L e t ~ be a stack of n - m o d u l e s on a collection $ of subsets of A. L e t
r 1
. . . r n E ~ ,~t E~(A) a n d consider the e q u a t i o n
r l v 1 -~- ... + rnv n ~-- ~, ( 1 )
where we seek a solution v I ... vnE~(A). If we set ~ i ( B ) = ~ | for
BES,
t h e n we m a y t h i n k of the n-tuple (r i ...rn)
as defining a m a p d : ~ 1 - ~ , b y d(# x ... tin) = r i f t i + ... +rn#,EZ(B)
for (#i ... #n) E ~ i ( B ). E q u a t i o n (1) has a solution if 2 E i m d. I f (1) has a solu- tion for every ). E ~(A), t h e n t h e sequence~I(A)~ ~(A)-->
0 is exact.There is a fairly well-known technique for s t u d y i n g t h e a b o v e situation. Suppose there is a sequence of stacks a n d stack h o m o m o r p h i s m s ,
*'''-->~n dn-1)~ ~ n - 1 tin-2 d x d
-..-+ ~:~ --~ ~ i --~ ~ - ~ 0 , (2) which is exact on a s u b d o m a i n $' of $. (Later in the section we shall show one w a y t h a t such a sequence m a y be obtained.) I f B = (B i ... Bn) is a complex whose elements arc in $', t h e n (2) induces the following d i a g r a m
~0
(~n (A) ~1 C~ ~n) ~ el(B, ~n) ~ "'" "+ CP( B, ~'~") ($o
~n I(A) ~l Co(B ' ~n-1) "-+ CI(B, ~n-1) . . .
CP(B,
~n-1) . . . .dl o
~ I ( A ) d C~ B, ~1) CI(B, ~1) "-> -+
CP( B,
~1) '->~ ( A ) ~-~ C~163 ~ C ~ ( B , ~ ) . . . . -+ C v ( B , Z ) . . . .
r r 4 r
0 0 0 0
(3)
All columns of this d i a g r a m are exact except possibly the first one. I t is t h e first column t h a t we wish to d r a w conclusions about. T h e following discussion of this d i a g r a m is derived f r o m s t a n d a r d double complex t h e o r y :
W h e n convenient in discussing d i a g r a m (3), we shall use c$ to d e n o t e a n y one of the m a p s i, (~o ... ~ .... a n d d to denote a n y one of the m a p s d, d 1 ... dr, .... W e shall also use the n o t a t i o n ~ o = ~ a n d C - I ( B , ~ v ) = ~ v ( A ) 9
D e / i n i t i o n 4.1. F o r the d i a g r a m (3), we m a k e t h e following definitions:
(a) D V - l = i l n 5 p 2 + i r a d p = C P - I ( B , ~ ) for p ~ 0 , where ~ - 2 = 0 a n d ~-1 = i ;
(b) A residue sequence a ~ I . . . a v . . . . is a sequence with a v e C V - i ( B , ~ v ) a n d cia p§ = ~a v for p ~>0.
LEMMA 4.1. (a) I] 2 E~.(A) then there is a residue sequence; a ~ a 1 .. . . . aV . . . u~ith 2 = a ~ (b) I / 2 = a ~ a 1, ..., a v .... and ~ = b ~ b I . . . . , b v . . . . are two residue sequences starting at 2, then a V - b V E D v-1 ]or each p .
(c) 2 7 / ~ = a ~ a 1, ..., a v . . . . is a n y residue s e q u e n c e / o r which a q E n q-l, then a P E D v 1 / o r p>~q.
Proo/. We construct the sequence a ~ a 1 ... a v .... b y induction on p. Suppose we h a v e a ~ 1 .. . . , a ~-1 defined with ; t = a ~ a n d (~aq=Sa q-1 for q < ~ p - 1 . Since cTSaP-l=Ot~a ~ - 1 =
&~aV-2=0, the exactness of t h e columns of (3) implies t h a t there exists a v E C V - I ( B , ~v) such t h a t claY= 6a v-1. This proves p a r t (a).
