Contemporary Mathematics Volume 8,1982
SEMIALGEBRA'IC TOPOLOGY OVER A REAL CLOSED- FIELD Hans D e l f s and Manfred Knebusch
We f i x a r e a l c l o s e d f i e l d R. By a v a r i e t y over R we always mean a scheme o f f i n i t e type over R. T h i s paper g i v e s a s h o r t survey about our theory o f s e m i a l g e b r a i c spaces over R. S e m i a l g e b r a i c spaces seem t o be the adequate g e n e r a l i z a t i o n o f the c l a s s i c a l n o t i o n o f s e m i a l g e b r a i c s e t s over R. Copying the c l a s s i c a l d e f i n i - t i o n we can c o n s i d e r a l s o s e m i a l g e b r a i c subsets o f a r b i t r a r y
v a r i e t i e s over R. I n t r o d u c i n g the category o f s e m i a l g e b r a i c spaces we g e t r i d o f the inconvenience t h a t every s e m i a l g e b r a i c s e t i s embedded i n a v a r i e t y . The b a s i c d e f i n i t i o n s a r e g i v e n i n §1. They can be found, as w e l l as a l o t o f f o u n d a t i o n a l m a t e r i a l , i n the paper [DK I I ] . The a p p l i c a t i o n t o the theory o f W i t t r i n g s o u t l i n e d i n §3 i s c o n t a i n e d i n [DK I ] . The r e s u l t s on t r i a n g u l a t i o n and cohomology o f a f f i n e s e m i a l g e b r a i c spaces a r e c o n t a i n e d i n the t h e s i s o f the f i r s t author ( [ D ] ) . We omit here n e a r l y a l l p r o o f s and r e f e r the reader to these papers. But we emphasize t h a t i n a l l these p r o o f s T a r s k i ' s p r i n c i p l e i s never used t o t r a n s f e r s t a t e - ments from the f i e l d IR t o o t h e r r e a l c l o s e d base f i e l d s .
Contents
§1 B a s i c d e f i n i t i o n s
§2 Some r e s u l t s i n the theory o f s e m i a l g e b r a i c spaces
§3 An a p p l i c a t i o n t o W i t t r i n g s
§4 T r i a n g u l a t i o n o f a f f i n e s e m i a l g e b r a i c spaces
§5 Cohomological dimension
§6 The homotopy axiom i n s e m i a l g e b r a i c cohomology
§7 The d u a l i t y theorems
§ 1 B a s i c d e f i n i t i o n s
For any v a r i e t y V over R we denote by V(R) the s e t o f R - r a t i o n a l p o i n t s o f V.
D e f i n i t i o n 1; L e t V = Spec(A) be an a f f i n e v a r i e t y over R. A subset M o f V(R) i s c a l l e d a s e m i a l g e b r a i c subset o f V, i f M i s a f i n i t e union o f s e t s
{x € V(R) | f ( x ) = 0, g j ( x ) > 0, j = 1,...,r}
w i t h elements f , g.. € A.
The o r d e r i n g o f R induces a topology on the s e t V(R) o f r e a l p o i n t s of every a f f i n e R - v a r i e t y V, hence on every s e m i a l g e b r a i c subset M of V. We c a l l t h i s topology the s t r o n g topology.
D e f i n i t i o n 2: L e t V, W be a f f i n e v a r i e t i e s over R and M, N be semi- a l g e b r a i c subsets o f V r e s p . W.
A map f : M -> N i s c a l l e d s e m i a l g e b r a i c w i t h r e s p e c t t o V and W, i f f i s continuous i n s t r o n g topology and i f the graph G(f) o f f i s a s e m i a l g e b r a i c subset o f V xD W. The s e m i a l g e b r a i c maps from M t o R
1
w i t h r e s p e c t to V and A_ = Spec R[X] a r e c a l l e d the s e m i a l g e b r a i c f u n c t i o n s on M w i t h r e s p e c t to V.
D e f i n i t i o n 3: A r e s t r i c t e d t o p o l o g i c a l space M i s a s e t M together with a f a m i l y 6(M) o f subsets o f M, c a l l e d the open subsets o f M, such t h a t the f o l l o w i n g c o n d i t i o n s a r e s a t i s f i e d :
i ) 0 € #(M) , M C £(M)
i i ) U1 e e( M ) , U2 € 6(M) => U1 U U2 € 6(M).
N o t i c e the e s s e n t i a l d i f f e r e n c e t o the u s u a l t o p o l o g i c a l spaces:
I n f i n i t e unions o f open subsets a r e i n g e n e r a l not open.
We c o n s i d e r every r e s t r i c t e d t o p o l o g i c a l space M as a s i t e i n the f o l l o w i n g sense:
The category o f the s i t e has as o b j e c t s the open subsets U £ 6(M) of M and as morphisms the i n c l u s i o n maps between such s u b s e t s .
