( .M.;~ a n d )
3 i ~ i J ~ F k
Thus, c(j,r) = ~ g' ® gp . Now, f = ~ h @ gp , w h e r e h is a
p P p P P
T l
m o r p h i s m o f F ~ ) e i t h e r f r o m (i,r) o r (k,r+l) to (j ,r ). S i n c e t h e r e is a n A u s l a n d e r - R e i t e n s e q u e n c e
(F (gp)) (F (gp))
0 - - > P(j,r) P> Q - P-> p ( j , r + l ) - - > 0 a n d s i n c e
0 = F(f) : v F(hp) r(gp) , P
we c a n f a c t o r ( F ( h ) ) : Q ÷ P ( j ' , r ' ) t h r o u g h (F(g'))
P P P P
t h e r e is a h o m o m o r p h i s m u : P ( j , r + l ) + P ( j ' , r ' ) s u c h t h a t
H e n c e ,
F(h ) = { F(g')
P P
And, s i n c e F is s u r j e c t i v e , w e can f i n d u : (j,r+l) ÷ (j',r') in
~(~,~) s u c h t h a t F(u) = u . O b v i o u s l y , the e l e m e n t s h p - u ~ gp ' lie in the k e r n e l of F , a n d t h e r e f o r e , b y i n d u c t i o n , t h e y b e l o n g to J . C o n s e q u e n t l y ,
f = ~ h p ® gp = [ (hp - u (9 gp) O gp + ~ u ® g'p ® gp i
P P P
a l s o b e l o n g s to J ; for, ~ u ® gp' @ gp u ® c(j r) P
5. P r o o f o f the t h e o r e m
The p r o o f of the t h e o r e m c o n s i s t s in i d e n t i f y i n g the a d d i t i v e s t r u c t u r e of ~(~) w i t h a f a c t o r of a s u b c a t e g o r y of ~(~,~)
I n d e e d , w e m a y c o n s i d e r b o t h ~ , ~ ) and P ~ , O ) d e f i n e d in s e c t i o n 4 as a b e l i a n g r o u p s f o r m i n g the d i r e c t s u m o f all H o m ( ( i , r ) , (j,s)).
D e n o t e b y ~(~,~) a n d ~ , ~ ) the r e s p e c t i v e s u b g r o u p s o f a l l H o m ( ( i , 0 ) , (j,s)) . Then, b o t h } ~ ) a n d ~ ) c o n t a i n a s u b - r i n g R = ~ H o m ( ( i , 0 ) , (j,0)) w h i c h is o b v i o u s l y i s o m o r p h i c to
i,j
R ~ , Q ) . F u r t h e r m o r e , u n d e r the c o m p o s i t i o n in ~ , ~ ) , H ( ~ O ) is a r i g h t R ~ O ) - m o d u l e ; for, if f : (i,0) ÷ (j,s) a n d a : (k,0) ÷
(i,0) f r o m R , t h e n fa -. (k,0) ÷ (j,s) in H ~ , O ) . P R O P O S I T I O N 5.1.
sum of all~preprojective multiplicity one).
H ( ~ ' O ) R ~ , ~ ) is isomorphic to the direct R ~ , O ) - m o d u l e s (each occurring with
Y = i n d e c o m p o s a b l e
P r o o f . U s i n g t h e n o t a t i o n o f s e c t i o n 3, t h e i n d e c o m p o s a b l e p r e p r o j e c t i v e R - m o d u l e s a r e P ( j , s ) , j ~ I , s > 0. I D p a r t i c u l a r , P ( j , 0 ) a r e t h e i n d e c o m p o s a b l e p r o j e c t i v e R - m o d u l e s a n d t h u s
R R = . • P ( i , 0 ) . F o r e v e r y R - m o d u l e X R , i £ I
X R = H o m ( R R R , XR) = H o m ( R [ ~ P ( i , 0 ) ] , X R) = 1
H e n c e ,
= [Hom(~ P(i,0)R, XR)] R : [~ Hom(P(i,0)R, XR)] R .
i i
P ( j , s ) : [ ~ H o m ( P ( i , 0 ) , P ( j , s ) ) ] R i
a n d t h u s u n d e r t h e i d e n t i f i c a t i o n o f P ( j , s ) w i t h (j,s) a n d H o m ( P ( i , 0 ) , P ( j , s ) ) w i t h t h e m a p s i n X ~ , ~ ) , w e g e t t h e s t a t e m e n t .
