Endomorphisms of finetely generated projective modules over a commutative ring
GERT ALMKVIST*
University of CaIifornia, Berkeley, U.S.A. and University of Ltmd, Sweden
Introduction
T h e origin o f this p a p e r is a m i s p r i n t (?) in B o u r b a k i ([4], p. 156, E x e r c i s e 13 d).
T h e r e it is s t a t e d t h a t i f f is a 2 • 2 - m a t r i x w i t h entries in a c o m m u t a t i v e ring a n d f 2 = 0 t h e n ( T r f ) 4 = 0 a n d 4 is t h e smallest i n t e g e r w i t h this p r o p e r t y . Using t h e C a y l e y - H a m i l t o n t h e o r e m we get f2 _ af ~- b l = 0 w h e r e a = T r f a n d b = d e t f . N o t i n g t h a t f 2 = 0 a n d t a k i n g t r a c e s we g e t a - T r f = a 2 = 2b.
M u l t i p l y i n g t h e f i r s t e q u a t i o n b y f gives bf = 0 w h i c h implies b 9 T r f = ba = O.
H e n c e a a = 2ab = 0 so 3 a n d n o t 4 is t h e smallest i n t e g e r a b o v e . E x p e r i m e n t i n g w i t h small m a n d n one soon m a k e s t h e c o n j e c t u r e : I f f is a n n •
w i t h f ~ + i = 0 t h e n ( T r f ) m"+l = 0. This is p r o v e d in a s o m e w h a t m o r e general s e t t i n g in 1.7 using e x t e r i o r algebra.
I n S e c t i o n 1 t h e c h a r a c t e r i s t i c p o l y n o m i a l 2t(f) is d e f i n e d for a n e n d o m o r p h i s m f : P - - ~ P w h e r e P is a f i n i t e l y g e n e r a t e d p r o j e c t i v e A - m o d u l e (A is a com- m u t a t i v e ring w i t h 1). I f P is free t h e n 2t(f) = d e t (1 ~- tf). T h e e x p o n e n t i a l t r a c e f o r m u l a (in case A contains Q)
~ Tr (f~) )
A,(f) = e x p - - ~ i ( - - t)~
c o n n e c t s At(f) w i t h t h e t r a c e s o f t h e powers o f f .
Various c o m p u t a t i o n s o f 2t(f) are m a d e in S e c t i o n 2. B y t h e i s o m o r p h i s m EndA (P) - - > P * @ A P w h e r e P * = t t o m A (P, A) e v e r y f: P---->P c o r r e s p o n d s t o a t e n s o r ~ x* @ x, w i t h x* e P * , x~ C P . L e t M ( f ) be t h e m a t r i x w i t h entries aq = <x*, xj>. T h e n At(f) = d e t (1 -~ tM(f)). E v e n t h e c o m p u t a t i o n o f ~t(1p)
* This research was partly supported by the Swedish Natural Science Foundation.
264 G E R T A L l V I K V I S T
where 1p is the identity map is not quite trivial. The result is 2t(lp) = ~ ] el(1 d- t) ~ where the e~:s are the idempotents given b y Ann A ~ P = (eo d- el -4- 9 9 9 d- e~_l)A.
I n Section 3 the behaviour of 2,(f) under change of rings and taking duals is studied. Some a t t e m p t s are made to connect the polynomials 2 , ( f ) , ~t(g) and 2,(f | g). I n the multiplicative group A = {1 ~- alt d- a2t 2 @ 9 9 .; a, e A } of formal power series with constant term 1 one can define a .-multiplication such t h a t 2,(f @ g) = 2,(f) 9 2t(g ). Then A becomes a ring (with ordinary multiplication as addition).
A formula for computing 2,(f) in terms of the minimal polynomial of f and some of t h e ' Tr (f~):s is given in Section 4.
I n Section 5 the definition of 2,(f) is extended to f: 21I--~ M where M is an A-module having a finite resolution of finitely generated projective modules.
Some of the results in Section 1 can be generalized to this case. Furthermore 2~(f) is defined for f = chain map of complexes (or map of graded A-modules).
Section 6 contains an a t t e m p t to classify all endomorphism of finitely generated projective A-modules, i.e. to compute the K-group K 0 (End ~(A)). The charac- teristic polynomial 2,(f) is sometimes a good enough invariant. This is the case if A i s a P I D o r A = K [ X , Y] where K i s a f i e l d o r A is a regular local ring of dimension at most two. Then K o (End ??(A)) is isomorphic (as a ring) with the direct product of K o ( A ) = Z and the ring of all ))rational functions))
1 + a l t @ . . . ~ - a m t'~
1 @ blt d- . . . -~ b~t ~
(under multiplication and ,-multiplication). This generalizes a result b y Kelley- Spanier ([8] p. 327) for A = field. The ring of ))rational functions)) is also isomorphic with a subring of the W i t t ring W ( A ) of A. Finally ))trace sequences~), (Tr (f~)~
are studied.
Finally I would like to t h a n k T. Farrell, G. Hochschild, M. Schlessinger and M. Sweedler for m a n y valuable discussions about this paper and mathematics in general.
1. The characteristic polynomial
First we fix some notation. A will always denote a commutative ring with u n i t y element 1. Spec A is the set of all prime ideals ~ of A. I f x C M where M is an A-module we denote b y x~ the image of x under the localization m a p M - ~ M ~ = M @ A A ~ for all p C S p e c A .
The category of all finitely generated projective A-modules will be denoted b y ~ ( A ) . I f P C ~ ( A ) then Po is a free Ao-module of finite rank --~ rk~P. We define r k P = max~ rk~P. This integer is equal to the minimal nmnber of generators of P . I f r k ~ P = r k P for all p E S p e e A we say t h a t P has c o n s t a n t r a n k . Let
E N D O M O R P I I I S M $ O F F I N I T E L Y G E N E R A T E D P R O J E C T I V E M O D U L E S 2 6 5
P * = HOmA ( P , A ) be the dual of P. Then for P E ~ ( A ) there are natural isomorphisms of A-modules
EndA (P*) -+ HomA (P* | P, A) --~ HomA (Enda P, A) (*) Let Tr be the image of 1~, under the composed map. We call Tr (f) the trace of f: P -+ P. This coincides with Bourbakis definition ([3] p. 112).
Definition 1.1:
L ( f ) = ~ Tr (A~f)t'
i = 0
Here t is an indeterminate, f : P - + P an endomorphism with P E ~ ( A ) , A~: A~P -+ A~P the induced endomorphism of the i:th exterior power of P and n = rkP. Observe t h a t A~P E ~ ( A ) ([4], p. 142).
R e m a r k 1.2. I f P is free thell Tr (f) is the usual trace of f and L(f) det ( 1 + tf) where 1 = identity of the free A[t]-module P @a A[t]. This is a well known formula ([9] p. 436).
PROPOSITION 1.3. Let f , g: P - + P with T h e n
(i)
(ii)
(iii) (iv)
P C ~ ( A ) and p E S p e c A be given.
(Trf)o -- Trf~
(2,(f)~ = 2,(L ), i.e. i f 2,(f) = 1 4- alt @ . . . @ a,,t" then
;t,(f) = 1 + alot + . . . + a.~t ~ L ( f o g) = 2,(9 o f )
L(h o f o h -1) = L ( f ) i f h: P --~ Q ks an isomorphism.
Proof.
(i) Localization commutes with everything in (,) since all modules involved (P*, E n d A (P*) etc.) are in ~ ( A ) ([4], p. 98).
(ii) Localization commutes with exterior powers, (A~)~ = A~f~, so (ii) follows from (i).
(iii) We have T r ( f o g ) = T r ( g o f ) ([3J, p. 112) and A ~ ( f o g ) = A ~ o A ~ g . (iv) B y (ii) it is sufficient to prove (iv) for P free (and hence Q is free), in
which ease it is well known.
CAYLEY-HAMILTON THEOREM 1.4. L e t ),,(f) = 1 + alt + . . . + a~t n and define qf(t) = t" - - alt ~-~ + . . . + ( - - 1)'%. Then qf(f) = O.
Proof. I t suffices to show
n _ _ a ~ n - 1 ~ 9 ~ - 0
(qf(f))~ = f~ l~j~ + . . . + ( - - 1) a., 1~
for all p E Spec A. B u t this follows from the ordinary Cayley-Hamilton theorem j o r f,: P ~ - + P ~ with P~ free since
266 G E I ~ T A L M K V I S T
t "~ - - alot ~ - i JF . . . d- ( - - 1)~a~ ---- t'-'i(%~)qf~(t).
