R E S I D U E S A N D A D E L E S

A . A . B e i l i n s o n UDC 513.60

In th is b r i e f a r t i c l e T a t e ' s c o n s t r u c t i o n in [1] fo r the r e s i d u e s of d i f f e r e n t i a l s on c u r v e s is g e n e r a l i z e d to the m u l t i d i m e n s i o n a l c a s e . The l o c a l r e s i d u e of P a r s h i n [2, 3] and E l z e i n [4] a r i s e s f r o m the c l a s s of c o - h o m o l o g i e s of s o m e i n f i n i t e - d i m e n s i o n a l L ie a l g e b r a . M o r e o v e r , i t i s shown that the g l o b a l m o r p h i s m of the G r o t h e n d i e c k t r a c e s p l i t s up into the s u m of r e s i d u e s . Hence we have the p - p a r t of the r e c i p r o c i t y l aws in [3].

1. C o n s i d e r the s e t T,. = (C . , Ai. , Bi.), c o n s i s t i n g of the c o m p l e x C , of v e c t o r s p a c e s o v e r the f i e ld k, and i t s s u b c o m p l e x e s A i B 1 , 1 < i < N. L e t S(n) be the s e t of p a i r s (P, Q) ,p , Q ~ {l . . . . ~} such that[ P N Q [ = n, [ P U Q I - - ~ r . F o r a - - - - ( P , q ) ~ S (n),~ P N Q - - { ~ . . . . . in}, i ~ < . . . < i n , s e t a (h 0 ) ~ ( P , Q \ { i j } ) , a ( / , l ) = ( P \ f f f , q )

S ( n - - t ) ; T , j ~ N Ai ~ B)" 0 (], r): o~S(n) " t~ .$<n+l i~e *~_Q *' r . a - - , r,~0.r)(r-~ 0J) a r e n a t u r a l e m b e d d i n g s . L e t r .~ ---- + r.~; 0~ ffi ~ (--W •

(0 (1, 0) - - a (], i)): r . ,+t ~ r . , ; ] ~0: r.o ~ c , be the s u m of e m b e d d i n g s . I t is e a s i l y s e e n tha t O,,o0n+x = 0, Oo0o = 0.

Suppose tha t we a r e g iven a n a s s o c i a t i v e k - a l g e b r a E and i t s t w o - s i d e d i d e a l s X i , y i , 1 _< i _< N s u c h tha t X i + y i = E fo r a l l i . We a s s o c i a t e wi th T = 0g, X i, y i ) the s e t T . = (C . , Ai . , Bi.), w h e r e C . i s a c o m p l e x of cha ins E c o n s i d e r e d a s a L ie a l g e b r a o v e r k: C r = A r E , the d i f f e r e n t i a l d r : Cr+~ --" C , i s g i v e n by the f o r - mu la

d, (e~ A . . . A e~+~) ffi ~ ( - I f +~+x (~ej - e~) A e, A. :. A ~ A . . . A ~ A . . . A e~+x;

_- y l A c,_~ a r e s u b c o m p l e x e s of C , . and Al = X i A C,_~, B, s

LEMMA. a) The s e q u e n c e 0 ~ T , N ~ . . . --- T , 0 ~ C , ~ 0 is exac t . Thus we have r r ~ IIo,- (I~N+I(C,),

L e t O be a c o m m u t a t i v e s u b a l g e b r a of E . I f 10 . . . . . l~ ~ ~ , then we deno te by r r if0 . . . . . IN) ~ ~ (r,~) the va lue of r T on the c y c l e lo A • • • A 1~.

b) T h e r e e x i s t s a (unique) m o r p h i s m Rest: fi~l~ --,Hx (r,N) s u c h tha t Res°(lodh A • • • A diN) = rr (to . . . . . . IN)"

2. Le t V be a N o e t h e r i a n s c h e m e , P(V) the s e t of poin ts of V wi th n a t u r a l o r d e r i n g [for ~ ~ P (v) nt > n~ m e a n s tha t ~ ~ ~ ], and l e t S(V) be the s i m p l i c i a l s e t connec t ed w i th P(V) [the n - s i m p l e x A ~ s (V)n i s the s e t (~0 > . . . > ~?n),*l~ ~ P(V~. If K ~ S ( D ~ , ~ I ~ P ( V ) , then

~K,, = {(~ > . . . > 'l~) ~ S (D , , -~ I (q > ~h > • • • > ~h) ~ K~}.

