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continued on page 335
Lecture Notes in Mathematics
Edited by A. Dold and B. Eckmann
1173
Hans Delfs
Manfred Knebusch
Locally Semialgebraic Spaces
UBR UBR UBR UBR UBR 069008388822
Springer-Verlag
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- - - ^ - 7 s 5 AuthorsHans Deifs Manfred Knebusch
Fakultat fur Mathematik, Universitat Regensburg Universitatsstr. 3 1 , 8400 Regensburg
Federal Republic of Germany
Univ.-Biblfothek Regensburg
S
Mathematics Subject Classification (1980): 14G30, 54E99, 55Q05, 57R05 ISBN 3-540-16060-4 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-16060-4 Springer-Verlag New York Heidelberg Berlin Tokyo
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To C h r i s t l a n d G i s e l a
P r e f a c e
The p r i m a r y o c c u p a t i o n o f r e a l a l g e b r a i c geometry, o r b e t t e r " s e m i a l g e - b r a i c geometry", i s t o s t u d y t h e s e t o f s o l u t i o n s o f a f i n i t e system o f p o l y n o m i a l i n e q u a l i t i e s i n a f i n i t e number o f v a r i a b l e s o v e r the f i e l d 3R o f r e a l numbers. One wants t o do t h i s i n a c o n c e p t u a l way, n o t a l w a y s m e n t i o n i n g t h e p o l y n o m i a l d a t a , s i m i l a r l y as i n a l g e b r a i c geometry, s a y o v e r €, where one most o f t e n a v o i d s w o r k i n g e x p l i c i t e l y w i t h the
systems o f p o l y n o m i a l e q u a l i t i e s (and n o n - e q u a l i t i e s f * 0) i n v o l v e d .
But a s e m i a l g e b r a i c geometry w h i c h d e s e r v e s i t s name s h o u l d be a b l e t o work - a t l e a s t - o v e r 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 R i n s t e a d o f t h e f i e l d JR . Such f i e l d s a r e u s e f u l and even u n a v o i d a b l e i n s e m i a l g e b r a i c geometry f o r much t h e same r e a s o n as a l g e b r a i c a l l y c l o s e d f i e l d s o f
• c h a r a c t e r i s t i c z e r o - a t l e a s t - a r e u n a v o i d a b l e i n a l g e b r a i c geometry
; o v e r <C, as soon as one t r i e s t o a v o i d t r a n s c e n d e n t a l t e c h n i q u e s o r e v e n I t h e n .
; I n o r d e r t o i l l u s t r a t e t h i s we g i v e a somewhat t y p i c a l example. L e t ' f : V->W be an a l g e b r a i c map between i r r e d u c i b l e v a r i e t i e s o v e r 1R . T h i s
y i e l d s , by r e s t r i c t i o n , a c o n t i n u o u s map f _ : V(]R) -+WCIR) between t h e
IK
s e t s o f r e a l p o i n t s . We assume t h a t W(3R) i s Z a r i s k i dense i n W w h i c h .means t h a t W(3R) c o n t a i n s non s i n g u l a r p o i n t s o r , e q u i v a l e n t l y , t h a t t h e [ f u n c t i o n f i e l d 3R(W) i s f o r m a l l y r e a l . The g e n e r i c f i b r e X o f f , i . e .
X = f 1 (n) w i t h n the g e n e r i c p o i n t o f W ( r e g a r d i n g V and W as s c h e m e s ) ,
\ i s an a l g e b r a i c scheme o v e r t h e f u n c t i o n f i e l d 3R(W) o f W, w h i c h c o n -
\ t a i n s a l o t o f i n f o r m a t i o n about f and fT O. But i t may be too d i f f i c u l t [ t o study X, s i n c e t h e f i e l d IR(W) i s u s u a l l y v e r y c o m p l i c a t e d . I n a l g e -
| b r a i c geometry one o f t e n r e p l a c e s X by the a l g e b r a i c v a r i e t y X o b t a i n e d I from X by e x t e n s i o n o f the b a s e f i e l d ]R(W) t o t h e a l g e b r a i c c l o s u r e C
o f 3E(W) . I t i s much e a s i e r t o s t u d y t h e " g e o m e t r i c g e n e r i c f i b r e " X
i n s t e a d o f X, and s t i l l one may hope t o e x t r a c t r e l e v a n t i n f o r m a t i o n a b o u t f from X. B u t i n s e m i a l g e b r a i c g e o m e t r y t h i s p r o c e d u r e i s n o t a d - v i s a b l e , s i n c e most r e a l phenomena i n X w i l l be d e s t r o y e d i n X. I n s t e a d o f X one s h o u l d s t u d y t h e v a r i e t i e s XQ, o b t a i n e d f r o m X by b a s e e x t e n - s i o n f r o m IR(W) t o t h e r e a l c l o s u r e s Ra o f IR(W) w i t h r e s p e c t t o t h e v a r i o u s o r d e r i n g s a o f t h e f u n c t i o n f i e l d IR(W) , and t h e s e t s o f r a t i o - n a l p o i n t s X (R ) . F o r e v e r y s u c h a we have R ^ ( VCT ) = C. Thus t h e R„
a ot a a a r e "as n e a r as p o s s i b l e " t o C and n e v e r t h e l e s s we may hope t o d e t e c t
some o f t h e r e a l phenomena o f X, and u l t i m a t e l y o f f , i n t h e s e t s
v v -
The v a r i e t y X i s t h e p r o j e c t i v e l i m i t o f t h e schemes f 1 (U) = V x^u w i t h U r u n n i n g t h r o u g h t h e Z a r i s k i - o p e n s u b s e t s o f W, s i n c e t h e s e U a r e the Z a r i s k i n e i g h b o u r h o o d s o f t h e g e n e r i c p o i n t ri i n W. S i m i l a r l y X i s t h e p r o j e c t i v e l i m i t o f t h e f i b r e p r o d u c t s V ^ u , w i t h r e s p e c t t o t h e e t a l e m o r p h i s m s cp : U -»W from a r b i t r a r y v a r i e t i e s U o v e r IR t o W (U * 0 , b u t U(IR) may be empty), s i n c e t h e s e morphisms cp a r e t h e e t a l e n e i g h b o u r - hoods o f n. How about t h e Xa? An o r d e r i n g a o f IR(W) c o r r e s p o n d s u n i - q u e l y t o an u l t r a f i l t e r F i n t h e B o o l e a n l a t t i c e )T(W ( IR ) o f s e m i a l g e - b r a i c s u b s e t s o f W(IR) such t h a t e v e r y A €F has a non empty i n t e r i o r A i n t h e s t r o n g t o p o l o g y (= c l a s s i c a l t o p o l o g y on W(IR)), w h i c h means t h a t A i s Z a r i s k i d e n s e i n W, c f . [B, 8 . 1 1 ] , [ B r , § 4 ] . (A r a t i o n a l f u n c - t i o n h €IR(W) i s p o s i t i v e w i t h r e s p e c t t o a i f and o n l y i f h i s d e f i n e d and p o s i t i v e on some s e t A C F ) . I t t u r n s o u t t h a t Xa i s t h e p r o j e c t i v e l i m i t o f t h e f i b r e p r o d u c t s V x^u w i t h r e s p e c t t o t h o s e e t a l e morphisms cp : U ->W s u c h t h a t cp(U(IR)) € F . (N.B. cp(U(IR)) i s s e m i a l g e b r a i c . ) T h i s i s due t o t h e f a c t t h a t Ra c a n be i n t e r p r e t e d as t h e u n i o n o f t h e r i n g s o f N a s h f u n c t i o n s &W( U ) on t h e v a r i o u s smooth open s e t s U € F , c f . [ R y ] .