The c o v e r i n g s (IL | i G I) of an open subset U of M are the f i n i t e systems o f open subsets o f M w i t h U = U Ü..
i € I 1
Example: L e t V be an a f f i n e R - v a r i e t y and M be a s e m i a l g e b r a i c sub- s e t o f V. L e t 6(M) be the f a m i l y o f a l l subsets o f M which are open i n M i n the s t r o n g topology and which a r e i n a d d i t i o n s e m i a l g e b r a i c i n V. (M, 6(M)) i s a r e s t r i c t e d t o p o l o g i c a l space. We c a l l £his topology of M the s e m i a l g e b r a i c topology (with r e s p e c t t o V) and denote t h i s s i t e by Mp .
sa
D e f i n i t i o n 4: A r i n g e d space over R i s a p a i r (M,0M) c o n s i s t i n g o f a r e s t r i c t e d t o p o l o g i c a l space M and a sheaf 0M of R-algebras on M.
A morphism (f,£) : (M,0M) -* (N^N> between r i n g e d spaces c o n s i s t s o f a continuous map f : M -+ N ( i . e . the preimages of a l l open sub- s e t s o f N are open) and a f a m i l y ($v) o f R-algebra-homomor-
1 V € 6(N)
phisms which are compatible w i t h r e s t r i c t i o n .
Example: L e t M be a s e m i a l g e b r a i c subset o f an a f f i n e R - v a r i e t y V equipped with i t s s e m i a l g e b r a i c t o p o l o g y . F o r every open semialge- b r a i c subset U o f M l e t O^(XJ) be the R-algebra o f s e m i a l g e b r a i c f u n c t i o n s on U w i t h r e s p e c t to V. Then (M,0^) i s a r i n g e d space over R. I t i s c a l l e d a s e m i a l g e b r a i c subspace o f V.
D e f i n i t i o n 5:
i ) An a f f i n e s e m i a l g e b r a i c space over R i s a r i n g e d space (M,(?M) which i s isomorphic t o a s e m i a l g e b r a i c subspace of an a f f i n e R - v a r i e t y V.
i i ) A s e m i a l g e b r a i c space over R i s a r i n g e d space (K,0^) which has a ( f i n i t e ) c o v e r i n g (M^Ji € I) by open subsets such t h a t (Mj/tfjyjMi)is for a11 i € I an a f f i n e s e m i a l g e b r a i c
space.
i i i ) A morphism between s e m i a l g e b r a i c spaces i s a morphism i n the c a t e g o r y o f r i n g e d spaces.
As i s shown i n [DK I I , § 7 ] , a morphism (f,#) : (M,0^) -» (N,0^) b e t - ween s e m i a l g e b r a i c spaces i s completely determined by f :
For V € <§(N) and g € 0 (V) we have t5 (g) = g ° f . Hence we w r i t e
simply f i n s t e a d o f {£,&). These morphisms are a l s o c a l l e d s e m i a l - g e b r a i c maps. The morphisms f : M -> N between s e m i a l g e b r a i c sub- spaces M,N o f a f f i n e v a r i e t i e s V,W are j u s t the s e m i a l g e b r a i c maps from M to N with r e s p e c t t o V and W ([DK I I , § 7 ] ) .
L e t M be a s e m i a l g e b r a i c space. We denote by 6(M) the s m a l l e s t f a m i l y o f subsets o f M which f u l f i l l s the f o l l o w i n g c o n d i t i o n s : i ) 6(M) c <*(M)
i i ) A € 6(M) => M - A € 6(M) i i ) A,B e 6(M) => A U B € 6(M) .
The elements of 6(M) are c a l l e d the s e m i a l g e b r a i c subsets o f M. In the s p e c i a l case M = V(R) t h i s d e f i n i t i o n c o i n c i d e s with D e f i n i - t i o n 1 .
Tne s t r o n g topology on M i s the topology i n the u s u a l sense which has 6(M) as a b a s i s o f open s e t s . Thus the open s e t s i n the strong topology are the unions o f a r b i t r a r y f a m i l i e s i n 6(M). In the s p e c i a l case t h a t M i s a s e m i a l g e b r a i c subspace o f v a r i e t y V t h i s topology i s o f course the same as the s t r o n g topology i n the pre- v i o u s sense.
Since now n o t i o n s l i k e "open", " c l o s e d " , "dense", ... always r e f e r to the s t r o n g topology. In t h i s terminology the s e t s U € 6(M) are the open s e m i a l g e b r a i c subsets o f M and t h e i r complements M-U are the c l o s e d s e m i a l g e b r a i c subsets of M.
I t i s easy t o show t h a t f o r any two s e m i a l g e b r a i c maps f : M -* N and g : L -» N the f i b r e product M xN L e x i s t s i n the category of s e m i a l g e b r a i c spaces. I f N i s the one p o i n t space we w r i t e simply M x L f o r t h i s product.