By induc t ion on n, we def ine a func to r A(Kn, • ) on the c a t e g o r y of q u a s i c o r e g e n t O v - b u n d l e s , a s the unique func to r c o m m u t i n g w i t h the induc t ive l i m i t s , and co inc id ing w i th II lira M ~ On/m ~ fo r n = 0 on c o r e g e n t

bundles M, and w i th ~I~P(V)TII lira A (,flKn, M @ O~/m~) for n -> 1 (m~? is a m a x i m a l i d e a l o f the l o c a l r i n g O~?). Then

A(Kn, • ) is an e x a c t func to r . I f Kn ~ K~, K~_,C0~(K~), we have the obvious m o r p h i s m s .4 (Kn, -) ~ A (K~, .),

A (K~_,, .) ~ a i(~n, .). Se t A n ( v , • ) = A(S(V)n, • ), and for a ~ s (v), s ! .)(a) = ,4 ({a}, .). C l e a r l y , ~i t u r n A*(V , . ) in to a s i m p l i c i a l c o m p l e x , and S( . )(A) into a s y s t e m of c o e f f i c i e n t s on S(V), and m o r e o v e r t h e r e e x i s t n a t u r a l e m b e d d i n g s A * ( V , . ) - - C*(S(V), S( . )), H°(V, • ) --.- H°{A*(V, • )). W e note tha t S(V), S( . ), A*(V, • ) depend func- t o r i a l l y on V. In p a r t i c u l a r , we have def ined a c o m p l e x of p r e b u n d l e s A * ( . ) on V: A* (M)(U) = A*(U, M IU), and a m o r p h i s m i: M - - H°(A*(IVI)).

LEMIVIA. A l l the Ai(M) a r e i n e r t b u n d l e s , __A* (M)= M ~ A * (Or) , i i s a n i s o m o r p h i s m , and HJ(_A*(M)) = 0 OV

fo r j ~ 0.

COROLLARY. H ' (V, M) = H ' (A*(V, M)).

Moscow S ta te U n i v e r s i t y . T r a n s l a t e d f r o m F u n k t s i o n a l ' n y i A n a l i z i Ego P r i l o z h e n i y a , Vol. 14, No. 1, pp. 44 -45 , J a n u a r y - M a r c h , 1980. O r i g i n a l a r t i c l e s u b m i t t e d May 8, 1979.

34 0016-2663 /80 /1401-0034 $07.50 © 1980 P l e n u m Pub l i sh ing C o r p o r a t i o n

R e m a r k . The complex of ade les A* is a " f o r m a l " ana log of the Godeman canon ica l s impl i c i a l reso lven t . The na tu ra l f a c to r s of A* a r e the Kuzen com plex and the complex C* f r o m [2].

H e n c e f o r w a r d , V wil l be a n N - d i m e n s i o n a l p r o p e r s c h e m e over the pe r f ec t field k. For b rev i ty , let MA = S(M)(A), ~ i = ~ / k " Fix A ~ s (~9~-. Then O A is a n A r t i n k - a l g e b r a , on each field K of r e s i d u e s o f w h i c h t he re is a na tu ra l N-d imens iona l loca l s t r u c t u r e over k (see [3]). This means that fo r N - 1, K is the field of f r ac t ions of a comple t e d i s c r e t e l y n o r m e d r ing R 1, whose field of r e s i d u e s K 1 has an ( N - 1 ) -d imens iona l local s t r u c t u r e , and for N = 0 K is f inite o v e r k (for R 1 we need to take the whole c l o s u r e of the image Oo0(A ) in K, and the loca l s t r u c t u r e on K 1 is defined induct ively) . Any N-d imens iona l loca l field is i s o m o r p h i c to k ' ( ( t N ) ) . . . ((tl)), w h e r e k' is finite ove r k {with the obvious loca l s t ruc tu re ) . In [2, 3] ReSK: ~ K N ---k was defined by the f o r m u l a aes x (aY[:~* dr, A . . . A dtN)= trk'/k(a) for l 1 = . . . = l N = - 1 , 0 in the opposi te c a s e (a ~ k'), and m o r e o v e r Res K does not depend on the i s o m o r p h i s m K = k' ((tN)) • • • ((tl)).