Much more c a n be s a i d about a g e o m e t r i c i n t e r p r e t a t i o n o v e r IR o f the f i e l d s Ra, t h e v a r i e t i e s XQ and t h e p o i n t s i nxa(Ra) * B ut t h i s w o u l d
t a k e u s t o o f a r a f i e l d . We o n l y m e n t i o n t h a t t h e r e a l s p e c t r a o f commu- t a t i v e r i n g s i n v e n t e d by M. C o s t e and M.F. C o s t e - R o y p r o v i d e e x a c t l y t h e r i g h t l a n g u a g e t o u n d e r s t a n d a l l t h i s , c f . [ C R ] , [ R y ] , and t h e
l i t e r a t u r e c i t e d t h e r e a n d , f o r an i n t r o d u c t i o n t o r e a l s p e c t r a , a l s o [L, § 4 , § 7 ] , [ B r , § 3 , § 4 ] , [ K ] , [BCR, Chap. 7 ] .
We have b e e n somewhat vague above. I n p a r t i c u l a r we d i d n o t make p r e - c i s e t h e v a r i o u s d i r e c t s y s t e m s w h i c h y i e l d t h e p r o j e c t i v e l i m i t s X and XQ. We o n l y wanted t o i n d i c a t e t h a t i n s e m i a l g e b r a i c geometry o v e r IR r e a l c l o s e d f i e l d s may come up i n a n a t u r a l and g e o m e t r i c way.
The p r e s e n t l e c t u r e n o t e s g i v e a c o n t r i b u t i o n t o a b a s i c b u t r a t h e r modest a s p e c t o f s e m i a l g e b r a i c geometry: t h e t o p o l o g i c a l phenomena o f
s e m i a l g e b r a i c s e t s i n V(R) f o r V a v a r i e t y o v e r a r e a l c l o s e d f i e l d R.
T h e r e i s a d i f f i c u l t y w i t h t h e word " t o p o l o g i c a l " h e r e . Of c o u r s e , V(R) i s e q u i p p e d w i t h t h e s t r o n g t o p o l o g y coming f r o m t h e t o p o l o g y o f t h e o r d e r e d f i e l d R. B u t , e x c e p t i n t h e c a s e R = I R , t h e t o p o l o g i c a l s p a c e V(R) i s t o t a l l y d i s c o n n e c t e d .
These p a t h o l o g i e s c a n be r e m e d i e d by c o n s i d e r i n g on V(R) a t o p o l o g y i n t h e s e n s e o f G r o t h e n d i e c k , where o n l y open s e m i a l g e b r a i c s u b s e t s U o f V(R) a r e a d m i t t e d a s "open s e t s " , and f o r s u c h a s e t U e s s e n t i a l l y o n l y c o v e r i n g s by f i n i t e l y many open s e m i a l g e b r a i c s u b s e t s o f U a r e a d m i t t e d as "open c o v e r i n g s " .
I t seems t h a t t h e c a t e g o r y o f s e m i a l g e b r a i c s p a c e s and maps o v e r a r e a l c l o s e d f i e l d R, w h i c h has been i n t r o d u c e d i n o u r p a p e r [ D K2] , p r o v i d e s t h e r i g h t framev/ork f o r t h i s " s e m i a l g e b r a i c t o p o l o g y " . A l r e a d y i n t h a t paper and l a t e r i n o t h e r ones ( [ D ] , [ D1] , [DK^], [DK^], [DK^]) we f o u n d a n a l o g u e s o f many r e s u l t s i n c l a s s i c a l t o p o l o g y . Sometimes t h i n g s a r e even n i c e r h e r e . T h i s i s n o t a s t o n i s h i n g s i n c e , i n t h e c a s e R =IR, t h e
s e m i a l g e b r a i c s e t s a r e r a t h e r tame f r o m a t o p o l o g i c a l v i e w p o i n t .
I n t h e c a s e R =IR t h e c a t e g o r y o f s e m i a l g e b r a i c s p a c e s c a n be compared w i t h t h e c a t e g o r y o f t o p o l o g i c a l s p a c e s , and t h i s a f f o r d s us a new p e r s p e c t i v e c o n c e r n i n g t h e two b r a n c h e s o f m a t h e m a t i c s i n v o l v e d , s e m i - a l g e b r a i c geometry and a l g e b r a i c t o p o l o g y , c f . t h e i n t r o d u c t i o n o f [ B ] , F o r example, a l o n g j o u r n e y a l o n g t h i s r o a d s h o u l d g i v e a t h o r o u g h un- d e r s t a n d i n g o f why so many s p a c e s o c c u r i n g i n u s u a l a l g e b r a i c t o p o l o g y a r e s e m i a l g e b r a i c s e t s .