Every s e m i a l g e b r a i c subset A G 6(M) o f a s e m i a l g e b r a i c space M i s i n a n a t u r a l way equipped with the s t r u c t u r e o f a s e m i a l g e b r a i c space:
The elements U € &(A) are those subsets o f A which are open i n A ( i n s t r o n g topology) and which are s e m i a l g e b r a i c i n M. A semialge- b r a i c f u n c t i o n f on U 6 6(A) i s a f u n c t i o n f : U -> R, which i s continuous i n s t r o n g topology and whose graph i s a s e m i a l g e b r a i c subset o f M x R.
A i s then a subobject o f M i n the category o f s e m i a l g e b r a i c spaces.
SEMIALGEBRAIC TOPOLOGY
§ 2 Some r e s u l t s i n the theory o f s e m i a l g e b r a i c spaces
D e f i n i t i o n 1; A s e m i a l g e b r a i c space M over R i s separated, i f i t f u l f i l l s the usual Hausdorff c o n d i t i o n : Any two d i f f e r e n t p o i n t s x and y o f M can be separated by open d i s j o i n t s e m i a l g e b r a i c neigh- bourhoods .
D e f i n i t i o n 2: A s e m i a l g e b r a i c space M over R i s c a l l e d complete, i f M i s separated and i f f o r a l l s e m i a l g e b r a i c spaces N over R the p r o j e c t i o n
M x N -* N
i s c l o s e d , i . e . , every c l o s e d s e m i a l g e b r a i c subset A o f M x N i s mapped onto a c l o s e d s e m i a l g e b r a i c subset o f N.
Since the c l o s e d s e m i a l g e b r a i c subsets o f the p r o j e c t i v e l i n e 1
PR(R) = R U {°°} which do not c o n t a i n the p o i n t 0 0 a r e f i n i t e unions of c l o s e d bounded i n t e r v a l s o f R, we o b t a i n immediately
P r o p o s i t i o n 2.1. L e t M be a complete s e m i a l g e b r a i c space over R.
Then every s e m i a l g e b r a i c f u n c t i o n f : M -> R a t t a i n s a minimum and a maximum on M.
Complete s e m i a l g e b r a i c spaces are the s u b s t i t u t e f o r compact spaces i n u s u a l topology.
Theorem 2.2 ([DK I I , 9.4]).
L e t M be a s e m i a l g e b r a i c space over the f i e l d 3 R o f r e a l numbers.
Then M i s complete i f and o n l y i f M i s a compact t o p o l o g i c a l space.
Theorem 2.3 ([DK I I , 9.6]).
L e t V be a complete R - v a r i e t y . Then the s e m i a l g e b r a i c space V(R) i s a l s o complete.
We c o n s i d e r every s e m i a l g e b r a i c subset M c Rn o f
AR = Spec R[X^,...,Xn] as a s e m i a l g e b r a i c subspace o f AR.
Theorem 2.4 ([DK I I , 9.4])
A c l o s e d and bounded s e m i a l g e b r a i c subset o f Rn i s a complete semi a l g e b r a i c space.
I f R i s not the f i e l d of r e a l numbers, the spaces Rn and hence a l l spaces V(R), where V i s an a f f i n e R - v a r i e t y , are t o t a l l y d i s - connected. To get reasonable connected components, one must f i n d another n o t i o n of connectedness.
D e f i n i t i o n 3: L e t M be a s e m i a l g e b r a i c space. A (semialgebraic) path i n M i s a s e m i a l g e b r a i c map a : [0,1] -* M from the u n i t i n t e r v a l i n R to M. Two p o i n t s P,Q of M are connectable, i f there i s a path a i n M with a(0) = P and a(1) = Q.
Every s e m i a l g e b r a i c space M s p l i t s under the equivalence r e l a t i o n
"connectable" i n t o path components.
Theorem 2.5 ([DK I I , 11.2]):
L e t M be a s e m i a l g e b r a i c space. Then M has a f i n i t e number of path components. Each of these components i s a s e m i a l g e b r a i c subset o f M.
I t f o l l o w s from a w e l l known theorem of T a r s k i t h a t the c l o s u r e N of a s e m i a l g e b r a i c subset N of a s e m i a l g e b r a i c space M i n s t r o n g topology i s again a s e m i a l g e b r a i c subset of M.
Theorem 2.6 (Curve S e l e c t i o n Lemma; [DK I I , 12.1]):
L e t N be a s e m i a l g e b r a i c subset of a s e m i a l g e b r a i c space M and l e t P be a p o i n t i n the c l o s u r e N of N i n M ( i n the s t r o n g t o p o l o g y ) . Then there e x i s t s a path a : [0,1] -> M with a(0) = P and
a(]0,1]) c N.