I D E A L T H E O R Y A N D L A P L A C E T R A N S F O R M S 275 N o t e t h a t if we consider the sequence O = a o - b o , a l - b 1 .. . . . a n - b . . . t h e n p a r t (b) follows f r o m p a r t (c) with q = 0. To prove p a r t (c), note t h a t if a ~ E D" 1 t h e n a ~ = ~b~+ ~c ~+~, where b" ~ C" ~(B, ~ ) a n d c "+ ~ ~ C" -I(B, ~ + ~). I t follows t h a t ~(a ~+~ - (~c "+~) = (~a" - (~c ~+~ =
&~b~=0. Hence, there exists c'+~EC~(B, ~ + ~ ) such t h a t ap+l-(~c~+l=c~c~+~; i.e., a ~ + l =
~tb "+~ + d c ~+~ if we set c ~+~ = b "+~. We conclude b y i n d u c t i o n t h a t p a r t (c) is true.
N o t e that, in t h e a b o v e lemma, D -~ is the image of d in ~(A). Thus, p a r t (c) of the l e m m a says in particular t h a t if ~ E i m d, t h e n a ' ~ D ~-~ for all p>~0. U n d e r a p p r o p r i a t e conditions on the c o h o m o l o g y groups H ' ( B , ~ ) , we can prove a n analogue of p a r t (c) which goes t h e other direction.
LEMMA 4.2. Suppose H ~ I(B, ~ p ) = 0 ]or l < p < m , and i*: ~ ( A ) ~ H ~ ~) and i*: ~ . I ( A ) - ~ H ~ ~1) are isomorphisms. I] ~ = a ~ a 1 ... a ~, ... is a residue sequence starting at ,~ E ~ ( A ) and a '~ E D m 1, then a ~ E D ~-1 /or 0 <~p <~ m. I n particular, ~ E im d under these circumstances.
Proo/. I f a m E D m-1 t h e n am=~bm~-c~c 'n+l, where bmEC'~-~(B,~m) a n d cm§
Cm-I(B, ~m+l)- I t follows t h a t O(a ~-1 -~bm)=d(am-(~bm)=~(~cm+l:O. I f m>~2 it follows f r o m the hypotheses t h a t there exists b ~ - I E C ~ - a ( B , ~m-1) such t h a t a rn-1 - d b m = ~ b m 1;
i.e., a m - l = S b m l +dc m if we set cm=b m. If m = l , t h e n a l = S b l +dc ~ implies i ~ b l = d a l = i a ~ Since ~ ( A ) L C~ ~) is one to one, we have db 1 = a ~ =~t. Hence ~tEim d, in this case. We n o w h a v e t h a t the l e m m a is true, b y induction.
De/inition 4.2. L e t A be a set a n d S a collection of subsets of A which is closed u n d e r finite intersection a n d contains O. L e t ~ be a m a p which assigns to each B E $ a ring
~ ( B ) with i d e n t i t y eB, a n d let 0 be a m a p which assigns to each pair (B, C ) c $, with B c C, a ring h o m o m o r p h i s m O B . c : ~ ( C ) - ~ ( B ) such t h a t OB.C=eB. We shall call .~ a stack of rings with i d e n t i t y on $ if ~ ( O ) = O ) , 0 satisfies t h e transitive law, a n d 0B. s = i d , as in Definition 3.1.
N o t e t h a t if ~ is a stack of rings with i d e n t i t y a n d (B, C)E $ with B ~ C, t h e n we m a y consider ~ ( B ) as an algebra over t h e ring ~(C) u n d e r t h e o p e r a t i o n / ~ ' v = ( 0 B . c # ) ' v for # E~(C), v E~(B). I n particular, ~ ( B ) is a n algebra over ~ ( A ) for e v e r y B E S, a n d t h e m a p s 0~. c are ~(A)-algebra h o m o m o r p h i s m s . If we consider only the additive s t r u c t u r e in ~ ( B ) for each B, t h e n ~ m a y be considered a stack of ~ ( A ) - m o d u l e s as in Definition 3.1.
De/inition 4.3. L e t A be a c o m p a c t c o n v e x subset of a topological v e c t o r space X a n d let ~ be a stack of rings with i d e n t i t y over t h e c o m p a c t convex subsets of A. L e t Q be a ~ ( A ) - m o d u l e which contains ~ ( B ) as a s u b m o d u l e for each B E S, in such a w a y t h a t the m a p s OB.c are inclusion m a p s for B4=O. If ~, considered as a stack of ~(A)-modules,
is a c o n v e x s t a c k of s u b m o d u l e s of Q a c c o r d i n g t o D e f i n i t i o n 3.4, t h e n ~ will be called a c o n v e x s t a c k of r i n g s w i t h i d e n t i t y in Q.