N e v e r t h e l e s s t h e c a t e g o r y o f s e m i a l g e b r a i c s p a c e s i s t o o r e s t r i c t i v e f o r some p u r p o s e s . A good i n s t a n c e where t h i s c a n be s e e n i s t h e t h e o r y o f s e m i a l g e b r a i c c o v e r i n g s . I f M i s a c o n n e c t e d a f f i n e s e m i a l g e b r a i c s p a c e o v e r R, and xq i s some p o i n t i n M, we c a n d e f i n e t h e f u n d a m e n t a l g r o u p ( M , xQ) i n t h e u s u a l way as t h e s e t o f s e m i a l g e b r a i c homotopy
* )
c l a s s e s o f s e m i a l g e b r a i c l o o p s w i t h b a s e p o i n t xQ ( c f . Ill,§6) . T h i s i s an h o n e s t t o g o o d n e s s g r o u p , g e n e r a t e d by f i n i t e l y many e l e m e n t s s a - t i s f y i n g f i n i t e l y many r e l a t i o n s . On t h e o t h e r hand we e v i d e n t l y have t h e n o t i o n o f an ( u n r a m i f i e d ) c o v e r i n g p :N -»M o f M, p b e i n g a l o c a l l y t r i v i a l s e m i a l g e b r a i c map w i t h d i s c r e t e (= z e r o - d i m e n s i o n a l ) f i b r e s . Of c o u r s e , one would l i k e t o c l a s s i f y t h e c o v e r i n g s o f M by s u b g r o u p s o f TT.j ( M , xQ) . B u t a z e r o - d i m e n s i o n a l s e m i a l g e b r a i c s p a c e i s n e c e s s a r i l y a f i n i t e s e t . Thus e v e r y s e m i a l g e b r a i c c o v e r i n g has f i n i t e d e g r e e . I t c a n be shown t h a t i n d e e d t h e i s o m o r p h i s m c l a s s e s o f s e m i a l g e b r a i c c o v e r i n g s o f M c o r r e s p o n d u n i q u e l y t o t h e c o n j u g a c y c l a s s e s o f s u b - g r o u p s o f f i n i t e i n d e x i n ( M , xQ) i n t h e u s u a l way. B u t t h e r e s h o u l d a l s o e x i s t c o v e r i n g s o f a more g e n e r a l n a t u r e w h i c h c o r r e s p o n d t o the o t h e r subgroups o f tt^ ( M ,xQ) . I n p a r t i c u l a r t h e r e s h o u l d e x i s t a u n i - v e r s a l c o v e r i n g o f M. T h e s e more g e n e r a l c o v e r i n g s c a n be d e f i n e d i n
*) T h i s means §6 i n C h a p t e r I I I o f t h i s book.
t h e c a t e g o r y o f " l o c a l l y s e m i a l g e b r a i c " s p a c e s and maps.
A f t e r s e v e r a l y e a r s o f e x p e r i m e n t i n g w i t h l o c a l l y s e m i a l g e b r a i c s p a c e s we a r e c o n v i n c e d t h a t t h e s e s p a c e s e x i s t " i n n a t u r e " . The c o v e r i n g s o f
a f f i n e s e m i a l g e b r a i c s p a c e s a r e r e g u l a r paracompact l o c a l l y s e m i a l g e - b r a i c s p a c e s , t o be d e f i n e d i n I , § 4 . R e g u l a r paracompact s p a c e s seem t o be t h e "good" l o c a l l y s e m i a l g e b r a i c s p a c e s , a n a l o g o u s t o t h e a f f i n e s p a c e s i n t h e s e m i a l g e b r a i c c a t e g o r y . F o r i n s t a n c e , f o r t h e s e spaces t h e r e e x i s t s a s a t i s f a c t o r y cohomology t h e o r y o f s h e a v e s , b a s e d on f l a b b y and s o f t s h e a v e s , w h i c h p a r a l l e l s t h e c l a s s i c a l t h e o r y f o r t o p o - l o g i c a l p a r a c o m p a c t s p a c e s . We w i l l n o t d e a l w i t h t h e s e m a t t e r s h e r e , e x c e p t f o r some b r i e f r e m a r k s i n A p p e n d i x A, b u t t h e y a r e q u i t e impor- t a n t f o r d e f i n i n g homology and cohomology g r o u p s o f v a r i o u s k i n d s f o r t h e s e s p a c e s , c f . [ D ] , [ D ^ , [ D2] .
A l t h o u g h r e g u l a r p a r a c o m p a c t s p a c e s a r e a v e r y s a t i s f y i n g s u b c l a s s o f l o c a l l y s e m i a l g e b r a i c s p a c e s one has t o f a c e t h e f a c t t h a t t h e r e e x i s t many l o c a l l y s e m i a l g e b r a i c s p a c e s i n n a t u r e w h i c h a r e n o t paracompact.
( I t seems t h a t r e g u l a r i t y may be assumed i n most a p p l i c a t i o n s . ) F o r example, s t u d y i n g open s u b s e t s o f q u i t e i n n o c e n t l y l o o k i n g r e a l s p e c t r a may l e a d t o r e g u l a r s p a c e s w h i c h a r e n o t paracompact, c f . A p p e n d i x A.
Thus i t i s n o t j u s t f o r f u n o r f o r s y s t e m a t i c r e a s o n s t h a t we s t u d y i n C h a p t e r I more g e n e r a l s p a c e s . I n t h e l a t e r c h a p t e r s we a r e f o r c e d t o r e s t r i c t t o paracompact s p a c e s , s i n c e o t h e r w i s e o u r d e e p e r t e c h n i q u e s break down.
There i s one phenomenon i n o u r t h e o r y w h i c h may seem somewhat u n u s u a l f o r a r e a d e r o f our p r e v i o u s p a p e r s . I n a s e m i a l g e b r a i c s p a c e M i t i s s t r i c t l y f o r b i d d e n t o work w i t h s u b s e t s o f M o t h e r t h a n t h e s e m i a l g e b r a i c s u b s e t s [DK2f § 7 ] . B u t i n a l o c a l l y s e m i a l g e b r a i c s p a c e M t h e r e e x i s t two n a t u r a l c l a s s e s o f a d m i s s i b l e s u b s e t s , t h e c l a s s T(M) o f l o c a l l y
s e m i a l g e b r a i c s u b s e t s o f M and t h e s m a l l e r c l a s s JT(M) o f s e m i a l g e b r a i c s u b s e t s o f M. The i n t e r p l a y between T(M) and T(M) i s a theme w h i c h r e - c u r s t h r o u g o u t t h e whole t h e o r y .
The g o a l o f t h e f i r s t volume o f o u r l e c t u r e n o t e s i s t o e s t a b l i s h t h e c a t e g o r y o f l o c a l l y s e m i a l g e b r a i c s p a c e s and maps o v e r 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 R on f i r m grounds, and t o p r o v e enough r e s u l t s a b o u t t h e s e s p a c e s and maps, t h a t t h e r e a d e r w i l l f e e l w e l l a c q u a i n t e d w i t h them and w i l l r e g a r d them as c o n c r e t e and a c c e s s i b l e o b j e c t s . The n e x t t o p i c s , t o be c o v e r e d i n t h e s e c o n d volume, a r e t h e t h e o r y o f l o c a l l y s e m i a l g e b r a i c f i b r a t i o n s and f i b r e b u n d l e s ( C h a p t e r IV) and t h e t h e o r y o f c o v e r i n g s ( C h a p t e r V ) .