An immediate consequence of the l a s t two theorems i s
C o r o l l a r y 2.7 The f i n i t e l y many path components of a semialge- b r a i c space M are c l o s e d and hence open s e m i a l g e b r a i c subsets of M
Theorems 2.5 and 2.6 can be e a s i l y d e r i v e d from the t r i a n g u l a t i o n theorem (4.1), but i t i s not necessary t o use such a s t r o n g r e s u l t . The p r o o f s i n [DK I I ] are r a t h e r easy and elementary.
D e f i n i t i o n 4 ( [ B ] , p.249): A s e m i a l g e b r a i c space M i s c a l l e d connected i f i t i s not the union o f two non empty open semialge- b r a i c subsets.
The images o f (path-)connected s e m i a l g e b r a i c spaces are o b v i o u s l y again (path-)connected. Since the u n i t i n t e r v a l i n R i s connected, we see t h a t any path connected space i s connected. C o r o l l a r y 2.7
i m p l i e s t h a t a l s o the converse i s t r u e .
C o r o l l a r y 2.8: A s e m i a l g e b r a i c space i s connected i f and o n l y i f i t i s path connected.
From now on we say simply "component" i n s t e a d o f "path component".
The number o f components i s i n the f o l l o w i n g sense a b i r a t i o n a l i n v a r i a n t .
Theorem 2.9 ([DK I I , 13.3]):
L e t V and W be b i r a t i o n a l l y e q u i v a l e n t smooth complete v a r i e t i e s over R. Then V(R) and W(R) have the same number o f components.
The proof o f Theorem 2.9 i n [DK I I ] i s a s t r a i g h t f o r w a r d a d a p t i o n o f c l a s s i c a l arguments and i l l u s t r a t e s t h a t i t i s p o s s i b l e by our theory to t r a n s f e r q u i t e a l o t o f geometric i d e a s , f a m i l i a r i n the case R =3R, to a r b i t r a r y r e a l c l o s e d base f i e l d s . A q u i t e d i f f e r e n t proof o f Theorem 2.9 has been given by M.F. Coste-Roy ( [ C ] ) .
§ 3 An a p p l i c a t i o n to W i t t r i n g s
L e t X be a d i v i s o r i a l v a r i e t y over an a r b i t r a r y ( n o t n e c e s s a r i l y r e a l closed) f i e l d k. (Notice t h a t a l l r e g u l a r and a l l q u a s i p r o j e c t i v e v a r i e t i e s over k are d i v i s o r i a l ) . We c o n s i d e r the s i g n a t u r e s o f X, i . e . the r i n g homomorphisms from the W i t t r i n g W(X) o f b i l i n e a r spaces over X t o the r i n g o f i n t e g e r s 2Z ( c f . [K] f o r the g e n e r a l theory and meaning o f s i g n a t u r e s ) .
I t i s shown i n [K, V.1] t h a t any s i g n a t u r e f a c t o r s through some p o i n t x of X, i . e . , there e x i s t s a commutative diagram
W(X)
W ( K ( X ) )
with W (X) -» W ( K ( X ) ) the n a t u r a l map from W (X) to the Witt r i n g of the r e s i d u e c l a s s f i e l d K ( X ) . Using our theory we are able to prove
Theorem 3.1 ([DK I ,5.1]):
Every s i g n a t u r e a of X f a c t o r s through a c l o s e d p o i n t x of X.
Theorem 3.1 i s f i r s t proved i n the case t h a t k = R i s a r e a l c l o s e d f i e l d . The proof runs e s s e n t i a l l y along the same l i n e s as the proof i n the s p e c i a l case t h a t R =3R [ K, Chap V ] . I t i s based on
Theorem 2.5 which s t a t e s t h a t X(R) has o n l y a f i n i t e number of components. Using the theory o f r e a l c l o s u r e s of schemes ( [ K ^ ] ) , i t i s p o s s i b l e to extend Theorem 3.1 to a r b i t r a r y base f i e l d s .
§ 4 T r i a n g u l a t i o n of a f f i n e s e m i a l g e b r a i c spaces
To avoid c o n f u s i o n what i s meant by a t r i a n g u l a t i o n we f i r s t give two d e f i n i t i o n s .
D e f i n i t i o n 1; An open non degenerate n-simplex S over R i s the i n t e r i o r ( i n s t r o n g topology) of the convex c l o s u r e of n + 1 a f f i n e independent p o i n t s e0,...,e i n some space Rm, c a l l e d the v e r t i c e s of S, i . e .
r n n 1
S = < I t . e. I t . € R, t . > 0, I t . = 1>.
li=0 1 1 1 i=0 1 J
r D e f i n i t i o n 2. A s i m p l i c i a l complex over R i s a p a i r (X, X = U S. )
k=1 K
c o n s i s t i n g of a s e m i a l g e b r a i c subset X of some a f f i n e space
R and a decomposition X = U S of X i n t o d i s j o i n t open nonde-
k=1 K
generate s i m p l i c e s S^., such t h a t the f o l l o w i n g c o n d i t i o n i s f u l - f i l l e d :
The i n t e r s e c t i o n S^ fl S^ of the c l o s u r e s of any two s i m p l i c e s
Sk ' S lis e itn er e mP t y or is a f a c e ( d e f i n e d as usual) of S^ as w e l l as o f S^.