BI~OYOSITION 1.5..Let
be a c o m m u t a t i v e d i a g r a m w i t h exact row a n d all
d d
( - - 1)i T r f~ = 0 a n d ]'-[
0 0
0 - - + P a - + . . . - + P i --+Po - ~ 0
0 - - > P d - - > 9 9 9 - - > P 1 - - > P o - - > 0
P~ r ~ ( A ) . T h e n
~,(L)(-~)" = i
P r o o f . Since localization is a n e x a c t f u n c t o r it is (using 1.3 (i), (ii)) s u f f i c i e n t to p r o v e t h e p r o p o s i t i o n w h e n all P~ are free. B u t t h e n it is well k n o w n a t l e a s t for d ~-- 2 (see [9], p. 402) a n d t h e general case follows b y splitting u p t h e long e x a c t sequence i n t o s h o r t ones.
COROLLAR~ 1.6.
T r ( f @ g ) : T r f + T r g a n d ~ ( f O g) ~-- ~,(f) " ~(g).
T H ~ O R ~ 1.7. JLet f: P - - > P be given w i t h
P e ~ ( A ) , r k P = n a n d
),,(f)
= 1 + a~t + . . . + ant ~.(i) A s s u m e that f is n i l p o t e n t w i t h f ~ + l = 0. T h e n a~'a[~.., a2, = O i f the weight vl + 2v: -~- . . . + nv~ ~ m n . T h e constant m n is best possible.
(ii) C o n v e r s e l y a s s u m e that a~l'a~ " . . . a: ~ = 0 w h e n rl d- 2v: -~- . . . d- n % > k.
T h e n f~+k = O. T h e integer n + k i s best possible.
P r o o f . (i) A f t e r localizing a n d using 1.3 (ii) we m a y a s s u m e t h a t P is free o f r a n k n (it is s u f f i c i e n t t o consider t h e case o f m a x i m a l r a n k ) . L e t P h a v e basis el, e2 . . . . , en. T h e n A n P is free w i t h basis el A e~ A . . . A e,,. N o w we claim t h a t a r e l A e ~ A . . . A e = ~ e l A . . . A f % A . . . A f % A . . . A f e i A . . . A e n (**)
il < i~ < : . . . < i r
B y d e f i n i t i o n we h a v e ar = T r (Art). L e t ell/~ el2/~ . . . A
eir
be a f i x e d basis e l e m e n t o f A r P (with i l < i 2 < . . . < i r ) . T h e nA r f ( e ~ A . . . A e i , ) = f % A . . . A f e ~ r = Ci,, .... # % A . . . A e # + o t h e r terms.
H e n c e
a~ ---- T r ( A ' f ) = ~ C,1 , .... ,,
il < i~ ~ . . . ~ i r
E N D O M O R P H I S M S O F F I N I T E L Y G E N E R A T E D P R O J E C T I Y E M O D U L E S 267 E x p a n d i n g the right h a n d side in (**) one easily gets
( ~ C,,, .... # / e l A e 2 A . . . A e ~
i l < i~ < : : " " < i r ]
~nd the claim is proved.
Using (**) several times we get
v I v~ . . . =
al a2 9 9 9 a ~ ( e l A e2 A A e~) ~ f ' % A f ~ % A . . . A f'.e~
where the sum is t a k e n over all s ~ , s 2 , . . . , s ~ such t h a t s i W s 2 + . . . + s ~ = v~ + 2v~ + . . . + nv~ which by assumption is larger t h a n ran. Hence each term contains an s~ > m a n d f f = 0. Therefore the right h a n d side is zero a n d the first p a r t of (i) is proved.
To see t h a t m n is best possible let A be the commutative ring generated b y 1,
~1 . . . . , ~ with the only relations ~ T + I = a~ +~ . . . a : + l = 0. L e t f be the map given b y the diagonal matrix
I
~ 1 0 ~ 2f = 0
Then f,~+l = 0 and
all sl, 8 2 , . . . . , 8 n
h + 2v2 + 9 9 9 + nv~ < m n hence a~ 1 . . . a ~ ~ # 0.
(ii) Assume t h a t a ~ ' . . . a : ~ = 0 if v l + 2 v 2 + . - - + n v ~ > k . H a m i l t o n theorem we have
= ~ ~2 ~ where the sum runs over
with S l + S ~ + . . . . + s n = v l + 2v~ + . . . + n v ~ . I f
f'~ = a l l ~-1 - - a 2 f "-2 + . . . ~ a n l , Multiplying by f and using Cayley-Hamilton again we get
f,~+l = a ~ f , , - 1 + . . . i ala,,1.
Repeating the procedure several times we get
f" : q,_~+lf n-1 + q r _ n + 2 f n-2 + . . . + q," 1
where q~ is a polynomial in al . . . . ,an of weight i. I f r : k + n then q , : q , - l : q , - ~ + l = O and we get f ' : O.
To show t h a t n + k is best possible let A : Z [ X 1 , X 2 . . . , X n ] / I where X1 . . . . , X~ are indeterminates and I is the ideal generated by all monomiMs in X1 . . . X~ of weight k + l . P u t
$1 s n
then there i s a t e r m ~1 . . . a n with all s ~ m and B y the Cayley-
2 6 8 G E R T ALIVII<VIST
f =
0 0 0 0 (-- 1) "-i a n l 1 0 0
0 1 0
0 -- a e
0 0 0 1 a i /
w h e r e at is t h e r e s i d u e o f X~. T h e n a c a l c u l a t i o n shows t h a t 2,(f) = 1 + alt -[- . . . + a.t"
a n d t h a t f,.+k-1 ~ O.
COROr~LiRV 1.8. f is nilpotent i f and only i f all coefficients ai(i ~ 1) of 2,(f) are nilpotent.
PROPOSlTIO~ 1.9. Given f: P --~ P with P C ~ ( A ) . I f f| = f @ f @ . . . | f = 0 t h e n i f = 0.
Proof. L o c a l i z i n g we m a y a s s u m e t h a t P is free o f r a n k n. L e t (%) b e t h e m a t r i x o f f in s o m e basis a n d I t h e ideal in A g e n e r a t e d b y t h e coefficients (%).
T h e e n t r i e s of t h e m a t r i x o f f| a r e j u s t all possible p r o d u c t s o f v o f t h e a~j:s.
Since f| = 0 we g e t I" = 0. T h e e n t r i e s (cij) of t h e m a t r i x o f f f are c e r t a i n s u m s o f p r o d u c t s o f v o f t h e %.:s. H e n c e c ; j 6 I " a n d % = 0 f o r all i, j a n d i f = 0 .
T~EOnnS~ 1.10 ( e x p o n e n t i a l t r a c e f o r m u l a ) . Let f: P - - > P P 6 ~ ( A ) . Then
d
- - t2,(f) -1 ~
2,(f) = ~
T r(f')(--
t)'1
be A-linear with
Proof. S e t t i n g b~ = T r ( - - f)~ a n d 2,(f) = 1 + alt + . . . -~- ant" we m u s t p r o v e - - (a~t + 2a2 t2 -~- . . . -~ na~t") --
(1 +
a~t @- . . . 4- a.t") ~ b~t ~1
C o m p a r i n g t h e coefficients of t ~ on b o t h sides one f i n d s b~ = O~(al . . . . , a~) w h e r e t h e Oi:s a r e c e r t a i n p o l y n o m i a l s w i t h i n t e g e r coefficients. L o c a l i z i n g a t ~ 6 Spee A we h a v e t o s h o w b~ = Q~(alv .. . . , a~). H e n c e it is s u f f i c i e n t t o s h o w t h e f o r m u l a w h e n P is free a n d f is a m a t r i x . T h e n b~ = Q~(al, . . . , an) b e c o m e s a p o l y n o m i a l i d e n t i t y (over Z) in t h e coefficients of t h e m a t r i x f. T h e r e f o r e it is e n o u g h to c o n s i d e r t h e case A - - Z [ X i i , . . . , X~,] w h i c h is a d o m a i n of c h a r a c t e r i s t i c zero.