As b a c k g r o u n d m a t e r i a l we assume o u r p a p e r s [DK23/ [DK^], [DK^], some s e c t i o n s o f [DK^], and Robson's p a p e r [ R ] . Here y o u f i n d n e a r l y e v e r y - t h i n g w h i c h we need about s e m i a l g e b r a i c s p a c e s , w r i t t e n up i n a s y s t e - m a t i c way c o m p a t i b l e w i t h t h e s p i r i t o f t h e s e l e c t u r e n o t e s . Of c o u r s e , i t w o u l d have been more c o m f o r t a b l e f o r t h e r e a d e r i f we h a d s t a r t e d t h e l e c t u r e n o t e s w i t h a r e v i e w o f t h e r e s u l t s o f t h o s e p a p e r s . B u t t h i s i s n o t r e a l l y n e c e s s a r y and would have made t h e l e c t u r e n o t e s t o o l o n g . Of c o u r s e , t h e book [BCR] o f Bochnak and t h e C o s t e s - as soon as i t has a p p e a r e d - w i l l c o n t a i n most b a s i c f a c t s w h i c h a r e n e c e s s a r y f o r an u n d e r s t a n d i n g o f t h e s e l e c t u r e n o t e s and much more.
A s u r v e y on some b a s i c r e s u l t s about s e m i a l g e b r a i c s p a c e s has b e e n g i v e n i n [DK]. A n o t h e r s u r v e y on b a s i c r e s u l t s about l o c a l l y s e m i a l g e - b r a i c s p a c e s , which, o f c o u r s e , a l l w i l l be c o v e r e d by t h e two volumes o f t h e s e l e c t u r e n o t e s , has been g i v e n i n [DKg] and [DK^].
We hope t h a t t h e s e l e c t u r e n o t e s , d e s i g n e d i n f i r s t p l a c e f o r t h e needs o f s e m i a l g e b r a i c geometry, a r e a l s o o f i n t e r e s t f o r t o p o l o g i s t s . The main r e s u l t s a r e u s u a l l y non t r i v i a l a l s o i n t h e c a s e R = 3R and n o t much e a s i e r t o be p r o v e d i n t h i s s p e c i a l c a s e . The c a t e g o r y o f l o c a l l y
s e m i a l g e b r a i c s p a c e s o v e r 3R l i e s somewhat " i n between" t h e c a t e g o r y TOP o f t o p o l o g i c a l H a u s d o r f f s p a c e s and t h e c a t e g o r y PL o f p i e c e w i s e l i n e a r s p a c e s , b e i n g l e s s r i g i d t h a n PL and, i n some r e s p e c t s , l e s s p a t h o l o g i c a l t h a n TOP.
The c e n t r a l r e s u l t o f t h e whole volume seems t o be Theorem 4.4 i n C h a p t e r I I , §4, w h i c h s t a t e s t h a t e v e r y r e g u l a r paracompact l o c a l l y s e m i a l g e b r a i c s p a c e M c a n be t r i a n g u l a t e d , and moreover a g i v e n l o c a l l y f i n i t e f a m i l y o f l o c a l l y s e m i a l g e b r a i c s u b s e t s o f M 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 . Thus we may r e g a r d e v e r y r e g u l a r paracompact space as a l o c a l l y f i n i t e p o l y h e d r o n w i t h some open f a c e s m i s s i n g ( c f . t h e d e f i - n i t i o n o f s t r i c t l y l o c a l l y f i n i t e s i m p l i c i a l complexes, i n I , § 2 , w h i c h i s s l i g h t l y d i f f e r e n t f r o m t h e c l a s s i c a l d e f i n i t i o n ) . B u t i n c o n -
t r a s t t o P L - t h e o r y , we may s u b d i v i d e s i m p l i c e s n o t o n l y l i n e a r l y b u t
" s e m i a l g e b r a i c a l l y " . N e v e r t h e l e s s , i n t h e s p e c i a l c a s e t h a t R= IR and M i s p a r t i a l l y c o m p l e t e , S h i o t a and Y o k o i have r e c e n t l y p r o v e d t h a t any two PL s t r u c t u r e s on M w h i c h r e f i n e t h e g i v e n s e m i a l g e b r a i c s t r u c t u r e a r e i s o m o r p h i c ([SY, Th. 4.1], t h e y p r o v e t h i s more g e n e r a l l y f o r s u i t a b l e l o c a l l y s u b a n a l y t i c s p a c e s ) . T h i s r e m a r k a b l e theorem can be e x t e n d e d t o p a r t i a l l y complete r e g u l a r paracompact s p a c e s o v e r any R, as we hope t o e x p l a i n i n t h e s e c o n d volume.
I f S i s 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 t h e n , as a consequence o f T a r s k i ' s p r i n c i p l e , we c a n a s s o c i a t e w i t h e v e r y l o c a l l y s e m i a l g e b r a i c space M o v e r R a l o c a l l y s e m i a l g e b r a i c s p a c e M(S) o v e r S by " e x t e n s i o n o f the base f i e l d R t o S", c f . 1.2.10. T h i s y i e l d s a v e r y good n a t u r e d f u n c t o r M ^ M ( S ) f r o m t h e c a t e g o r y o f r e g u l a r paracompact s p a c e s o v e r R t o the c a t e g o r y o f r e g u l a r paracompact s p a c e s o v e r S, w h i c h i s o f c r u - c i a l i m p o r t a n c e f o r o u r whole t h e o r y . The homotopy groups ( c f . I l l , § 6 ) , t h e homology groups ( c f . I l l , §7) and a l s o t h e v a r i o u s K-groups o f M
( o r t h o g o n a l , u n i t a r y , s y r a p l e c t i c , c f . C h a p t e r IV i n t h e s e c o n d volume) a r e p r e s e r v e d under b a s e f i e l d e x t e n s i o n f r o m R t o S. These a r e examples
o f t h e main message o f o u r whole t h e o r y , t h a t o v e r a c o m p l i c a t e d r e a l c l o s e d f i e l d t h e l o c a l l y s e m i a l g e b r a i c s p a c e s a r e i n many r e s p e c t s n o t more c o m p l i c a t e d t h a n o v e r a s i m p l e f i e l d , as t h e f i e l d IR o r the f i e l d RQ o f r e a l a l g e b r a i c numbers. We b e l i e v e t h a t t h i s message i s by no means t r i v i a l . I t may be r e g a r d e d as a v a s t g e n e r a l i z a t i o n o f T a r s k i ' s p r i n c i p l e f o r t o p o l o g i c a l s t a t e m e n t s . As soon as one l e a v e s t h e c a d r e o f s e m i a l g e b r a i c t o p o l o g y and works, s a y w i t h a l g e b r a i c f u n c t i o n s t h e n t h e a n a l o g u e o f o u r message seems t o h o l d o n l y u n d e r s e v e r e r e s t r i c t i o n s F o r example, i t i s w e l l known t h a t , i n g e n e r a l , s e m i a l g e b r a i c f u n c t i o n s on t h e u n i t i n t e r v a l [0,1] i n R c a n n o t be a p p r o x i m a t e d u n i f o r m l y by p o l y n o m i a l s , i n c o n t r a s t t o t h e S t o n e - W e i e r s t r a B t h e o r e m f o r R= 3R .