We can now s t a t e the t r i a n g u l a t i o n theorem which says t h a t f i n i t e l y many s e m i a l g e b r a i c subsets of an a f f i n e s e m i a l g e b r a i c space can be
t r i a n g u l a t e d s i m u l t a n e o u s l y .
Theorem 4.1 ([D, 2.2]):
L e t " fMj } j ^ j be a f i n i t e f a m i l y of s e m i a l g e b r a i c subsets o f an a f f i n e s e m i a l g e b r a i c space M. Then t h e r e e x i s t s a s e m i a l g e b r a i c
isomorphism r
<f> : X = U S, U M. , k=1 K j € J 3
such t h a t each s e t M. i s a union of c e r t a i n images (J>(S,) of s i m p l i -
3 K ces S^ of X.
In the proof of Theorem 3 one e a s i l y r e t r e a t s to the case t h a t the s e t s Mj are bounded s e m i a l g e b r a i c subsets o f some Rn. Then i n d u c - t i o n on n i s used, the case n = 1 being t r i v i a l . The main problem i s to f i n d a s u b s t i t u t e f o r the a n a l y t i c t o o l s used i n the
c l a s s i c a l p r o o f s f o r R = 3R ( c f . [ H ] ) .
§ 5 Cohomological dimension
The f o l l o w i n g " i n e q u a l i t y of L o j a s c i e w i c z " can be proved with h e l p of Theorem 2.4.
Lemma 5.1 ([D, 3.2]): L e t M be a c l o s e d and bounded s e m i a l g e b r a i c subset o f Rn and f , g be s e m i a l g e b r a i c f u n c t i o n s on M. Assume t h a t f o r a l l x € M f ( x ) = 0 i m p l i e s g(x) = 0 . Then there i s a c o n s t a n t C > 0, C € R, and a n a t u r a l number m, such t h a t f o r a l l x € M
| f ( x ) | > C | g ( x ) |
From Lemma 4.1 one can d e r i v e i n a s i m i l a r way as i t i s done i n a s p e c i a l case f o r R = 1R i n [BE]
Theorem 5.2 ([D, 3.3]):
L e t M be a s e m i a l g e b r a i c subset o f an a f f i n e R - v a r i e t y V = Spec A and l e t U be an open s e m i a l g e b r a i c subset o f M. Then U i s a f i n i t e union o f s e t s
{x € M | f.(x) > 0, i = 1,...,r}
with f . € A , i = 1 , . . . , r .
Other p r o o f s o f t h i s f a c t have been g i v e n by D e l z e l l [ D e , Chap.II]and M.F. Coste-Roy ([C]).
A sheaf F on a s e m i a l g e b r a i c space M a s s i g n s t o every open s e m i a l - g e b r a i c subset U o f M an a b e l i a n group F(U) such t h a t the u s u a l c o m p a t i b i l i t i e s with r e s p e c t t o r e s t r i c t i o n and the sheaf c o n d i - t i o n are f u l f i l l e d . Since we admit o n l y f i n i t e c o v e r i n g s i n the s e m i a l g e b r a i c topology M o f M ( c f . § 1 ) , the sheaf c o n d i t i o n
sa
must hold o n l y f o r f i n i t e c o v e r i n g s .
I t i s c l e a r from G r o t h e n d i e c k1s d e f i n i t i o n what the cohomology groups Hq(M,F) o f M with c o e f f i c i e n t s i n an a b e l i a n sheaf F on M are ( [ G ] , [ A ] ) :
Choose a r e s o l u t i o n
0 F -+ J° -* J1 -» J 2 ->
o f F by i n j e c t i v e sheaves J , apply the g l o b a l s e c t i o n f u n c t o r and take the cohomology groups o f the a r i s i n g complex:
Hq(M,F) = K e r ( Jq( M ) - Jq+1 (M) ) / I m ( Jq~1 (M) - Jq( M ) ) . I t f o l l o w s from Theorem 5.2 t h a t an a f f i n e s e m i a l g e b r a i c space has s i m i l a r s e p a r a t i n g p r o p e r t i e s as a u s u a l paracompact t o p o l o g i c a l space. Using t h i s f a c t we can prove
Theorem 5.3 ([D, 5.2]):
L e t M be an a f f i n e s e m i a l g e b r a i c space and F be an a b e l i a n sheaf on M. Then the c a n o n i c a l homomorphism
HP(M,F) HP(M,F)
from Cech- t o Grothendieck cohomology i s f o r a l l p > O an i s o - morphism.