L e t K be t h e q u o t i e n t field o f K a n d R t h e a l g e b r a i c closure o f K . O v e r R t h e f o r m u l a is e a s y to p r o v e . I f 21 . . . 2, a r e t h e e i g e n v a l u e s o f f we h a v e
E N D O N O R P t t I S I ~ I S O F F I N I T E L Y G E N E R A T E D P I 1 , O J E C T I V E N O D U L E S 269 2,(f) ---- ]-[]=1 (1 4- ~t). T a k i n g t h e l o g a r i t h m i c d e r i v a t i v e , e x p a n d i n g 2~(1 4- 2~t) -1 into p o w e r series a n d using T r (fl) = ~ = 1 2~ we get t h e desired f o r m u l a . ~
Remark 1.11. I n t h e t h e o r y o f d i f f e r e n t i a l e q u a t i o n s t h e r e is a )>eontinuous~>
a n a l o g u e o f t h e f o r m u l a above: L e t U(t) a n d B(t) be n • w i t h real entries, d e p e n d i n g on a p a r a m e t e r t, satisfying
d
d~ U ( t ) = B(t)U(t) a n d T h e n
u ( 0 ) = 1.
t
d e t U(t) = e x p
f
T r B(s)ds.0
I t is well k n o w n t h a t i f T r ( f l ) = 0 for i = 1 , 2 . . . n w h e r e f is a n n X n - m a t r i x o v e r a field o f c h a r a c t e r i s t i c zero t h e n f is n i l p o t e n t . O u r n e x t r e s u l t is a g e n e r a l i z a t i o n o f this.
W e will call t h e ring A torsion-free if it is torsion-free as a n abelian group, i.e. n a = 0 w i t h n E Z a n d a C A implies n = 0 or a = 0 .
PRO~OSITIO~ 1.12. Assume that A is torsion-free. Let f: P - ~ P be A-linear where P C ~ ( A ) has rank n. I f T r ( f i ) = 0 for n consecutive i:s then f is nilpotent.
Proof. A s s u m e t h a t T r (if) = T r (fl+r) __ . . . = T r ( i f + " - ' ) = 0. M u l t i p l y i n g C a y l e y - H a m i l t o n b y f r we get
- - a r
ff+~ alf "+r-1
a2f "§ 4- . . . ~ nfT a k i n g t r a c e s on b o t h sides we get T r (if+r) = 0. R e p e a t i n g t h e p r o c e d u r e we get T r ( f f ) = 0 for all v > r . P u t g = f r . T h e n T r ( g ~ ) = 0 for v = 1 , 2 . . . Using t h e e x p o n e n t i a l t r a c e f o r m u l a for g we f i n d -dr 2,(g) d = 0 w h i c h implies 2,(g) = 1 since A h a s no torsion. C a y l e y - H a m i l t o n a p p l i e d to g gives g n = 0, i.e. f n r = O .
Remark 1.13. T h e p r o p o s i t i o n is t r u e i f A has no s-torsion for s < r k P . Remark 1.14. I f A is a field o f c h a r a c t e r i s t i c 2 t h e n T r l ~ = 0 for P free o f r a n k 2.
Remark 1.15. I f we a s s u m e t h a t A is torsion-free we can give a n o t h e r p r o o f of t h e f a c t t h a t f| = 0 ~ f n i l p o t e n t ( c o m p a r e 1.9). P u t b; = T r (f~). T h e n (ff)| = ( f | implies T r ( ( f ~ ) | ( T r ( f ~ ) ) ~ = b ~ = 0 for i = 1 , 2 , . . . . C o m p a r i n g coefficients in t h e e x p o n e n t i a l t r a c e f o r m u l a we get a 1 = hi, 2a 2 = b~ - - b 2 , . . . . Since A has no t o r s i o n all a~:s are n i l p o t e n t . T h e n f is also n i l p o t e n t b y 1.8.
2 7 0 G E R T A L M K V I S T
2. Some computations
First a generalization 1.3 (iii):
PROPOSITION 2.1. Given f: P -+ Q and g: Q --+ P with P, Q 6 ~ ( A ) . Then T r ( f o g ) - - - - T r ( g o f ) and At(f o g) -~ A,(g o f )
Proof. After localization we m a y assume t h a t P and Q are flee. The formula for the trace is t h e n easily proved and
Wr A~(f o g) --~ Tr (A~f o A~g) = Tr (A~g o A~f) = Tr A~(g o f ) finishes the prooL
We continue with describing a m e t h o d for computing 2,(f). P denotes always module in ~ ( A ) of rank n.
THEOREhl 2.2. We have EndA (P) ~ P* | P. Let f: P --+ P correspond to
~ = 1 xi* @ xl in P* @A P. Let M ( f ) be the re• with entries (xi*, xj}
at place (i, j). Then
2,(f) ---- det (1 + tM(f))
I n particular the right hand side is independent of the choice of representatives for the tensor. The x~:s can be chosen as a minimal generator set of P.
Proof. First we reduce to the case when P is free. L e t p be a prime ideal in A.
Localizing at p we get a commutative diagram
%
P* @A P ~ EndA P
P~ | ~ ~ EndA~P~
where the star in the south west corner means HomA~ (., Av). Hence if f: P -+ P corresponds to ~ T xi* Q xl t h e n f~: P , -+ P~ corresponds to ~ T (x~*)~ | x~, and by using 1.3 (ii) we m a y ~ssume t h a t P is free. L e t now Y l , - - - , Y~ be a basis for P and h I . . . h. a d u a l b a s i s for P*, i.e. (h~,yj}~--(5fj. Given f : P - + P let it correspond to
| yj) = ( 5 aj,h,) 0 = | Yj
i,j j = l i = l j = l
Hence the (j, k):th entry in the matrix is
in i.e.
i = 1
n
i = l
E N D O M O R P H I S M S O F F I N I T E L Y G E N E R i T E D P R O J E C T I V E M O D U L E S 2 7 1
Now u: P * | P --> EndA P is given b y x * | so f : u ( ~ ai,h , @ Yi) means f(xk) = ~. ai,<h,, xk>y~ --- ~ aiky r
i,i ~,j j
I t follows t h a t f has the matrix (ajk) in the basis Yl . . . . , y~. Thus the formula is true if the x~:s form a basis for P .
Let now ~ l x l | x~ be another representation of f. Assume t h a t m
Then
x , = ~ cliy i and x * = ~ d , k h k
j = l k = l
X ~ @ X i : ~ CJ id i k h k @ YJ --- E Cj idlk h k ~ Y j
i = 1 i : 1 j,k j,k i
where
( a i k ) = C D with C = (c~i) and D = ( d ~ k )
= ~ a j A
| yjj,k
(here C and D are n X m - and m X n - m a t r i c e s respectively). The (i, k):th entry of the matrix in the formula is
~.i j=l
Thus this matrix is D C and we are done since X,(f) = det (1 + tCD)) b y the first part of the proof and det (1 + tCD) = det (1 + tDC) b y 2.1.
N e x t we compute 2t(lr) where ]e is the identity map of P 6 ~ ( A ) .
T~EO~E~ 2.2 (Goldman). (i) Tr (lp) = ~ i e , and i,(1~) = ~.~e,(1 + t)' where eo, q . . . e. are orthogonal idempotents with e o + e 1 ~ - . . . - @ e n ~ 1.
(ii) Ann ( A ~ P ) = (e 0 + q + . . . + e~_l)A. Furthermore the ei:s are uniquely determined by P .
.Remark. Some of the ei:s might be zero, e.g. if P is constant rank n, then
e 0 ~ e 1 ~ . . . = e n _ 1 = O.
Proof. (i) L e t Z have the discrete topology. Then rk: Spec A --~ Z given b y p --> rkvP is a continuous function. Hence Xi = (p 6 Spee A0; rk~P = i} is b o t h open and closed. I t follows t h a t Spec A ~ X o U X 1 U . . . U X . where the union is disjoint. B u t to this covering of Spec A corresponds a unique ))partition of u n i t p l = e o + q + . . . - 4 - e , where e,(x)={10 i f x E X ,
otherwise i.e. 1 - - e i E p for all O E X ~ and el 6 p for all p ~X~: (see Swan [12] p. 140). This means t h a t the ei:s are orthogonal idempotents.
N o w we claim t h a t i,(lp) = ~ el(1 + t / .