N o t e t h a t w i t h Q = ~ ' ( A ) , t h e c o r r e s p o n d e n c e
B ~ ( B ) ,
of S e c t i o n 2, defines a c o n v e x s t a c k of rings in Q. I n fact, D e f i n i t i o n 4.3 a b s t r a c t s p r e c i s e l y t h o s e p r o p e r t i e s of t h e cor- r e s p o n d e n c e A - ~ ( A ) which a r e n e e d e d t o c a r r y o u t t h e r e s u l t s of t h i s section.I f ~ is a s t a c k of rings w i t h i d e n t i t y , t h e r e is a c a n o n i c a l w a y of c o n s t r u c t i n g a se- q u e n c e like (2). This is t h e K o s z a l c o m p l e x , which we d e s c r i b e below. F o r e a c h B E S we set ~ I ( B ) = ~ n ~ ) ~ ( B ) . F o r m > 1, we l e t ~ ( B ) = A
m~l(B)
b e t h e re,fold e x t e r i o r p r o d u c t of ~ I ( B ) o v e r t h e r i n g ~ ( B ) . This space m a y be d e s c r i b e d as follows: L e t e b e t h e i d e n t i t y of ~ ( B ) a n d for i = 1 . . . n set ei = (0 . . . e . . . 0) E ~ I ( B ) , w i t h e a p p e a r i n g in t h e i t h posi- tion; t h e e l e m e n t s e I . . . e~ f o r m a basis for "~I(B) o v e r ~ ( B ) . F o r m > l , we let ~m(B) be t h e free ~ : ( B ) - m o d u l e h a v i n g a s g e n e r a t o r s t h e s y m b o l s e~, A ... A e ~ , w h e r e we m a k e t h e i d e n t i f i c a t i o n s : e a A ... A e~m=(--1)*e~, A ... A e ~ if (j~ . . . ]m) is a p e r m u t a t i o n of(i~ ... ira)
w i t h ~ = 1 if t h e p e r m u t a t i o n is o d d a n d e = 0 if t h e p e r m u t a t i o n is even, a n d e~, A ... A e~m = 0 if t h e s u b s c r i p t sil ... im
a r e n o t all d i s t i n c t . N o t e t h a t~m(B)
is a free ~ ( B ) - m o d u l e of d i m e n s i o n (~). I n p a r t i c u l a r , ~ m ( B ) = ( 0 ) form > n
a n d ~ ( B ) is i s o m o r p h i c to ~ ( B ) ; i.e.,~n(B)
is a free ~ ( B ) - m o d u l e w i t h a single g e n e r a t o r e 1 A ... A e~.F o r
# E ~ ( B ) , v E ~q(B)
w i t ha n d r = Y. #j ... ~ e~, h ... A e~q,
,q,...Jq
we set /~ A v = ~ /~ ... ~ - vj ... ~e~, A ... A e~ A ey~ A ... A e~
~1, -..,)'q
W e t h e n h a v e # A~E~p+q(B) a n d ~u A r = ( - 1 ) P % Aft. U n d e r t h e m u l t i p l i c a t i o n i n d u c e d b y t h e wedge p r o d u c t # A ~, t h e space ~ = 0 | is a n a s s o c i a t i v e a l g e b r a o v e r ~ ( B ) , w h e r e we set ~ 0 ( B ) = ~ ( B ) a n d / ~
A r = # ' v
for #, v E ~ ( B ) .I f (B, C ) = $ w i t h
B = C ,
t h e n t h e m a p 0a.c:~(C)---.'-~(B)
i n d u c e s a h o m o m o r p h i s m 0B, c: Y . ~ - 0 ~ ( C ) - ~ = 0 | w h i c h p r e s e r v e s w e d g e p r o d u c t s a n d carries ~ ( C ) i n t o~ ( B ) . I n p a r t i c u l a r , if w e c o n s i d e r ~p(C) a n d ~ ( B ) a s ~ ( A ) - m o d u l e s , t h e n
OB.c
is a ~ ( A ) - m o d u l e h o m o m o r p h i s m of ~ ( C ) i n t o ~ ( B ) . I n o t h e r words, we m a y consider ~p to b e a s t a c k of ~ ( A ) - m o d u l e s on $.I f #1 . . . #nE~:(A), t h e n we m a y define a s t a c k h o m o m o r p h i s m dr: ~ + 1 - + ~ as fol- lows: F o r
~ , = Z ~,, ... j , + , e,, A . . .