Every s e m i a l g e b r a i c space M has a c e r t a i n w e l l d e f i n e d dimension ([DK I I , § 8 ] ) . I f M i s a s e m i a l g e b r a i c subset o f an R - v a r i e t y V, dim M i s simply the dimension o f the Z a r i s k i - c l o s u r e o f M i n V.
Since every a f f i n e s e m i a l g e b r a i c space can be t r i a n g u l a t e d (Theorem 4.1) and Grothendieck-cohomology c o i n c i d e s w i t h Cech- cohomology,similar arguments as i n [Go, 1 1 . 5 , 1 2 ] show t h a t the cohomological dimension does not exceed the t o p o l o g i c a l dimension.
Theorem 5.4 ([D, 5.9]):
L e t M be an a f f i n e s e m i a l g e b r a i c space o f dimension n. Then f o r a l l a b e l i a n sheaves F on M
Hq(M,F) = 0 f o r a l l q > n.
§ 6 The hojtiotopy axiom i n s e m i a l g e b r a i c cohomology
L e t M be a s e m i a l g e b r a i c space and G be an a b e l i a n group. G y i e l d s
the c o n s t a n t sheaf GM on M: F o r an open s e m i a l g e b r a i c subset U o f M
G (U) = TT G ,
•nO(U>
where n 0( U ) i s the f i n i t e s e t o f components o f U.
We denote by [0,1] the u n i t i n t e r v a l i n R.
In the c l a s s i c a l theory homo t o p i c maps induce the same homomorphisms i n cohomology. T h i s i s a l s o t r u e i n s e m i a l g e b r a i c topology over an a r b i t r a r y r e a l c l o s e d f i e l d , a t l e a s t i n the a f f i n e case.
Theorem 6.1 ([D, 7.1]): L e t fQ, f^ : M 4 N be homotopic semialge- b r a i c maps between a f f i n e s e m i a l g e b r a i c spaces, i . e . there e x i s t s a s e m i a l g e b r a i c map H : M x [0,1] -> N, such t h a t H(-,0) = f ,
H(-,1) = f . Then f and induce the same homomorphisms i n coho- mology:
f * = f * : Hq(N,GN) -> Hq(M,GM) .
We proof Theorem 6.1 by use of Alexander-Spanier-cohomology which i s d e f i n e d i n a s i m i l a r manner as i n the c l a s s i c a l case ( c f . [ S ] ) . The sheaves of A l e x a n d e r - S p a n i e r - c o c h a i n s y i e l d a r e s o l u t i o n of GM
by f l a s k sheaves (defined as i n the t o p o l o g i c a l c a s e ) . F l a s k r e s o l u t i o n s can be used to determine cohomology. Thus Alexander- Spanier-cohomology i s the same as Grothendieck-cohomology. The e x i s t e n c e of i n f i n i t e l y small elements i n a non archimedian f i e l d r i s e s many d i f f i c u l t i e s i n the proof of Theorem 6.1, compared with the c l a s s i c a l case. For example, we cannot make i n t e r v a l s and t r i a n g l e s " a r b i t r a r i l y s m a l l " by b a r y c e n t r i c or even " l i n e a r " sub- d i v i s i o n . Our proof i s based on a c a r e f u l i n v e s t i g a t i o n of the r o o t s of a system of p o l y n o m i a l s .
We use Theorem 6.1 to i d e n t i f y the s e m i a l g e b r a i c cohomology groups Hq(M,GM) with c e r t a i n s i m p l i c i a l cohomology groups. For the r e s t of t h i s s e c t i o n l e t M be an a f f i n e s e m i a l g e b r a i c space and G be a f i x e d a b e l i a n group. Consider a t r i a n g u l a t i o n
r
4> : X = U S. M i=1 1
of M (§4) . For t e c h n i c a l reasons we have to assume t h a t (j) i s a b a r y c e n t r i c s u b d i v i s i o n of another t r i a n g u l a t i o n o f M.
We then a s s o c i a t e to <j) the f o l l o w i n g a b s t r a c t s i m p l i c i a l complex K:
The s e t V(K) of v e r t i c e s o f K c o n s i s t s o f a l l v e r t i c e s of X l y i n g i n X. (Notice t h a t X i s not n e c e s s a r i l y c l o s e d ) .
A subset e~,...,e of V(K) i s a simplex of K, i f e_,...,e are
u q L O q
the v e r t i c e s of a simplex S ^ i G {1,...,r}.
In the u s u a l way we form the s i m p l i c i a l cohomology groups Hq(K,G).
P r o p o s i t i o n 6.2 ([D,8.4]):
Hq(M,GM) = Hq(K,G) f o r a l l q > 0.
In p a r t i c u l a r , the s i m p l i c i a l cohomology groups Hq(K,G) do not depend on the chosen t r i a n g u l a t i o n of M.