272 al~t~T ALltlKVIST
F i x a prime p C X , T h e n t h e localization a t 13 of t h e left h a n d side is (2t(lp))~ = 2t(lp~) = (1 + t) ~ since P , is free of r a n k i. To c o m p u t e t h e localization of t h e r i g h t h a n d side we need ekv. B u t e l e j = O w i t h e ~ p implies e j ~ = 0 in A~ for j # i . F u r t h e r m o r e e ~ ( 1 - - e ~ ) = 0 w i t h e ~ p implies % = 1 in A~.
T h u s ~ o ej(1 ~- t)i)~ (1 + t) ~ (2~(le))~ a n d we are done since p C Spee A was a r b i t r a r y .
(ii) A~P is in ~ ( A ) a n d t h u s A n n ( A i p ) = e A where e is a u n i q u e l y d e t e r m i n e d i d e m p o t e n t (Goldman [6] p. 33). N o w (A~P)v = 0 i f a n d o n l y if r k ~ P < i if a n d o n l y if ~ C X o U X 1 U . . . U X I _ 1. This is t h e case if a n d o n l y i f eA = A n n ( A ~ P ) q k p if a n d o n l y i f e ~ p . T h u s e ( x ) = 0 if a n d o n l y if x C X i U . . . U X ~ (and hence e ( x ) - - 1 otherwise). B u t e 0 ~ e l ~ - . . . @ e i _ l is a c a n d i d a t e satisfying these conditions. B y uniqueness we get
e = e o @ e 1 @ . . . @ ei_l.
P u t t i n g i = 1 we get e 0 uniquely. Sinee e 0 @ e 1 is u n i q u e e~ is unique etc.
D e f i n i t i o n 2.3: W e define t h e determinant of f b y d e t f = 2 ~ ( f - - l e ) for f : P - + P w i t h P ~ ( A ) .
F i r s t we note t h a t det 1 e = hi(0 ) = 1. I f P is free t h e n det (f) coincides w i t h the usual d e t e r m i n a n t of a m a t r i x for f. I f r k P = n t h e n there exists Q such t h a t P | Q = F where F is free of r a n k n. Clearly Q ~ ~ ( A ) . Localizing a t
;3 ~ Spec A we get P , | Q~ = F~ where P~, Q~, F~ are free A~-modules of r a n k r --- rk~P, n - - r a n d n, respectively. W e get (det ( f | 10))~ = det (f~ | 1o~) = d e t f v , det le~ = d e t f ~ = (detf)~. H e n c e we could also h a v e d e f i n e d d e t f as det (f @ 1~) where t h e last det is t h e o r d i n a r y d e t e r m i n a n t of a m a t r i x for f | 1 o.
Thus d e t f is t h e same as Goldman.s d e t e r m i n a n t ([6] p. 29). W e s t a t e some properties of d e t (f).
Pt~OPOSITIOI~ 2.4. (i) d e t ( f o g) -- d e t f d e t g.
(ii) f is an i s m o r p h i s m i f a n d only i f d e t f is invertible i n A .
W e now collect some formulas for L ( f ) = 1 + alt -~- . . . @ a~t n where f: P --> P w i t h P C ~ ( A ) a n d r k P = n .
n - - 1 n
P~oPosITIosr 2.5. (i) 2,t(Akf) = 1 ~- akt ~ . . . @ a~(k-i)t (k). I n particular (ii) 2,(A'[f) = 1 + ant.
( i i i ) ) , , ( A ~ - l f ) = 1 + a~_lt -~- an_2a~t 2 ~- a~_3a2~t 3 ~- . . . -~- ala:-2t ~-1 @ a"~-lt ~.
(iv) 2,(f2) = 1 + (a~ -- 2az)t -t- (2a4 -- 2axa3 -~- a~)t 2 -~ . . . + a2nt ~.
P r o @ Since ~, a n d A ~ c o m m u t e w i t h localization we m a y assume t h a t P is free. Using t h e t e c h n i q u e e m p l o y e d in proving t h e e x p o n e n t i a l trace f o r m u l a 1.10 we m a y even assume t h a t A is an algebraically d o s e d field. I f
E N D O M O I ~ P H I S M S O F F I N I T E L ~ x r G E N E R A T E D I ~ I ~ O J E C T I V E M O D U L E S 2 7 3
m n
;t,(f) = I l( 1 + 2,t) = 1 +
a~t +
. . . -}- a.t"1
we h a v e
2,(A~f) = ] - [ (1 + ~ @ . . .
~,~t).
1 _ < i 1 < 1 2 < . . . < i k < n
The first t w o formulas n o w follow easily.
rt
(iii) W e m a y assume t h a t a~ = ]--[2~ 4= O. T h e n we h a v e
1
(iv)
2 t ( A " - l f ) = 1 @ ~ t = c~,t" 1 + " 4,
1
= a . ~ l + a~. ~ t + a z - ~ + . . . @ . - L . - t @ =
~ n t a n
---- 1 -}- a,~_lt + a , ~ _ 2 % t 2 + . . . + a l a ~ , - 2 t + a,"-l"~ .
Set t ~ - - s 2. T h e n
L ( f 2) = d e t (1 -- serf) --- det (1 -- i f ) . d e t (1 + i f ) = 2_,(f)}~,(f) =
= (1 - - a~s @ a2s ~ - - + . . . d - ( - - 1 ) ~ a ~ s ~ ) ( 1 + a l s + a2s ~ + 9 . . 4 - a~s") =
= 1 + (2a~ - a~)a ~ + (2c~ -
2a~a~ +
c~)~ ,~ + . . . + ~ , ~ ( - a~) '~W e keep t h e n o t a t i o n f r o m above a n d f u r t h e r m o r e % e ~ , . . . , e~ are t h e i d e m p o t e n t s in t h e o r e m 2.2.
n C ~
PROPOSITION 2.6. (i) d e t f = ~ o ~e~ where a o = 1.
(ii) I f f is invertible then d e t f is a u n i t i n A a n d L ( f -1) = ~ d k t k where
~ - 1 ~ ( i . e . i f 0
& = ~$=kC,_ke, w i t h c, given by ( d e t f ) -* = (E0 a,e3 = E,=0 e~e~ e, 4=
t h e n c~e~ is t h e inverse o f afe~ in t h e subring Aei).
P r o o f . (i) Localization a t p C Xi (for t h e n o t a t i o n see t h e p r o o f of 2.2) gives ( ~ ajei) ~ = ai~ej, = aj~ since ej, = dq.
0 0
B u t (detf)~ = (21(f-- 1p)~ ~ 21(f~ - - lp~) = det (f~) since P~ is free. F u r t h e r m o r e P~ has r a n k i (since ~ E X ~ ) a n d hence ( L ( f ) ) ~ - - - - L ( f ~ ) = 1 + . . . + d e t f ~ . t i a n d a~ = detf~: This proves (i).
(ii) I t is sufficient to show t h e f o r m u l a locally. F i x a ~0 E X~. T h e n P~ is free o f r a n k v a n d we get
274 G E l ~ T & L I ~ I K V I S T
(2,(f-1),
= 2,(f~ ) = a c t (1 +-1 tf2 ~)
= (aerie) -~ d e t (t- 1,~) d e t (1 +t-~f~) = 9 " = a j ~-j
sinceejl 3
= (5#.= %t-J %
0 j = O j = 0
On the other h a n d
(~, dktk)o = 4or ~ --- (~. %aq_k)l~)t k = GO%_oot = % ~ aiS-~
with j = v - -o o ~=o ~=~ ~=o j=o
H e n c e the localizations of b o t h sides agree.
3. The behaviour of 2t under change ot rings, taking duals and forming of tensor products
1)t~oPoslTIO~ 3.1.
Let
rA---> B be a ringhomomorphism (with
r = 1)and f : P - - + P an A-linear map with P E ~ ( A ) . Then P | is in ~(B) and
~ ( f Q 1B) = r
Proof.
The first s t a t e m e n t is well known. SinceA~(P | B)
is n a t u r a l l y isomorphic as B - m o d u l e to(A~P) @A B
it is sufficient to p r o v e Tr~ ( f @ 1B) = r ) which is well known.P~OPOSITION 3.2.
Every f : P - - ~ P with P in ~ ( A ) where
P * = HomA(P, A) is in ~(A). Furthermore
T r f * = T r f
and 2~(f*) = 2,(f).
induces
f*: P * - + P *Proof.