The book has two a p p e n d i c e s . A p p e n d i x B ( t o C h a p t e r I) c o n t a i n s some e a s y b u t f u n d a m e n t a l r e s u l t s i n t h e t h e o r y o f b a s e e x t e n s i o n . They have n o t been i n c l u d e d i n t o C h a p t e r I s i n c e some o f t h e t e c h n i q u e s n e e d e d t o d e r i v e them seem t o have t h e i r n a t u r a l p l a c e i n C h a p t e r I I . A p p e n d i x A i s o f d i f f e r e n t k i n d . Here we draw t h e c o n n e c t i o n s between o u r t h e o r y and " a b s t r a c t " s e m i a l g e b r a i c g e o m e t r y w h i c h , s t a r t i n g f r o m t h e n o t i o n o f t h e r e a l s p e c t r u m , now i s i n a p r o c e s s o f r a p i d d e v e l o p - ment. A p p e n d i x A i s n o t n e e d e d f o r o u r t h e o r y i n a t e c h n i c a l s e n s e , b u t t h e r e we w i l l f i n d t h e o c c a s i o n t o e x p l a i n some more p o i n t s o f our p h i l o s o p h y a b o u t t h e " r a i s o n d ' e t r e " o f l o c a l l y s e m i a l g e b r a i c s p a c e s .
We t h a n k t h e members o f t h e f o r m e r R e g e n s b u r g e r s e m i a l g e b r a i c group, i n p a r t i c u l a r R o l a n d Huber and Robby Robson, f o r s t i m u l a t i n g d i s c u s s i o n s and c r i t i c i s m a b o u t t h e c o n t e n t s o f t h e s e l e c t u r e n o t e s . S p e c i a l t h a n k s a r e due t o J o s e M a n u e l Gamboa and R. Huber f o r a p e n e t r a t i n g (and v e r y s u c c e s s f u l ) s e a r c h f o r m i s t a k e s i n t h e f i n a l v e r s i o n o f t h e m a n u s c r i p t .
We t h a n k M a r i n a R i c h t e r f o r h e r p a t i e n c e and e x c e l l e n c e i n t y p i n g the book and R. Robson f o r e l i m i n a t i n g some o f t h e most annoying g r a m m a t i c a l m i s t a k e s . We a r e w e l l aware t h a t we c o u l d have w r i t t e n a b e t t e r book
i n o u r n a t i v e l a n g u a g e , b u t s i n c e t h e book i s d e s i g n e d as a " t o p o l o g i e g e n e r a l e " f o r s e m i a l g e b r a i c geometry w h i c h s h o u l d be u s e f u l as a w i d e l y a c c e p t e d r e f e r e n c e , we have w r i t t e n i n t h a t l a n g u a g e w h i c h w i l l be u n d e r s t o o d by the most.
R e g e n s b u r g , J u l y 1985
Hans D e l f s , M a n f r e d K n e b u s c h
TABLE OF CONTENTS
page
C h a p t i e r I - The b a s i c d e f i n i t i o n s 1
§1 - L o c a l l y s e m i a l g e b r a i c s p a c e s and maps 1
§2 - I n d u c t i v e l i m i t s , some examples o f l o c a l l y s e m i a l g e -
b r a i c s p a c e s 1 1
§3 - L o c a l l y s e m i a l g e b r a i c s u b s e t s 27
§4 - R e g u l a r and paracompact s p a c e s 42
§5 - S e m i a l g e b r a i c maps and p r o p e r maps 54
§6 - P a r t i a l l y p r o p e r maps 63
§7 - L o c a l l y complete s p a c e s 75
Chaptter I I - C o m p l e t i o n s and t r i a n g u l a t i o n s 87
§1 - G l u i n g paracompact s p a c e s 87
§2 - E x i s t e n c e o f c o m p l e t i o n s 94
§3 - A b s t r a c t s i m p l i c i a l complexes 99
§4 - T r i a n g u l a t i o n o f r e g u l a r paracompact s p a c e s 1 °6
§5 - T r i a n g u l a t i o n o f w e a k l y s i m p l i c i a l maps, maximal
complexes 113
§6 - T r i a n g u l a t i o n o f amenable p a r t i a l l y f i n i t e maps 124
§7 - S t a r s and s h e l l s 13 8
§8 - Pure h u l l s o f dense p a i r s 146
§9 - Ends o f s p a c e s , t h e L C - s t r a t i f i c a t i o n 156
§10 —Some p r o p e r q u o t i e n t s 178
§11 — M o d i f i c a t i o n o f p u r e ends 189
§12 — T h e S t e i n f a c t o r i z a t i o n o f a s e m i a l g e b r a i c map 198
§13 — S e m i a l g e b r a i c s p r e a d s 211
§14 — H u b e r ' s theorem on open mappings 219
C h a p t e r I I I - Homotopies 22 6
§1 - Some s t r o n g d e f o r m a t i o n r e t r a c t s 22 6
§2 - S i m p l i c i a l a p p r o x i m a t i o n s 23 2
§3 - The f i r s t main theorem on homotopy s e t s ; mapping s p a c e s 24 3
§4 - R e l a t i v e homotopy s e t s 24 9
§5 - The s e c o n d main theorem; c o n t i g u i t y c l a s s e s 257
§6 - Homotopy groups 26 5
§7 - Homology; t h e H u r e w i c z theorems 278
§8 - Homotopy groups o f ends 286
Appendix A - A b s t r a c t l o c a l l y s e m i a l g e b r a i c s p a c e s 295 Appendix B - C o n s e r v a t i o n o f some p r o p e r t i e s o f spaces and maps
under base f i e l d e x t e n s i o n 309
R e f e r e n c e s 315 L i s t o f symbols 319
G l o s s a r y 322
C h a p t e r I . - The b a s i c d e f i n i t i o n s
The g o a l s o f t h i s c h a p t e r a r e modest. We p r e s e n t t h e b a s i c d e f i n i t i o n s and some e l e m e n t a r y o b s e r v a t i o n s needed by anyone who wants t o work w i t h l o c a l l y s e m i a l g e b r a i c s p a c e s . The c o v e r i n g s mentioned i n t h e p r e - f a c e w i l l be a s p e c i a l c l a s s o f t h e " p a r t i a l l y p r o p e r maps" c o n s i d e r e d i n § 6 .