Set nesk = z. ~ nes~: ~ - . k, w h e r e l is the length of 0170, and the s u m is taken over al l f ie lds of r e s idues

OA. F u r t h e r m o r e , we denote by R e s k the c o m p o s i t i o n ~ - ~ A N (V, fire)-. H~v (V, ~N)~ k, w h e r e T r is the m o r - p h i s m of the Gro thend ieck t r ace .

3. We now show how to connec t the t r ip le (E, X i, y i ) f r o m Sec. 1 wi th the N-s imp lex . Le t ~ = (~],~> ~ > . . . > ~,) ~ s (v)~, A' = 0o (A) = (~, > : . . > r,,). F o r each finite O~?0-module M, let ¢ (M) = {L C M [ L is a finite O~l -modu le , L - 0~? 0 = M~.

I f Li ~ (I)(M), L t ~ La, then L I a , ~ M a and (Lt/L~) a, : LI±,/L~a,. F o r pa i r s M1, M 2 of finite O~0-modules , we define induct ively i n n ~H~(M,, M2)= 1! ~ H o m k ( M l a , M ~ a ) ] V L I ~ ( D ( M t ) , L2~P(M2) ~tLI~EP(M1), L 2 ~ ¢ ( M 2 ) : LI" ~ LL L~ D L, ,[ ( L~ ' ) C L,a,, / ( L~a,) C L~a': and for n _> 1 the induced m o r p h i s m f. ( n~/ Li) ~. -~ ( L~/ L~)~. belongs to t l a,( L,/ L~, L~/L,)} , and X ~':- " n -> i > 1, X ~ ( r e spec t ive ly , y i ) cons i s t s of those ! ~ Ha , fo r which all the f in the def in i t ion of HA belong to

" v v . ~ i-1 X ~ I ( L 1 / L 1 , L 2 / L 2) [ r e spec t ive ly , YA' (" ' ")]" C l e a r l y , //~_D Horn% (M~a, M,~). Le t 5 ~ S (~)~-. Cons ide r the a s -

soc i a t i ve k - a l g e b r a Ea = ~/~ (~o, ~0) ~ Ead~ (~a, ~a) and its two-s ided ideals x~ ---- x~(~0, ~ ) , y~. Then X~ +

YA = E A f ° r a l l i ~ N , ZA_~$a. We apply the c o n s t r u c t i o n of See. 1 t o T = (EA, X , Y ) ( incidental ly, T depends only on the loca l s t r u c t u r e on the f ields of r e s i d u e s ~ ).

THEOREM. a) All the e l emen t s Q x~ ~ Y~ a r e f ini tely potent [1], and the t r a c e def ines a m o r p h i s m

tr: H, (r.N) --~ k. b) aes~ = Res~ = troRes~ra.

COROLLARY (formula for r e s i d u e s in P a r s h i n ' s form). Fo r 5 ' ~ s (v)N_i, (~ ~ ~ 7 ~ m s ~ 0 . A~A'

I a m deeply g ra te fu l to A. N. P a r s h i n , whose c o n s e r v a t i o n s wi th me f o r m e d the bas ic s o u r c e for this a r t i c l e , and to I.. N. Be rnsh t e in and Yu. I. Manin for the i r a t t en t ion to the a r t i c l e and the au thor .

1.

2. 3. 4.

L I T E R A T U R E C I T E D

J. Tate , Matemat ika , 1__33, No. 1, 5-14 (1969). A. N. P a r s h i n , Izv. Akad. Nauk SSSR, Ser . Mat. , 40, 736-773 (1976). A. N. P a r s h i n , Dokl. Akad. NaukSSSR, 243, No. 4 , 355-358 (1978). F. E lze in , Compos i t i o Math. , 29, 9-33 (1974).

35