F o r e v e r y p C Spec (A) we get a n a t u r a l A~-isomorphism (P*)o = ( I t o m a (P, A))o ~ , h tIomAo (Po, A~) = (P~)*a n d we h a v e a c o m m u t a t i v e diagram h
(P*)~ ~ (P~)*
l(f*)v h l(f~)*
(P*)~ ~ * (P~)*
H e n c e (f*)~ = h -1 o (f~)* o h. I t follows
(2,(f*))v = 2,((f*)~) = 2,(h -1 o (fv)* ~ h) --- 2,((f~)*) b y 1.3 (iv). B u t (P,)* is free a n d
E N D 0 1 ~ O R P I I I S M S O F F I N I T E L Y G E N E R A T E D P R O J E C T I V E M O D U L E S 275 2,((fv)*) = det (1 d- (fv)*) = det (1 d-fo) = t,(fo) = (l,(f))v.
This proves the formula for i, and taking the coefficient of t we get the formula for the trace.
N e x t we t u r n to the tensor product of two A-linear maps f: P - - > P and g: Q--> Q with P , Q in ~ ( A ) . F o r completeness we quote
PROPOSITmN 3.3. Tr (f | g) ---- T r f . Tr g.
There is a corresponding formula for i, b u t it is more complicated. I t is convenient to introduce some notation:
L e t A denote the set of all formal power series 1 d- ait -d- a2t ~ -4- 9 9 9 over A with constant term 1. Then A is an abelian group under multiplication. We define
~)*-mult@lication~) in A such t h a t the following formula is valid
1,(f | g) =
1,(f) 9 t,(g).This defines 9 for all polynomials in .4- since 1 + ait + . . . + a~t ~ = I t ( f ) where f: An--> A n is given b y the matrix
o o . . . o• \ 1 0 T a , _ l ~
f = o 1 . . . + ~ " - ~ /
o . . . o ' ~ ~,
PROPOSITION 3.4. I f I t ( f ) = 1 d- air -4- . . . -4- a . t ~ a n d It(g) = 1 -4- bit d- 9 9 9 d - brat m t h e n
it(f @ g) = (1 -~- ait -4- 9 9 9 -~- ant n) * (1 + bit + . . . -{- b,~t m) ---- 1 -4- dlt - 4 - . . . + d,,nt'"
w h e r e dl = ~b~
d2 = a~b2 -4- a2b~ - - 2a2b~
d a = a~b3 + asb~ --~ aza2bzb2 - - 3aza2b a - - 3aablb~ + 3aaba
cla = a~a2blbs -[- a~aab~b~ - - alaab~bs + a~ba -4- a4b~ d - 4axaab4 + 4a4b~ba - - 2a~aab~ - - - - 2a2blba d- 2a22b4 -4- 2a4b~ - - 4aaba - - 4a~a2b4 - - 4aab2b2 d- a~b~
dmn i = a ' ~ - l a -- n n - - I b ' - l b m m - - I d,~, = a'~b:.
2 7 6 GEI~T A L ~ K V I S T
P r o o f . J u s t a s in t h e p r o o f o f 1.10 w e m a y a s s u m e t h a t A is a n a l g e b r a i c a l l y c l o s e d f i e l d o f c h a r a c t e r i s t i c z e r o . T h e n
n
= ]7 (1 + = (-[ (1 + jt)
1 1
a n d
s | g) = ~ (1 +
~,~jt) i,j
U s i n g f o r m u l a s f o r s y m m e t r i c f u n c t i o n s (see [1] p. 258) i t is p o s s i b l e t o c o m p u t e d v d2, d 3 . . . . A b e t t e r w a y is t o u s e t h e e x p o n e n t i a l t r a c e f o r m u l ~ 1.10.
P u t p~ = T r f ~, q~ : T r gl a n d r~ : T r ( f | g)~. T h e n r~ : plq~ since T r ( f @ g)~ : T r ( f i | g~) : T r f l T r g*. T h e e x p o n e n t i a l t r a c e f o r m u l a a p p l i e d t o f g i v e s c h t + 2 a 2 s n : (1 + a l t + . . . + a ~ t n ) ( p l t - - p 2 t 2 + p a t 3 - . . . )
a n d h e n c e
a 1 : ~91
2a2 = a i P i - - 1o2 3a3 = a2Pi -- alP2 + ~3 4aa = a 3 p l - - a2P2 ~ - all~ - - P4
, 9 9
S o l v i n g f o r t h e pi:s w e g e t
I01 = a i
P2 ---- a ~ - - 2a2
3 3 a l a 2 + 3a a l~ --- (~i - -
4 4a~a 2 + 4 a l a a + 2a~ ~ 4a a P4 = ai --
9 ~ 9
T h e r e a r e s i m i l a r f o r m u l a s c o n n e c t i n g t h e bi:s a n d qi:s (dim a n d rim). T h e l a t t e r g i v e
di - - ri : Piqi : aibi
: 2 2 2 2
2d 2 d~r~ - - r 2 = a~b~ - - i02q2 = a~b~ - - (a~ - - 2a2)(b ~ - - 2b2) = 2(a262 -~- a2b~ - - 2a2b2) 3da : d2ri - - dir2 + r3 = d2Piqi - - dlp2q~ + I03q3 : albi(a~b2 + a2b~ - - 2a2b2) - -
- - a~bx(a ~ - - 2a2)(b ~ - - 2b2) + (a~ - - 3aia2
+ 3a3)(b~
- - 3b~b~+ 3b3)
= : 3(a~b a + aab~ - - 3aia2b a - - 3aablb 2 + 3aaba + aia2bib~)W e o m i t t h e c a l c u l a t i o n o f d 4.
E N D O M O R P B I S M S O F F I N I T E L Y G E I ~ E I % A T E D P R O J E C T I V E M O D U L E S 277 3 a a~blb2 w o u l d be missing W e could i m m e d i a t e l y h a v e seen t h a t t h e t e r m s a l b l ,
in d a since t h e y w o u l d o c c u r in (1 ~ % t ) 9 (1 + bit + b2t~) w h i c h o n l y has degree 1 9 2 = 2. S i m i l a r l y alb2b ~ will n o t occur.
T o get t h e last t e r m s one can use
( 1 ~ % t + . . . + a n t n) * ( l + blt + . . . -~ b.~t m) - -
(
%-~- t - l + ~ t-2 -~- . . . 9 1 + - ~ + --b-~- t-2 -[- . . .)
= a'~b~t mn 1 + an an
I n p a r t i c u l a r t h e n u m b e r o f m o n o m i a l s occurring in d~,_~ is t h e same as in dl. L e t sk d e n o t e t h e n u m b e r o f m o n o m i a l s in dk for large m, n (say m, n > k).
T h e c o m p u t a t i o n o f sk seems to be q u i t e a p r o b l e m . B y f o r m a l l y f a c t o r i n g
1 + alt + % t 2 + . . . + ant" = (1 + ~t)(1 + fit)(1 + 7 t ) ( t + St) . . . b4b~, o f
we f i n d t h a t t h e t e r m containing, s a y 2 2
(1 + a / - ~ - . . . ) , ( 1 + b l t + b ~ t 2 - 4 - . . . ) =
= (1 + bx~t + b~fl2t ~ ~- . . . )(1 + b~flt + b2fi2t ~ + . . . ) . . .
is - - ~ f l a ~ . Using t h e large f o l d - o u t t a b l e s of F a a de B r u n o : T h e o r i e des f o r m e s binaires, T u r i n 1876, we f i n d t h e following results s 1 = 1, s2 = 3, s a = 6, s 4 = 15,
% = 28, s 6 = 64, 8 7 = 116, a s = 234, s 9 ~ 373, s l 0 = 814, s 1 1 = 1508.
T h e m e t h o d b a s e d on couning zeroes in tables c a n n o t be g e n e r a l i z e d t o k larger t h a n 11.
N o w b a c k t o defining . - m u l t i p l i c a t i o n in A. B y t h e c o m p u t a t i o n s a b o v e it is clear t h a t i f we cut o f f t h e p o w e r series in t h e left h a n d side o f
( l + a ~ t + . . . ) . ( l + b ~ t + . . . ) = 1 + d~t + . . . + dkt ~ + . . .
a n d t a k e . o f t h e r e m a i n i n g p o l y n o m i a l s of degree n a n d m r e s p e c t i v e l y , t h e n d k = t h e coefficient o f t k will n o t d e p e n d on n a n d m if n , m ~ ] c , t t e n c e w e c a n d e f i n e dk in this w a y . T h e n z] b e c o m e s a c o m m u t a t i v e ring w i t h o r d i n a r y m u l t i p l i c a t i o n as 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 as m u l t i p l i c a t i o n . T h e u n i t y e l e m e n t is 1 ~ t. Clearly A is t o r s i o n f r e e (as a b e l i a n group). F u r t h e r m o r e 2~(f) ~-> 2 t ( A k f ) induces a 2-ring s t r u c t u r e on A (it is e v e n a special 2-ring, see [1], p. 257).