The proof of P r o p o s i t i o n 6.2 uses the d e s c r i p t i o n of H4(M,GM) as Cech-cohomology and Theorem 6.1 which i m p l i e s t h a t the c o v e r i n g
{ S t ( e ) }e evo f M, c o n s i s t i n g of the s t a r neighbourhoods of the v e r t i c e s of M w i t h r e s p e c t to c|), i s a L e r a y - c o v e r i n g f o r the sheaf
I t i s now p o s s i b l e to d e f i n e a l s o homology groups.
D e f i n i t i o n 1: H (M,G) := H (K,G) i s c a l l e d the q-th homology group q q
of M w i t h c o e f f i c i e n t s i n G.
T h i s d e f i n i t i o n does not depend on the chosen t r i a n g u l a t i o n of M as f o l l o w s from the corresponding f a c t f o r cohomology ( P r o p o s i t i o n 6.2).
The homology groups H (M,G) are f u n c t o r i a l i n M s i n c e , a c c o r d i n g to Theorem 4.1, every s e m i a l g e b r a i c map between a f f i n e s e m i a l g e b r a i c spaces can be "approximated" by a s i m p l i c i a l map ( c f . [D, §8]).
P r o p o s i t i o n 6.2 i m p l i e s a l s o t h a t f o r R = 1R the s e m i a l g e b r a i c coho- mology groups c o i n c i d e w i t h the usual ( s i n g u l a r ) cohomology groups determined w i t h r e s p e c t to s t r o n g topology. For homology t h i s i s t r u e by d e f i n i t i o n .
Another immediate consequence of the s i m p l i c i a l i n t e r p r e t a t i o n of homology and cohomology i s t h a t the groups are i n v a r i a n t under change of the base f i e l d . We w i l l i l l u s t r a t e t h i s i n a s p e c i a l case.
L e t L be a r e a l c l o s e d f i e l d c o n t a i n i n g R. Assume, M i s a s e m i a l - g e b r a i c subset of Rn. We choose a d e s c r i p t i o n of M by f i n i t e l y many polynomial i n e q u a l i t i e s and e q u a l i t i e s . L e t denote the s e m i a l g e b r a i c subset of Ln d e f i n e d by the same i n e q u a l i t i e s and e q u a l i t i e s . By use of T a r s k i!s p r i n c i p l e - here c l e a r l y l e g i t i m a t e and unavoidable - we see t h a t the s e t i s independent of the c h o i c e of the d e s c r i p t i o n of M.
I t a l s o f o l l o w s from T a r s k i ' s p r i n c i p l e t h a t the t r i a n g u l a t i o n r
cj> : X = U S . M
i=1 1
y i e l d s a t r i a n g u l a t i o n
*L = XL = Si L — ML
of MT . The a s s o c i a t e d a b s t r a c t complex of <j>T i s a l s o K and we get with Prop. 6.2 and Def. 1:
Hq(M,G) = Hq(ML,G), HfM,G) = H (M_,G).
q q L
Example: The n-sphere s£ = {(x.,...,x ) E RN | X x. = 1} has the i=0 1
same homology and cohomology as the n-sphere S^ over IR, s i n c e SR and S^ both can be obtained from SR by base e x t e n s i o n , w i t h RQ the r e a l c l o s u r e of Thus °
HQ(SR,G) 2 H n(SR ' G ) S H°(SR,G) Hn( SR, G ) a* G, Hq( SR, G ) = 0, Hq( SR, G ) = 0 f o r q ^ 0,n.
§ 7 The d u a l i t y theorems
As an example o f our theory we want to e x p l a i n t h a t i n a c e r t a i n sense the c l a s s i c a l d u a l i t y theorems f o r manifolds remain t r u e over an a r b i t r a r y r e a l c l o s e d f i e l d . We c o n s i d e r a s e m i a l g e b r a i c space M over R.
D e f i n i t i o n 1. M i s an n-dimensional s e m i a l g e b r a i c m a n i f o l d i f every p o i n t x € M has an open s e m i a l g e b r a i c neighbourhood which i s isomorphic to an open s e m i a l g e b r a i c subset of Rn.
Example 1: I t f o l l o w s from the i m p l i c i t f u n c t i o n theorem f o r p o l y - nomials ( c f . [DK I I , 6.9]) t h a t every open s e m i a l g e b r a i c subset U of the s e t V(R) of r e a l p o i n t s of an n-dimensional smooth R - v a r i e t y V i s an n-dimensional s e m i a l g e b r a i c m a n i f o l d ([DK I I , §13]).
From now on we assume t h a t M i s a f f i n e and complete.
We choose a t r i a n g u l a t i o n
r
<j) : X = U S. M i=1 1
and a s s o c i a t e to (f> an a b s t r a c t s i m p l i c i a l complex K as i n §6.
Since M i s complete, X i s c l o s e d and bounded i n i t s embedding m
space R .