Aej,+,e~+,(a),
J t , . . . , i p + l
I D E A L T H E O R Y A N D L A P L A C E T R A N S F O R M S 277 we set
p + l
d~,v = ~. ~.. ( - 1)'~+',u~ "~,~ ... ~),+, e~, A ... A eS~ A . . . A ej,+,.
I~, ...,Jp + 1 k = l
A simple calculation shows t h a t d~_~ od~ = 0 a n d dp+q+l(Y A ~)): (d~v) A ~)+ (--1)~ A (dq~) :[or ~ E ~ p + i ( B ) a n d ~E~q+i(B).
LEMMA 4.3. I / the equation/.~1~,1-}-... + # n v ~ = e can be solved/or ~1 ... v~E~(B), then the sequence O--->~n(B) ~ ~ . . . ~ c s is exact.
Proo/. The hypothesis says t h a t d o is onto, since do(vial + ... +~,~e~)=fflri + ... +ff~v~.
Since d,_id,=O , we need only show t h a t if d~_ig=O t h e n 9=d~,~ for some 2 E ~ ( B ) . W e set 2 = v A ~ , where V=~lel+...+vne~ a n d H i ~ l + . . . + f f n ~ = e . I f d ~ _ l ~ = 0 , t h e n d , 2 = (doV) A ~ - f - r A (d~_l~) = e A 0 + r A 0 = 0.
Thus, if $' is a subcollection of S which is closed u n d e r finite intersection, B is a complex whose elements are in S', a n d t h e e q u a t i o n
/.~1 721 - ~ - . . . - ~ n Y n = e (/A 1 . . . jt$ n E ~(A)) (4)
is solvable in ~(B) for each B E S ' , t h e n we h a v e a sequence 0 ~ d,-1 . . . . -~ ~ l - - ~ ~ - * 0 do which is exact on S', a n d the results of L e m m a s 4.1 a n d 4.2 a p p l y for t h e complex B.
I f ~ is a convex stack of rings with i d e n t i t y in Q, t h e n we define Qp = {~j ... jpvj ... jp ejl A ... A ejp: ~j ... jp EQ}, with the appropriate identifications a m o n g t h e symbols ejl A ... A ejp.
W e m a y consider Qp to be a ~ ( A ) - m o d u l e containing ~ ( B ) as a s u b m o d u l e for each B e $.
Since ~ is just t h e (;)-fold direct sum of copies of the stack ~, it follows t h a t ~ is a convex stack of submodules of Qp for each p. We m a y n o w a p p l y the results of the previous section a n d L a m i n a s 4.1 a n d 4.2 to o b t a i n specific theorems concerning equation (4) for c o n v e x stacks.
L e t ~ be a c o n v e x stack of rings in Q, on a c o m p a c t c o n v e x set A = X. I f x E X a n d ffi ... fin E ~(A), we shall s a y t h a t e q u a t i o n (4) is solvable locally at x if there is a convex n e i g h b o r h o o d U of x such t h a t (4) can be solved for ~l .... , ~ E ~ ( B ) , where B = 0 N A.
If ~ is the stack ~ of Section 2, t h e n L e m m a 2.5 says e x a c t l y t h a t (4) is solvable locally at x E A if (4) is solvable in ~({x}).
THEOREM 4.1. Let ~ be a convex stack o/rings on A and fll ... #nE~(A). I / (4) is.
solvable locally at each point o / A , then (4) is solvable in ~(A).
Proo]. Since A is compact, we h a v e t h a t t h e t o p o l o g y of A is t h e weak t o p o l o g y g e n e r a t e d b y t h e f a m i l y of linear functionals on X. I t follows t h a t if (4) is locally solvable at each p o i n t of A, t h e n we m a y choose hyperplanes P1 ... Pk such t h a t if B = {B1 ... Bz}
is t h e polygonal decomposition of A induced b y P1 ... Pk, t h e n (4) is solvable in ~(B~)
for i = 1 ... l. Also, if $' is the collection of all finite intersections of elements of B, then (4) is solvable in ~ ( B ) for every BE S'. Hence, by L e m m a 4.3, we have a sequence of the form (2) which is exact on $', and L e m m a 4.1 applies. This gives us a sequence e = a ~ ...
a p ... with a~eCP-I(B, ~ ) and ~a~=~Ta ~§
B y Theorem 3.2, H~(B, ~ q ) = 0 for p > 0 , and i : ~ q ( A ) ~ H ~ ~q) is an isomorphism for all q. Hence, the hypothesis of L e m m a 4.2 is satisfied for all m. Thus, if a m E D m-1 for some m, then e E D -1 = i m do, and (4) is solvable in ~(A). However, Cm-I(B, ~ m ) = 0 = D m-1 if m > n or m > l + 1. This completes the proof.