N o w , d e f i n e t h e m a p h : T ~ + [ ~ , ~ ) a s f o l l o w s . F i r s t , t h e m o r p h i s m s i n [ ~ ) c a n b e d e s c r i b e d i n t h e f o l l o w i n g w a y : F o r a n ( u n o r i e n t e d p a t h ) w = i n + 1 - in - ... - i 2 - i I o f ~ , c a l l
÷ i t 1 < t < n , i n Q t h e l a y e r t h e n u m b e r o f a r r o w s i t + 1
l(w) o f w . T h e n , t h e m o r p h i s m s i n [ ~ , ~ ) a r e t h e e l e m e n t s o f t h e t e n s o r p r o d u c t s
r r 2 r 1
nM. {9 ... {9 . M. {9 . M. ,
i n + 1 i n 1 3 12 12 l 1
- i 0 ÷ i t
w h e r e r t = 2 1 ( i t i t - i - ' ' ' - i2 - il) + [i i f i t + l ÷ i i f i t + 1 t N O W , t h e m a p h is d e f i n e d b y
r _ r l r r 2 r 1
r n Q {9. .{9 2 ~9 -~-> n M {9...(9. M. O . M. ,
M. {9 ... {9 . M. {9 . M. " . .
i n + 1 1 n 13 12 12 1 1 i n + 1 1 n 13 12 12 11
w h e r e r Q a r e t h e m a p s o f L e m m a 1 . 2 f o r M = .M. a n d N = .M..
1 ] j 1
F r o m t h e d e f i n i t i o n o f ~ ) , i t i s c l e a r t h a t ~ , ~ ) i s j u s t t h e i m a g e o f T ~ u n d e r 4. A l s o , A is o b v i o u s l y R ~ , Q ) - l i n e a r .
L E M M A 5.2. h ( < c > ) = J ~ ~ )
P r o o f . B y d e f i n i t i o n , c = Z (~ c i) = ~ c(j) ; n o t e t h a t
j i 3 j
c(j) = e. c e. , w h e r e e. is t h e i d e m p o t e n t o f T ~ c o r r e s p o n d i n g
3 ] ]
to the identity of F. ; thus <c> is the ideal generated by all ]
I
c(j) s. Hence, the statement follows from Lemma 1.2 taking into account that, by definition,
A(I ~ 1 ~ ... ~ c(j) ~ ... Q i) = 1 ~ 1 ~ ... ® c(rM) ~ ... ~ 1 . Now, from Lemma 5.2, it follows that ~ defines an isomorphism of ~ = T ~ / < c > onto ~ ( ~ ) = ~ , ~ ) / J N ~ , ~ ) . This completes the proof of the theorem.
The corollaries follow from the results in [2].
[z]
[2]
[3]
[4]
[5]
[6]
[7]
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Gelfand, I.M. and Ponomarev, V.A.: Model algebras and represent- ations of graphs. Funkc. anal. i prilo~. 13 (1979), 1-12.
Ringel, C.M.: Report on the Brauer-Thrall conjectures:
Rojter's theorem and the theorem of Nazarova and Rojter.
These Lecture Notes.
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Department of Mathematics Carleton University Ottawa, Ontario KIS 5B6 Canada
Fakult~t f~r M a t h e m a t i k Universit~t
D-4800 Bielefeld West Germany