We f o l l o w i n our n o t a t i o n the book of Maunder on a l g e b r a i c t o p o l o - gy ([M]). F o r any p o i n t x € M we denote by N^Cx) the union
U ^(S^) o f a l l c l o s e d " s i m p l i c e s " of M w i t h r e s p e c t to (J)
* "1( x ) € S i .
c o n t a i n i n g x. The union Lk. (x) := N. (x) ^ ( U <j>(S.)) o f those
* (x)€ S±
" s i m p l i c e s " o f N.(x) which do not c o n t a i n x i s c a l l e d the l i n k o f x.
D e f i n i t i o n 2: M i s c a l l e d a homology-n-manifold i f f o r a l l x € M , 7Z q = 0,n
Hg( L ^ ( x ) , B ) = { 0 G ^ 0 / N
D e f i n i t i o n 2 does not depend on the chosen t r i a n g u l a t i o n cj>
( c f . [ M ] ) .
Example 2: Every a f f i n e complete n-dimensional s e m i a l g e b r a i c mani- f o l d i s a homology-n-manifold ([D, §10]).
Subexample 2 a. The Z a r i s k i - o p e n subset U o f the m-dimensional p r o - j e c t i v e space IP1? over R, o b t a i n e d by removing the hyper s u r f ace
2 2 2 m
X^ + X. + ... + X = 0 fromlP^, i s a f f i n e and has the same r e a l o 1 m R
p o i n t s as 3PR. Hence the space V(R) o f r e a l p o i n t s o f a p r o j e c t i v e R - v a r i e t y V i s a f f i n e . I t i s a l s o complete by Theorem 2.3. Thus i f V i s p r o j e c t i v e , smooth and has dimension n the space V(R) i s a homology-n-manifold.
We r e t u r n t o our a f f i n e and complete s e m i a l g e b r a i c space M and assume i n a d d i t i o n t h a t M i s a homology-n-manifold.
D e f i n i t i o n 3: I f M i s connected, we c a l l M o r i e n t a b l e i f
Hn(M, ffi) =ZZ. In g e n e r a l M i s c a l l e d o r i e n t a b l e , i f each component of M i s o r i e n t a b l e .
Example: The n-sphere SR over R i s an o r i e n t a b l e homology-n-mani- f o l d Cef. §6) .
Theorem 7.1 ( P o i n c a r e - d u a l i t y ) :
Assume M i s o r i e n t a b l e . Then t h e r e are c a n o n i c a l isomorphisms Hq(M,2Z) H (M,ffi ) -
n-q
I f M i s not o r i e n t a b l e , there a r e s t i l l isomorphisms Hq(M,2Z/2) ^> H (M,ZZ/2) .
n-q
The proof i s very easy: Consider a r e a l i z a t i o n I K^ of the a b s t r a c t complex K over JR, i . e . , a c l o s e d s i m p l i c i a l complex over M w i t h a s s o c i a t e d a b s t r a c t complex K. Then I K L i s an ( o r i e n t a b l e ) homo- logy-n-manifold over ]R and the c l a s s i c a l Poincare'-Duality a p p l i e s to I K L. But s e m i a l g e b r a i c (co-)homology of M and s i n g u l a r (co-)
JK
homology o f IKI c o i n c i d e both with the (co-)homology o f the ab-
IK
s t r a c t complex K.
In a s i m i l a r way other d u a l i t y theorems can a l s o be t r a n s f e r r e d to an a r b i t r a r y r e a l c l o s e d f i e l d .
Using r e l a t i v e homology- and cohomology groups, one d e r i v e s from P o i n c a r e - d u a l i t y
Theorem 6 . ( A l e x a n d e r - d u a l i t y )
Let A be a s e m i a l g e b r a i c subset o f the n-sphere S^. Then there a r e isomorphisms
where Hq( r e s p . H ) denotes the reduced cohomology (resp.homology) group.
As an a p p l i c a t i o n we get the g e n e r a l i z e d Jordan curve theorem over any r e a l c l o s e d f i e l d .
C o r o l l a r y 7. L e t M be a s e m i a l g e b r a i c subset o f SR (n > 1 ) . Assume M i s a homology-n-manifold w i t h k components.
Then S^+^ -M has k+1 connected components. In p a r t i c u l a r : I f M
n4* 1
i s s e m i a l g e b r a i c a l l y isomorphic to sj? then S_ - M has 2 compo-
R K
nents. In t h i s case M i s the common boundary o f these two compo- nents .
Proof: From A l e x a n d e r - d u a l i t y we o b t a i n
H ( S £+ 1 -M,ZZ) ^ ffn(M,ZZ) ^ Hn(M,2Z)
and by P o i n c a r e - d u a l i t y
HN( M, 2 Z / 2 ) HQ( M, 2Z/ 2 ) £ TT 7Z/2.
The statement c o n c e r n i n g the boundary i s proved by s i m i l a r arguments.
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[Cl
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