W e d e n o t e b y ~V(A) = {a ~ A0; a is n i l p o t e n t } t h e nilradical o f a ring A.
:PRoPosITION 3.5. (i) I f A i s t o r s i o n f r e e t h e n _N(A) c N ( A ) = {1 + a l t + % t ~ + . . .;
(ii) I f A i s n o e t h e r i a n t h e n X ( A ) c N ( A ) .
a, c
2 7 8 GElZT A L M K V I S T
Proof. (i) A s s u m e t h a t (1 + alt +
a2t2@...).k
1. T h e left h a n d side is 1 + cat -[- c2t ~ + . . . w i t h ca = a~ a n d in general c, ~ m~a~ -r a p o l y n o m i a l o f w e i g h t n/c c o n t a i n i n g a t least one o f a 1, a 2 .. . . . a~_t. H e r e m~ is a n integer.W e p r o c e e d b y i n d u c t i o n o v e r n. W e t h a t a 1 , a ~ , . . . , a n _ l E N ( A ) . Since c~
since A is t o r s i o n free.
(ii) I f A is n o e t h e r i a n t h e n 2V(A) p r o d u c t o f a n y k e l e m e n t s o f 2V(A) t h a t all m o n o m i a l s o c c u r r i n g in cn
a 1, . . . , an E N ( A ) . I t follows t h a t (1 + alt -t- 9 9 .),k W e will r e t u r n t o t h e ring A in S e c t i o n 6.
h a v e a~ = 0 so a 1 E N ( A ) . A s s u m e n o w
= 0 we get m n a ~ E N ( A ) a n d a ~ E N ( A ) is n i l p o t e n t , s a y N ( A ) k ---- 0. H e n c e t h e
is zero. T h e c o m p u t a t i o n a b o v e shows c o n t a i n a t least k f a c t o r s a m o n g t h e
1 .
PROI'OSITIO~I 3.6. Given f'. P ---> P , g:
an induced map
H o m ( f i g ) : HomA (P, Q) - + H o m ~ defined by u ~ g o u o f . Then
T r H o m (f, g) = T r f - T r g
Q -+ Q with P , Q E ~ ( A ) . Then we have (P, Q) where H o r n A (P, Q) E W(A)
and )l,(Hom (f, g)) = ~df) * ~,(g).
Proof. W e h a v e a n a t u r a l i s o m o r p h i s m Q =~_ Q** w h i c h induces n a t u r a l i s o m o r p h i s m s
HomA (P, Q) ~ HomA (P, Q**) ~ HomA ( P | Q*, A) = ( P | Q*)*
H e n c e we get T r (Horn (f, g) =- T r ( f | g*)* a n d 2,(Horn (f, g)) ---- ~,(f | g*)*).
Using 3.2 twice a n d t h e d e f i n i t i o n o f . - m u l t i p l i c a t i o n we g e t t h e desired formulas.
4. Relations between 2,(f) and minimal polynomials of f
PROPOSITION 4.1. Let f: M-->M be A-linear with M a finitely generated A-module.
Then there is a m o n i c polynomial q 6 A[t] of m i n i m a l degree such that q(f) = O.
(q will be called a m i n i m a l p o l y n o m i a l o f f ) . The degree of q is at most equal to the m i n i m a l number of generators of M .
s u r j e e t i o n A n s u c h t h a t
Proof. L e t n b e t h e m i n i m a l n u m b e r o f g e n e r a t o r s o f M .
M v 0 . Since A n is free we c a n f i n d g : A n An U ) - M ~ 0
Yg
I f
A n > M ~'0
T h e n we h a v e a ). A n
E N D O M O I ~ F H I S M S O F F I N I T E L Y G E N E I ~ A T E D P I ~ 0 J E C T I V E I ~ I O D U L E S 279 commutes. N o w g satisfies a monic p o l y n o m i a l qi
H a m i l t o n theorem. Using this in t h e d i a g r a m gives A . 7~ ~ M ) ' 0 0 - - qi(g) 9 ~ q~(f)
A " ~ ' M ~ 0
; g
f r o m which it follows t h a t q i ( f ) = O.
of degree n b y t h e Cayley-
l ~ e m a r k 4.2. The p o l y n o m i a l q is n o t unique in general. I f A = Z/(4) f = (20~) satisfies b o t h f 2 = 0 a n d f z + 2f = 0.
t h e n
PROPOSITIO~ 4.3. G i v e n f : P --> P w i t h P i n ~ ( A ) . A s s u m e t h a t f h a s m i n i m a l 2 o o l y n o m i a l q a n d ? u t ~(t) ~ ( - - t)~q(-- t -i) w h e r e v = degree o f q. T h e n L ( f ) s a t i s f i e s the f o l l o w i n g d i f f e r e n t i a l e q u a t i o n i n A [ t ]
d ~ . ~f (rood t ~+1) t2'(f)-I ~ ) " ( f ) = ~l
w h e r e ~v(t) = blt - - b2t 2 + bat s . . . w i t h b~ = T r f ( I f q(O) = 0 w e m a y t a k e (mod t ~) i n the f o r m u l a above.
P r o o f . Assume t h a t q(t) = t ~ + clt ~-1 + . . . + ekt "-k. T a k i n g t h e trace of O = f~ + e i f ~ - l + . . . + ckk ~-k w e get O = by + clb~_l + . . . + ckk~_k where in case k = v we p u t b o = T r 1p. Multiplying b y f a n d t a k i n g traces again gives b ~ + 1 - 4 - c l b ~ + . . . + e k b v _ k + x = O etc. N o w ~ ( t ) = 1 - - c l t + c 2 t 2 - . . . • k a n d
q ( t ) ~ ( t ) = (1 -- clt -+- c~t 2 - - . . . • ckt~)(blt - - b~t ~ + b a t 3 . . . ) =
= (terms of degree < v) • (bv + Clb~_ I + . . . + ckb~_k)t j - (b~+ l + q b + . . . + ckb~_k+~)t ~ + 1 + . . .
H e r e all t e r m s of degree higher t h a n v vanish a n d t h e coefficient of V is zero unless k = v in which case it is (-- 1)"-lck Tr 1p. T h e e x p o n e n t i a l t r a c e f o r m u l a gives
d
t 2 ' ( f ) - x dt L ( f ) ~ ~v(t) a n d m u l t i p l y i n g b y q(t) finishes t h e proof.
B e m a r k 4.4. I f A contains t h e r a t i o n a l n u m b e r s Q t h e n 2,(f) is d e t e r m i n e d b y a m i n i m a l p o l y n o m i a l q of f a n d b~, b 2 . . . . b~_ 1 where v = degree of q.
E x a m p l e 4.5. A s s u m e t h a t A ~ Q. L e t f: P - + P h a v e m i n i m a l p o l y n o m i a l q(t) = t~ - - t, i.e., f is a non-trivial i d e m p o t e n t in E n d A P . T h e n ~(t) = 1 + t a n d i f we a p p l y 4.3 we get (since q(0) = 0)
2 8 0 G E R T A L 1VIKVIST
d ((1 + t)(bit - - b 2 t 2 . . . ) ) ( r o o d t 2) b,t
t2t(f)-* -~ 2,(f) = 1 -~- t -- 1 + t
which implies %(f) = (1 + t) b* = (1 + t) vrf.
I f f8 = f, i.e. q(t) ---- t 3 - - t one finds similarly
b~+b, b2--b,
% ( f ) = ( l + t ) 2 . ( l _ t ) 2
E x a m p l e 4.6. Let G be a finite group of order n and A[G] the group algebra.
Let f: A[G] --> A[G] b e given b y left multiplication with a E G. I f a has order k then the minimal polynomial of f is q(t) = t k - 1 and q(t)--- 1 + (-- 1)kt k.