§1 - L o c a l l y s e m i a l g e b r a i c s p a c e s and maps
Our i d e a i s t o d e f i n e l o c a l l y s e m i a l g e b r a i c s p a c e s as s u i t a b l e r i n - ged s p a c e s i n t h e sense o f G r o t h e n d i e c k . Then t h e more d e l i c a t e q u e s t i o n how t o d e f i n e l o c a l l y s e m i a l g e b r a i c maps becomes t r i v i a l . These maps w i l l be s i m p l y a l l t h e morphisms between t h e l o c a l l y s e - m i a l g e b r a i c s p a c e s i n t h e c a t e g o r y o f r i n g e d s p a c e s . We used t h e same p r o c e d u r e a l r e a d y t o d e f i n e s e m i a l g e b r a i c s p a c e s i n [DK2,§7].
D e f i n i t i o n 1. A g e n e r a l i z e d t o p o l o g i c a l space *) i s a s e t M t o g e t h e r w i t h a s e t 'f(M) o f s u b s e t s o f M, c a l l e d t h e "open s u b s e t s " o f M, and a s e t C o vM o f f a m i l i e s ( Ual a € I ) i n ^ ( M ) * * ) , c a l l e d t h e " a d m i s s i b l e c o v e r i n g s " such t h a t t h e f o l l o w i n g p r o p e r t i e s h o l d :
i ) 0 e 7(M) , M € tf(M) .
i i ) I f U1 € *(M) , U2 € T(M) , t h e n fl U2 € f ( M ) and U1 U U2 € f ( M ) . i i i ) E v e r y f a m i l y ( Ual a € I ) i n ^f(M) w i t h I f i n i t e i s an element
o f C o vM.
M
i v ) I f (UglaGI) i s an e l e m e n t o f Cov^, t h e n t h e u n i o n U := U ( Ual a € I ) o f t h i s f a m i l y i s an element o f jP(M) .
*) T h i s term s h o u l d be r e g a r d e d as ad h o c .
**) In o r d e r t o g u a r a n t e e t h a t Cov^. i s r e a l l y a s e t one s h o u l d o n l y a l l o w s u b s e t s o f some f i x e d l a r g e s e t as index s e t s I . We w i l l i g n o r e a l l s e t - t h e o r e t i c d i f f i c u l t i e s h e r e .
F o r any U € T(M) we d e n o t e t h e s u b s e t o f a l l (UglaGI) € C o vM w i t h U ( Ual a € I ) = U by Cov^dJ) and we c a l l t h e s e c o v e r i n g s t h e " a d m i s s i b l e c o v e r i n g s o f U".
v) I f ( Ual a e i ) i s an a d m i s s i b l e c o v e r i n g o f U € T(M) and i f V € f(M) i s a s u b s e t o f U, t h e n ( UaD V l a € I ) i s an a d m i s s i b l e c o v e r i n g o f V.
v i ) I f an a d m i s s i b l e c o v e r i n g (Ualcx€I) o f U € T(M) i s g i v e n and f o r e v e r y a € I an a d m i s s i b l e c o v e r i n g ( Vap l 3 £ Ja) o f Ua i s g i v e n , t h e n (V^^I a € 1 , 3G JQ) i s an a d m i s s i b l e c o v e r i n g o f U.
v i i ) I f (U |a€I) i s a f a m i l y i n f(M) w i t h U ( Ual a € I ) = U € fCM) a n d i f (Vpl3€J) € C o vM( U ) i s a r e f i n e m e n t o f ( Ual a € I ) { i . e . t h e r e e x i s t s a map A : J-» I w i t h V ^ c u ^ ^ ^or ever^ 3 € J} , t h e n
( Ual a € I ) € C o vM( U ) .
v i i i ) I f U € f ( M ) , (U l a € I ) 6 C o vM( U ) and i f V i s a s u b s e t o f U w i t h a M
V n Ua € T(M) f o r e v e r y a € I , t h e n V € f (M) [and t h u s , by (v) , (V n U l o c € I ) i s an a d m i s s i b l e c o v e r i n g o f V] .
Comments.
a) We u s u a l l y j u s t w r i t e "M" f o r t h e t r i p l e (M, T(M) , Cov^) . A g e n e r a - l i z e d t o p o l o g i c a l space M i s a ( r a t h e r s p e c i a l ) example o f a s i t e i n t h e s e n s e o f G r o t h e n d i e c k . Thus we have G r o t h e n d i e c k ' s t h e o r y o f
s h e a v e s o v e r such spaces M a t o u r d i s p o s a l , c f . [ A ] , [SGA4, Exp. I I ] . L e t u s r e c a l l t h e n o t i o n o f an ( a b e l i a n ) s h e a f h e r e i n o u r s p e c i a l s i t u a t i o n . A p r e s h e a f ? on M i s an a s s i g n m e n t Uv+^U) o f an a b e l i a n g r o u p ^(U) t o e v e r y U G CT(M) e q u i p p e d w i t h a r e s t r i c t i o n homomorphism r ^ : T(U) -> ^(V) f o r e v e r y p a i r o f open s e t s U,V w i t h V c U s u c h t h a t r ^ = i d and r X ° r ^ = rr^ f o r U D V D W . A p r e s h e a f T i s a s h e a f i f i n
U W V W
a d d i t i o n f o r any a d m i s s i b l e c o v e r i n g ( U ^ l a E I ) o f any U € T(M), t h e u s u a l sequence
o - T ( u ) - TT7(\J) Z 1 r T ( u n u J
<x€I ( a, 3) € I x i a 1 3
i s e x a c t . Thus a s h e a f i s v e r y much the same n o t i o n as f o r u s u a l t o - p o l o g i c a l s p a c e s , e x c e p t t h a t now o n l y s e t s i n T(M) a r e a l l o w e d as open s e t s and o n l y a d m i s s i b l e c o v e r i n g s a r e a l l o w e d as open c o v e r i n g s . b) The l a s t two axioms ( v i i ) and ( v i i i ) i n the d e f i n i t i o n o f a gene- r a l i z e d t o p o l o g i c a l space a r e l e s s s u b s t a n t i a l t h a n the o t h e r s . Axiom ( v i i ) i s j u s t a t e c h n i c a l d e v i c e t o make f o r m a l arguments smoother. N o t i c e t h a t i f axioms ( i ) - ( v i ) a r e f u l f i l l e d , then by e n r i c h i n g t h e s e t s C o vM( U ) , U € T(M), by a l l f a m i l i e s (U |a€I) i n ?(M) which have t h e u n i o n U and w h i c h admit r e f i n e m e n t s l y i n g i n C o v ^ d J ) , we o b t a i n a new s i t e (M,?(M),Cov^) w h i c h f u l f i l l s ( i ) - ( v i i ) and which has t h e same sheaves as t h e o r i g i n a l s i t e (M, (M) , C o vM) . The r o l e o f axiom ( v i i i ) i s more s u b t l e and w i l l be d i s c u s s e d a t t h e end o f t h i s s e c t i o n .