The above theorem shows t h a t local solvability implies global solvability in ~(A) for the equation/~lv~ + ... +/znv~ =e. I t would be useful to have a similar theorem for the e q u a t i o n / ~ l v l § with e=4:~E~(A). To prove such a theorem b y our present methods would require constructing a sequence . . . . ~ . . . . ~ 1 ~ 0 - > 0 which is exact in sufficiently small neighborhoods of each point of A, where ~ is a convex stack for p ~> 1 and ~0(B) is the s u b m o d u l e / ~ I ~ ( B ) +... + # ~ ( B ) of ~(B) for each B ~ S . We have not been able to do this in the case of the stack ~ if Section 2. A solution to an analogous problem for the sheaf of germs of analytic functions is presented in IV.F.5. of [2]. This solution is quite involved and requires a great deal of information concerning the local structure of the sheaf.
I n the n e x t two theorems we will be concerned with the case where (4) is locally solvable on a subset of A. Using Theorem 3.3, we obtain a particularly useful result in the case where (4) is locally solvable on the b o u n d a r y of an m-simplex S = A.
THEOREM 4.2. Let ~ be a convex stack o/ rings on A , S an m-simplex in A , and /z 1 ... / ~ E ~ ( A ) . I//UlV 1 + ... +/a~rn = e is locally solvable at each point o / ~ S , then
(a) i] n < m the equation/~1~1 + ... +/~nv~ =e is solvable in ~ ( S ) ;
(b) i / n = m there is an element Q EQ such that,/or each 2 E~(A), the equation/~lVl § ... + Iz~vn=~ is solvable in ,~(S) i / a n d only i/,~o =0.
Proo]. Let B = { B 0 ... Bin} be the complex consisting of the ( m - 1 ) - f a c e s of S. Since (4) is locally solvable at each point of ~S, Theorem 4.1 implies t h a t (4) is solvable in ~ ( B ) for each compact convex set B which is a subset of some Bi. Hence, b y L e m m a 4.3, we have t h a t L e n m a 4.1 applies for the complex B. B y Theorem 3.3, HP(B, ~ q ) = 0 for 0 < ~ o < m - 1 and H~ ~ q ) = ~ ( S ) for all q, provided m > l . Hence, for m > l , L e m m a 4.2 also applies.
Let e = a ~ a I ... a p .... be the residue sequence guaranteed b y L e m m a 4.1. I f n < m then ~ m = 0 and a m = 0 E D m-1. Hence, b y L e m m a 4.2, e E i m d a n d #lVl+...+/~n~n=e is solvable in ~(S).
I D E A L T H E O R Y A N D L A P L A C E T R A N S F O R M S 279 I f n = m > l , t h e n T h e o r e m 3.4 implies t h a t (~ : H n - I ( B , ~n)--->Qn=Q is one to one.
If ~t E ~ ( A ) t h e n ~t = 2a ~ 2a 1, .:., )~a p .... is a residue sequence, since (~ a n d a are ~ ( A ) - m o d u l e h o m o m o r p h i s m s . I t follows f r o m L e m m a 4.2 that/ZlV 1 + ... + / ~ v . = 2 is solvable in ~(S) if a n d o n l y if ~ta ~ E D n 1. However, Cn-I(B, ~ + 1 ) = 0 , since ~n+l = 0 , a n d so D n-1 = B n l(B,~n).
I t follows that/ZlV ~ + ... + # ~ v ~ = 2 is solvable i n ~(S) if a n d only if 25oan~(~o2an=O. T h e proof is complete for m > l if we set ~ = ~0a ~.
F o r m = n = l , we use t h e last s t a t e m e n t of T h e o r e m 3.3. This gives us t h e d i a g r a m 0
i do
0 -~ ~:l(Z) -~ ~~ ~ ) ~ Q
l ~ l ~
i 60
0 -~ ~ : ( s ) -~ C~ ~:) -~ Q,
0
with exact rows a n d exact second column. If ~tc~0al=0 t h e n 2=db, where i b = 2 a 1. Thus,
= 50 a 1 is the required element of Q in t h e case m = n = 1.