Using 4.3 and the fact t h a t b 1 - - b ~ - - . . . = b ~ _ ~ = 0 and b k - ~ n we get
n
%(f) = (1 -- (-- l)ktk) k
5. Endomorphisms of modules having finite resolutions of finitely generated projective modules
L e t 9(~(A) denote the category of A-modules M such t h a t M has a finite resolution in ~ ( A ) . We want to define ~t(f) for f: M - + M when M E~)(~(A).
For this we need some preparations.
Definition 5.1. Let E n d ~ ( A ) denote the category of endomorphisms of modules in ~(A), i.e. the objects are endomorphism f: P - + P with P E ~ ( A ) and a morphism u from f to g: Q - ~ Q (where Q E (~(A)) is a commutative diagram
P ~ Q
P ~-Q
Then K o (End ~(A))
morphism classes of) the objects in b y all If] -- [f'] -- If"] where
0 ~ P '
+ f
0 + P 'is defined as the free abelian group generated b y (the iso- E n d ~ ( A ) modulo the subgroup generated
) P )~P" ~ 0
f f"
) P ~ P " ~ 0
is commutative with exact row. Similarly we define E n d 9(~(A) and K o (End ")C(A)).
PROrOSITION 5.2. The embedding E n d ~ ( A ) --> E n d 9(~(A) m o r p h i s m i: K o (End ~(A)) ~ K 0 (End ~)C(A)).
induces an iso-
E N D O M O R P H I S M S O F F I N I T E L Y G E N E R A T E D P R O J E C T I V E M O D U L E S 2 8 1
Proof. The usual proof does not apply since f: P - ~ P with P E ~ ( A ) is not a projective object in the abelian category of all endomorphisms (which is iso- morphic to the category of modules over A[t]). F o r t u n a t e l y Swan has formulated a theorem general enough for our purposes (see [12] p. 235. Theorem 16.12). P u t
= E n d ~ ( A ) and c)]Z = E n d ~)(~(A). Then the assumptions in 16.12 are fulfilled.
Indeed,
(1) Clearly E n d ~ ( A ) and ~(~(A) are closed under direct sums
(2)
I f 0 v P ' - % ~ ' P V ~'P" > 0q~ "i" V
0 > P ' - ) - P ~ ' P " ~ 0
is exact and commutative then P , P " E C ~ ( A ) implies P ' E ~ ( A ) and P, P" E 9C(A) implies P ' E ~)C(A) (see Bass [2], p. 122, Proposition 6.3).
(3) Given a n y f: M ~ M with M E ~)(~(A) there exists a finite resolution in E n d ~CP(A), i.e.
O - - ~ P a > . . . . > /)1 ~'Po ~ M > O
O ~ - Pa ~" . . . ~ P1 ~ P o - - - ~ M > O is commutative with exact row and all P~ E ~ ( A ) . This is easily proved.
Now the inverse ~0 of i: K 0 (End ~(A)) - ~ K o (End ~ ( A ) ) is given b y
d
v([f]) = Z ( - 1)'[f,]
0
(*)
and it is shown in [12] t h a t the right hand side is independent of the choice of the resolution (.).
THEOaE~ 5.3. Given f: M ~ M with M E ccF(A). Consider the resolution (,) in End,CP(A) above. T h e n
d d
0 0
are indepe~dent of the choice of the resolutions and the liftings f~ of f.
Proof. For f: P - + P with P C ~ ( A ) , f~--~ )~,(f) is a map from (isomorphism classes in) E n d ~ ( A ) to A. I f O - - > ( P ' , f ' ) - - ~ ( P , f ) - - ~ ( P " , f " ) - - ~ O is exact in E n d W(A) we have (by (1.5) 2 , ( f ) = 2,(f'),~,(f").
Hence b y the universal p r o p e r t y of K o (End ~ ( A ) ) we have a factorization
282 G E R T A L M K V I S T
[]
E n d W(A) > K o (End W(A))
A Assume now t h a t (M,
f)
and
in E n d ~ ( A ) .
and thus
in E n d 9(~(A) has two resolutions 0 - + (Pd, f~) - + . . . ---> (Po, fo) - ~ , (M, f ) - + 0
0 -+ (P~,, ]j,) - + . . . -+ (P'o, fg)--~ (M, f ) --> 0 B y the proof of 5.2 we have
d d"
(-- 1)i[~] = ~ (-- ])i[f~] in K o (End ~ ( A )
0 0
d d"
2,(fj)(-1)J = i . d .
0 o
The statement about the trace follows from taking the coefficient of t in the formula for 2,.
Now we can safely make the
Definition 5.4. For f : M - + M with M in 9((A) we define
d d
x(f) = Z (-- 1)' T r f , and M f ) = ]-T
Mf,) (-1)'
where the fi:s are given in (,).
PROI'OSlTIO~ 5.5. Let
0 - - - M k - - - ~ . . . ---~ M1 ---~ M0 --~ 0
be a commutative diagram with exact row and all M~ in DC(A). Then
k k
~. (-- 1)'X(fi) = 0 and ~ ~ t ( ~ ) ( - 1 ) ' = 1
0 0
Proof. Consider the diagram (see the proof of 5.2) K o (End ~ ( A ) ) + - : ~ + K o (End 9((A)) i
ENDOI~IOI:tPHIS2r163 O F F I N I T E L Y G E N E I ~ A T E D I ~ O J E C T I V E :~IODITLES 283 where we denote ~, b y ~, on E n d g ~ ( A ) . The definition of ~ and ~,
exactly t h a t ~ = 2t o ~. Now given an exact sequence o -+ (M~, fk) - ~ . . . - ~ (3/0, fo) -> o
k
in E n d 9((A) we get ~ (-- 1)i[f~] ~- 0 in _R7 o (End ~9(~(A)) and hence
0
k
]7- 7t ,[/;](-1/---- 1
0
Taking the coefficien~ of t we get the formula for Z.
means
COROLLARY 5.6. z ( f O g) = z(f) + X(g) and 2,(f 9 g) = M f ) ' M g ) . N e x t we generalize the exponential trace f o r m u l a
PROPOSITION 5.7. I f f: M --> M with M 6 ~ ( A ) then d
_ t~,(f)-i ~ ~,(f) = ~ z ( f ) ( - t)' in d .
1
Proof. L e t 0--~ (Pa, fa) - + . . . ~ (Po,%) -*" ( M , f ) -+ 0 be a E n d ~(A). Taking logarithmic derivatives of
d
2,(f) = ] - [ 2,(fj) ( - #
j - 0
we get (using the exponential trace formula)
- t,W) -1 a,(f) = ~ ( - 1 / - tx,(L) -1 ~,(fj) =
)
-- 5 (-- 1 ) i ~ ( - - 1 ) i Tr (f~)t ~ -
-- -- (-- 1/ ( 1)J T r (f~)
j = 0 i = l i = l
since
0 -->- (Pd, f~) - - > . . . --> (Po, f~) --> (M, f ' ) -->- 0
r e s o l u t i o n in
= ~ (-- 1)~z(f')t '
i ~ l
is a resolution of (M, fi).
THEOREM 5.8. Let f: M --~ M with M E 9((A) be nilpotent, f,~+l = O. Then, there is a resolution
o --> (P~, f~) - - > . . . - ~ (Po, fo) --> ( M , f ) --> o in E n d ~ ( A ) such that all f,~+l ~_ O.
2 8 4 GERT ALMKVIST
A s s u m e t h a t r k P i = n~ a n d L ( f ) = 1 + ~ cl 9 T h e n all the ci:s are n i l p o t e n t t i
v v 2 v k
a n d clc~ . . . c~ = O i f the w e i g h t ~l ~- 2v~ + . . . + kvk > m ~ a o n . I t f o l l o w s t h a t L ( f ) i s a p o l y n o m i a l o f degree
na i f d is e v e n
<~ no + ran1 + n2 + tuna + " " " + tuna i f d is odd.