From now on R d e n o t e s a f i x e d r e a l c l o s e d f i e l d .
D e f i n i t i o n 2 . A r i n g e d space o v e r R i s a p a i r (M,0 ) c o n s i s t i n g o f a g e n e r a l i z e d t o p o l o g i c a l space M and a sheaf 0?^ o f commutative R- a l g e b r a s . A morphism (cp,*f) : (M,C?M) -* ( N , ©N) between r i n g e d s p a c e s
(M,#M) and (N,C?N) o v e r R i s d e f i n e d i n t h e o b v i o u s way: cp i s a c o n - t i n u o u s map from M t o N, i . e . e v e r y open s e t V i n N has an open p r e -
-1 -1 image cp (V) and f o r e v e r y (V la€I) e C o vN( V ) the f a m i l y (cp ( Va) l a € I )
i s an a d m i s s i b l e c o v e r i n g o f cp"1 (V) . The second component 3 i s a I s o - morphism from t h e s h e a f (0^ t o t h e sheaf t P * ^ r e s p e c t i n g the R - a l g e - b r a s t r u c t u r e s . In o t h e r words, f o r any open s e t s U i n M and V i n N with cp(U) czv we have an R - a l g e b r a homomorphism
* U, V!VV> - < V ° >
with the u s u a l c o m p a t i b i l i t i e s w i t h r e s p e c t t o the r e s t r i c t i o n maps.
Example. L e t M be a s e m i a l g e b r a i c space o v e r R as d e f i n e d i n [DI^,
§ 7 ] . Choose f o r 7(M) t h e s e t ?(M) o f a l l open s e m i a l g e b r a i c s u b s e t s o f M, and f o r U G f(M) d e f i n e t h e s e t C o vM( U ) t o be t h e s e t o f a l l f a m i l i e s ( L L l i G I ) i n f(M) such t h a t t h e u n i o n U ( u \ | i € I ) = U and s u c h t h a t f i n i t e l y many U ^ i € 1 , a l r e a d y c o v e r U.
The axioms ( i ) - ( v i i i ) a r e c l e a r l y f u l f i l l e d . Thus M i s a g e n e r a - l i z e d t o p o l o g i c a l s p a c e . (N.B. In t h i s way e v e r y " r e s t r i c t e d t o p o - l o g i c a l space", as d e f i n e d i n [DK>, § 7 ] , can be r e g a r d e d as a gene- r a l i z e d t o p o l o g i c a l s p a c e ) . The sheaves on t h i s g e n e r a l i z e d t o p o l o - g i c a l space M a r e t h e same as t h e sheaves on t h e " r e s t r i c t e d t o p o l o - g i c a l space" M c o n s i d e r e d i n [DI<2,§7]. In p a r t i c u l a r on M t h e r e i s a s h e a f C?M o f R - a l g e b r a s d e f i n e d as f o l l o w s : I f U € flM) , t h e n #M(U) i s the R - a l g e b r a o f s e m i a l g e b r a i c f u n c t i o n s f:U R. F o r open s e m i a l - g e b r a i c s e t s V c U t h e r e s t r i c t i o n map r^r from (5> (U) t o (V) i s the
r V M M
o b v i o u s r e s t r i c t i o n o f f u n c t i o n s f »-> f | v . T h i s r i n q e d space (M,#M) o v e r R i s r e a l l y t h e same o b j e c t as t h e s e m i a l g e b r a i c space M and w i l l be i d e n t i f i e d w i t h i t .
I f (M,<^) i s a r i n g e d space o v e r R t h e n f o r any U £ 7(M) we o b t a i n
"by r e s t r i c t i o n " a r i n q e d space (U,^|U) o v e r R as f o l l o w s : cr(U) c o n s i s t s o f a l l V G tf(M) w i t h V c U . Cov^ c o n s i s t s o f a l l f a m i l i e s
(V laGI) £ Cov., w i t h V c ( ] f o r e v e r y aGI, and C?|TJ i s t h e r e s t r i c t i o n
a M a J M
o f t h e sheaf C? t o U, i . e . (C? |U)(V) = ® ^ (V) f o r e v e r y V G 7(U) .
M M M
These r i n g e d s p a c e s (U,0MIU) a r e c a l l e d t h e open s u b s p a c e s o f (M,0 ).
An open s u b s e t U o f M i s c a l l e d an open s e m i a l g e b r a i c s u b s e t i f (U,0MIU!
i s a s e m i a l g e b r a i c space o v e r R, as d e f i n e d i n t h e example above.
D e f i n i t i o n 3. A l o c a l l y s e m i a l g e b r a i c space o v e r R i s a r i n g e d space (M,C>M) o v e r R w h i c h p o s s e s s e s an a d m i s s i b l e c o v e r i n g (M^laGI) ECov^'M) such t h a t a l l M a r e open s e m i a l g e b r a i c s u b s e t s o f M.
L e t us l o o k a t t h e s e d e f i n i t i o n s more c l o s e l y . Assume t h a t (M,C?^) i s a l O ' C a l l y s e m i a l g e b r a i c space and t h a t (M^lcxEI) i s an a d m i s s i b l e c o - v e r i n g o f M by open s e m i a l a e b r a i c s u b s e t s M^. What a r e the o t h e r open s e m i a l g e b r a i c s u b s e t s o f M? C l e a r l y; i f U € f(M) i s s e m i a l g e b r a i c , t h e n
( U H M ^ I a E I ) i s an a d m i s s i b l e c o v e r i n g o f the s e m i a l g e b r a i c space U, and t h u s U i s c o n t a i n e d i n the u n i o n o f f i n i t e l y many s e t s M^. Con- v e r s e l y i f U € 7(H) and UcMot^ U ... U M0 r f o r f i n i t e l y many i n d i c e s
, ..., ar € I t h e n t h e s e t Ma^ U ... U Ma r = W i s open s e m i a l g e b r a i c i n M by t h e v e r y d e f i n i t i o n o f s e m i a l c r e b r a i c s p a c e s . Moreover U £ T(W) . T h u s (U,C?M|U) i s a l s o a s e m i a l g e b r a i c s p a c e . The open s e m i a l g e b r a i c s u b s e t s o f M a r e p r e c i s e l y t h o s e s e t s U € f(M) which a r e c o n t a i n e d i n t h e u n i o n o f f i n i t e l y many s e t s M . We w i l l h e n c e f o r t h d e n o t e t h e s u b s e t o f ^?(M) c o n s i s t i n g o f a l l open s e m i a l g e b r a i c s u b s e t s o f M by
T(M) . N o t i c e t h a t ?-(M) = f(M) i f and o n l y i f M i t s e l f i s s e m i a l g e b r a i c .