Definition 4.3. T h e element ~)EQ, given b y T h e o r e m 4.2, will be called t h e residue o f t h e system/~1 ... #~ on t h e n-simplex S.
A n i m p o r t a n t feature of the residue ~ is t h a t it is an element of t h e ~ ( A ) - m o d u l e Q.
I n working with ~ ( A ) , Q will be ~ ' ( A ) , a n d s o Q will be a measure which is locally in our original subalgebra N of M(G). We will show, in the n e x t section, t h a t ~ m u s t be a v e r y s m o o t h absolutely continuous measure. Hence, T h e o r e m 4.2 gives a strong connection between spectral properties in ~ ( A ) a n d t h e existence of absolutely continuous measures in N.
THEOREM 4.3. Let r163 be a convex stack o/rings on A. Let B and C be compact semipoly- gonal subsets o / A such that A = B 0 C. Suppose that i~1 ... ~ , ,~ E~(A) and B and C satis/y the/ollowing conditions:
(1) B is connected and ( B ) = A ;
(2) the equatio~ # ~ 1 +... +#nUn =e is locally solvable at each point o / B ;
(3) the equation ]u1~1+ ...+ l ~ , n = 2 is solvable in ~ ( ( E~ ) /or each connected c~mponen~
E o / B A C :
Then the equation/~lV 1 + ... +l~nV~ =2 is solvable in ~ ( A )
1 8 - - 682904 A c t a mathematica. 121. I m p r i m ~ le 6 d ~ c e m b r e 1968
Proo[. Since B and C are scmipolygonal sets, we m a y choose a polygonal decomposi.
tion .,4 of A such t h a t A~, Ac, and ABnc are polygonal decompositions of B, C, and B n C respectively. As a first step in the proof, we reinterpret conditions (1), (2), and (3) in terms of the complexes AB, ~4c, and ~4~c.
In view of Lemma 3.6, condition (1) is equivalent to (1') i*: ~(A)~H~ ~)=Z~ ~) is an isomorphism.
B y L e m m a 4.3, condition (2) implies t h a t 0-~ ~ . ~"-~ ' " . - > ~ t ~ ~ O is an exact sequence on $', where $' is the collection of all compact convex subsets of B. In view of Lemma 4.1, we have that condition (2) implies
(2') there are residue sequences : t = b ~ .. . . ,b ~ .. . . and l = a ~ .. . . . a ~,... with b~6 C~-I(As,Z~) and a~ 6 C~-' ( A , n c , ~ ) .
If Et ... E~ are the components of B f l C , then i: ~ p ( < E i > ) ( ~ . . . ~ p ( < E k > ) -'>
H ~ Y~)=Z~ ~,) is an isomorphism for each p. Condition (3) implies that d: ~1(<E1>) (~... ( ~ 1 (<Ek>)--> ~(<EI>) (~--- ( ~ < E k > has the element t I<~,> | | in its image. In other words, the element a'6C~ ~t), of the residue sequence
= a 0, a 1 . . . a ~ ... m a y be chosen from Z~ ~,). Thus e~a* = 0 and we m a y choose a * =0. Hence, condition (3) implies
(3') there is a residue sequence t = a ~ a* ... a ~ .... for the complex Aca~ such t h a t a~=O.
Recall, from the proof of Theorem 3.1, t h a t the sequence
0 - ~ ~ , 7s #B-~, ~Bnc -~ 0
is exact on a subdomain containing A, As, .,'iv, and Ashy, and it induces the exact sequence
This, in turn, induces the exact sequence
... ~ H~(A, ~) , I F ( A s , ~)@HP(Ac, ~) , H~(ABnc, ~) --~ HP+I(A, ~) -~ . . . .
Since H~(A, ~:)= 0 for p > 0, we have that f l * - f l * is an isomorphism for p > 0. This implies t h a t f18 - fig carries Bn(j4B, Z) @Bn(~4c, ~:) onto BP(~4snc, ~) for each p > 0. These considerations hold for each ~:q as well as for Z. I n particular, for p = 1 and q = 2, 3 this yields the following commutative diagram:
IDEAL THEORY AND LAPLACE TRANSFORMS 281
C ~ (A~, 7;~) ~ C~(Ac, 7;~)
, Ol(A~nc, 7:~) -~ O.