P r o o f . The existence of t h e projective resolution such t h a t f 7 +~ = 0 is precisely P r o p o s i t i o n 6.2, p. 653 in Bass [2]. N o w 2t(f) is a p r o d u c t of factors
L ( f , ) = 1 + aat -+- . . . ~- a.it"i
o r their inverses. B y 1.7 a n y m o n o m i a l in t h e aj:s vanishes p r o v i d e d its weight is larger t h a n mn~. I n v e r t i n g t h e p o l y n o m i a l L(f;) ~-- 1 + a l t + 9 9 9 + a.~t "~ we f i n d t h a t 2,(fl) -~ is a p o l y n o m i a l o f degree a t m o s t mn~ a n d t h e coefficient of t" is a p o l y n o m i a l in t h e aj:s where e v e r y t e r m has weight v. T a k i n g t h e a l t e r n a t i n g p r o d u c t of t h e L(fi):s we get L ( f ) = 1 - ? c l t + c p t 2 + . . . where c, is a s u m o f t e r m s of t h e t y p e
r l 9 a : ; 0 b~ 1 1) snd ( * * )
a 1 . . . . . . . . n d
i f 2t(fo) : 1 + alt + . . . + a % t % . . . , L ( f d ) : 1 -{- b~t + . . . + bo~t ~d.
F u r t h e r m o r e t h e weight of t h e m o n o m i a l (**) is
v : r 1 + 2r~ ~ . . . + n o r % + . . . + s 1 + 2s 2 + . . . + nas%.
v 1 v 2 V k
:Let n o w c = c ~ c ~ . . . c k be a m o n o m i a l i n t h e ci:s o f w e i g h t d
731 -~ 2V 2 -~ 9 9 9 -~ kvk > m ~ . n ~ .
i=0
T h e n c is a s u m of monomials o f t y p e (**) such t h a t t h e i r weight
d
0
H e n c e a t least one of t h e factors
(bl b2 . . . 9 . a~ ) . . . . bs"d~
- - n d !
h a s w e i g h t > m n ~ , . . . , m n ~ respectively a n d this factor is zero b y 1.7.
The e s t i m a t e of t h e degree of L(f) is clear f r o m t h e previous considerations.
COROLLARY 5.9. A s s u m e t h a t the r i n g A i s r e d u c e d, i.e. the n i l r a d i c a l N ( A ) = 0. T h e n L ( f ) ~ 1 f o r all n i l p o t e n t f : M - ~ M w i t h M E%V~(A).
E N D O I ~ O R P K I S M 8 O F F I N I T E L Y G E N E R A T E D P I ~ 0 J E C T I V E N O D U L E S 285 We denote the projective dimension of an A-module M with dhAM.
PI~OI'OSlTION 5.10. Let A be a local noetherian ring with maximal ideal m, residue field k = A / m , and M a finitely generated A-module. I f d = d h ~ M is finite then M 6 r and 2A(1M) = (1 + t)ZA(1M) where
d
ZA(1M) = ~ (-- 1)' dimk Tor~ (M, k)
i = 0
Proof. Choose a minimal free resolution
O-+ Pa--+. . . - ~ P1---~ Po--+ M - - ~ O
with nl = rkAPi = dim~ Tor~ (M, k) (see Serre [10] p. IV -- 47). Then
But
d
d d X ( - - 1 / , , i
),t(1M) = ~ 2t(1p,) ( ' / = ]--[ (1 + t) (-1~hi = (1 + t) ~
0 0
d d
Z(1M) = ~ (-- 1 / T r lv, = ~ (-- 1/%.
0 0
PROPOSlTIO~ 5.11. Let A be a regular local noetherian ring with residue field k.
Then k
6~-2(A)
and , ~ ( l k ) = 1.Proof. P u t t i n g M = k in 5.10 we get
/ ( l k ) = ~ (-- 1 / d i m k Tor~ (k, k) = (-- 1/ = (1 -- 1) a = 0
0 0
d i m k T o r ~ ( k , k ) = ( d / where d = g l o b a l d i m e n s i o n o f A i f A i s a r e g u l a r since
local noetherian ring. V /
PRO:POSITI02~- 5.12. Let r A - + B be a f l a t ring homomorphism, i.e. B is flat as an A-module. I f f: M -+ M with M 6 ~ ( A ) , then M @A B E C~(B) and
2 ~ ( f Q IB) = 4 ( 2 { ( f ) ) . Proof. Let
0 - - + P a - - + . . . + P o - + M --+ 0
f~ fo f
"f "t" Y
O -+ P a--~ . . . -+ P o-+ M - - ~ O
286 G E : R T A L ] Y I K V I S T
be a projective resolution. T h e n t h e exactness is preserved after t a k i n g 9 @A B since B is A - f l a t . F u r t h e r m o r e each Pi @A B is B-projective a n d f i n i t e l y g e n e r a t e d as B-module. H e n c e M @A B e 9(~(B) a n d since r = 2B(f~ @ 1B ) b y 3.1 we finish t h e p r o o f b y t a k i n g a l t e r n a t i n g products.
COROLLARY 5.13..Let A be an integral domain and K
= ~, ( f | K l s ) Proof. The inclusion A - + K is flat.
its quotient field. Then
COI~OLLARY 5.14. Let A be an integral domain and f: M - + M where M is a torsion module in ~)C(A). Then ~,(f) = 1.
Proof. Since M is torsion we h a v e M @A K : 0 a n d hence
~ ( f ) = ~ ( f @ 1K) = ~ ( 0 ) = 1 b y 5.13.
COROLLARY 5.15. Let A be a Dedekind ring and f: M --~ M A-linear where M is finitely generated. Then M = T • P where T is a torsion module and P is projective and torsion free.
.Furthermore f ( T ) c T and ~t(f) = ~,(fv) where f~: P ~ P is the )>torsion free part)) of f .
Proof. F i r s t we n o t e t h a t M E 9((A) since A is n o e t h e r i a n a n d gl. dim A < 1.
T h e n M = T Q P is j u s t B o u r b a k i [5] p. 79, Corollaire. N o w Horn A (T, P) = 0 so we get t h e following d i a g r a m rising m a t r i x r e p r e s e n t a t i o n
O--->T--~T | P---.'-P--~O
f:(fTh [
O-+ T - + T | P - + P - ~ O.
(:)
F r o m 5.6 a n d 5.14 it follows t h a t
~,(f) = R,(fT)':',(fP) = 1 9 ~,(fp) = 2,(fp).
W e n o w e x t e n d t h e definitions of Z a n d )~t to e n d o m o r p h i s m s of g r a d e d modules a n d complexes.
E N D O 1 K O E P H I S M S O F F I N I T E L Y G E N E E A T E D 1 J B O J E C T I V E M O D U L E S 2 8 7
Definition 5.16. Let M = | Mi be a graded A-module with all Mi E 9d(A).
I f f: M -+ M is a hornomorphism of degree zero, i.e. f ( M i ) c Mf, we p u t f / = the restriction of f to M~ and define
d d
Z~'(f) = ~ (-- 1)'z(f, ) and 2,g'(f) = ]--[ 2,(fi) (-1/.
0 0
:Note t h a t Zg'(f) and ~,~(f) in general do not agree with z(f) and 2t(f) where M is considered just as an A-module.
Similarly if
0 ~" C a - C d _ 1-->- . . . --->- C 1 - C O ) 0
Y 8 Y (~1
0 ~Cd -d + C d _ I - - > . . . ~ C 1 > Co ~'0
for short f: C--->C is a chainmap of a finite complex C with all C i in <)g(A), we define
d d
Z(f) = ~ (-- 1) i T r f i and as(f) = T-[ 2,(fi) (-1)/.
0 0
P~OeOSITIO~ 5.17. Let f'. C --+ C be as above. A s s u m e that all homology modules Hi(C ) are in 9 ( ( A ) . Then
g ( f ) = Zg'(H.(f)) and 2t(f) = 2,g'(H.(f))
where H . ( f ) : H . ( C ) --* t I . ( C ) is the induced endomorphism of the graded homology module H . ( C ) = | ).
Proof. P u t K~ = Ker ~i and B~ = I m 51+1. Then we have exact sequences O --~ K ~ --~ C I ---~ B i_ I ---> O
O--* Bi--> K,--> H,(C ) -->0.
:Now B o = C o e g ( ( A ) and C l e g ( ( A ) so K 1 e g g ( A ) b y Bass [2] p. 122, Proposition 6.3. Since HI(C ) E 9 ( ( A )
B i, K i C 9 ( ( A ) . We get induced maps 0 - * Ki --* Ci - * B~_I --> 0
0 --~ K i ~ ff~ --~ B i _ l ~ 0,
we also get B 1 E c)((A). By induction all 0 --, Bi ---> K , --, HI(C) - * 0
,Hi(f)
0 ---> Bi --> K i "-~ HI(C) ---> O.
Using 5.5 several times and taking alternating products all 2,(gl) and 2,(hi) cancel and we get the w a n t e d formula for 2,(f).