We c l e a r l y have t h e f o l l o w i n g r e l a t i o n s between the s e t s r(M) and CT(M) .
a) A s u b s e t W o f M b e l o n g s t o IT(M) i f and o n l y i f f o r e v e r y
( U}I A € A ) € C o vM( U ) , U € f(M) , a l l the i n t e r s e c t i o n s W D a r e e l e m e n t s o f ?(M) and W flU i s c o v e r e d by f i n i t e l y many s e t s W D .
b) A s u b s e t U o f M b e l o n g s t o f(M) i f and o n l y i f U PI W € y-(M) f o r e v e r y W€ f(M) . A l s o U € f(M) i f and o n l y i f U O M ^ fi(M) f o r e v e r y a € I .
I t i s a l s o an e a s y consequence o f our d e f i n i t i o n s , i n p a r t i c u l a r o f the axioms v i ) , v i i ) , v i i i ) i n D e f i n i t i o n 1 , t h a t T(M) d e t e r m i n e s the s e t Cov^ i n t h e f o l l o w i n g way:
c) A f a m i l y ( U ^ I A € A ) i n T'(M) b e l o n g s t o Cov^ i f and o n l y i f f o r e v e r y W € T(M) the i n t e r s e c t i o n W D U o f Wwith the u n i o n U o f the f a m i l y i s c o v e r e d by f i n i t e l y many s e t s W D U^,AGA. In f a c t , i t s u f f i c e s t h a t f o r e v e r y a € l t h e i n t e r s e c t i o n n U i s c o v e r e d by f i n i t e l y many s e t s M n u > , A € A .
D e f i n i t i o n 4. A f a m i l y ( X^ | A € A ) o f s u b s e t s o f M i s c a l l e d l o c a l l y f i n i t e i f any W€ s{M) meets o n l y f i n i t e l y many X^, i n o t h e r w o r d s „ i f W n X^ * 0 f o r o n l y f i n i t e l y many A € A . A g a i n i t i s o n l y n e c e s s a r y t o c h e c k t h a t , f o r e v e r y a € I , t h e s e t Ma meets o n l y f i n i t e l y many .
As a s p e c i a l c a s e o f o u r o b s e r v a t i o n ( c ) , we have
P r o p o s i t i o n 1 . 1 . E v e r y l o c a l l y f i n i t e f a m i l y i n T(M) i s an e l e m e n t o f Cov^. I n p a r t i c u l a r , t h e u n i o n o f t h i s f a m i l y i s an e l e m e n t o f T(M) .
F o r e v e r y x € M, t h e s t a l k ^ i s a l o c a l r i n g and t h e n a t u r a l map from R t o t h e r e s i d u e c l a s s f i e l d 0 /+* o f (9„ i s an i s o m o r p h i s m .
M,x M,x M,x
Indeed, t h i s i s known t o be t r u e f o r a l l t h e s e m i a l g e b r a i c s p a c e s ( Ma,<0Ml Ma) and t h u s a l s o h o l d s f o r ( M , ©M) . We i d e n t i f y ©M X/ * *M x w i t h t h e f i e l d R. I f U 6 f(M) and f GO (U) t h e n f y i e l d s an R - v a l u e d f u n c t i o n f : U -> R, which maps e v e r y x C U t o t h e n a t u r a l image o f f i n
©M xAl'M x« T n e e l e m e n t f € ©M( U ) i s u n i q u e l y d e t e r m i n e d by t h i s f u n c - t i o n f , s i n c e t h e c o r r e s p o n d i n g f a c t i s known t o be t r u e f o r a l l t h e r e s t r i c t i o n s r^J p M ( f ) € ( 9M( u n Ma) o f f . We i d e n t i f y f w i t h f . Thus
a
we r e g a r d a s a s u b s h e a f o f t h e s h e a f o f a l l R - v a l u e d f u n c t i o n s on M. I n p a r t i c u l a r t h e r e s t r i c t i o n maps r ^ : ®M( U ) -><9M(V) a r e now t h e n a i v e r e s t r i c t i o n maps f »->f|v f o r f u n c t i o n s . From now on we w i l l c a l l t h e e l e m e n t s o f ^M( U ) t h e l o c a l l y 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 t o M). N o t i c e t h a t , i n t h e s p e c i a l c a s e where U i s a s e m i a l g e b r a i c open s u b s e t o f M, t h e s e f u n c t i o n s a r e j u s t t h e s e m i - a l g e b r a i c f u n c t i o n s on t h e s e m i a l g e b r a i c s p a c e ( U , ©MI U ) . Thus i n t h i s c a s e t h e f CO^(\J) w i l l a l s o be d e s i g n a t e d as t h e " s e m i a l g e b r a i c f u n c - t i o n s o n U" .
Now l e t (N,0N) be a s e c o n d l o c a l l y s e m i a l g e b r a i c s p a c e o v e r R.
D e f i n i t i o n 5. A l o c a l l y s e m i a l g e b r a i c map from (M,^) t o (N,£>N) i s a m o r p h i s m ( f : (M,e?M) (N,#N) i n the c a t e g o r y o f r i n g e d s p a c e s o v e r R ( c f . D e f . 2 a b o v e ) .
The f o l l o w i n g theorem i s known f o r s e m i a l g e b r a i c s p a c e s [DK2,Th.7.2]
and e x t e n d s i m m e d i a t e l y t o l o c a l l y s e m i a l g e b r a i c s p a c e s .
Theorem 1.2. L e t ( f , J) : (M, 6?M) -> (N,O^) be a l o c a l l y s e m i a l g e b r a i c map. F o r any open s e t s U and V o f M and N w i t h f (U) c V and any h e e ?N( V ) we have
v( h ) (x) = h ( f (x) ) f o r a l l x € U.
Thus (f,?) i s d e t e r m i n e d by i t s f i r s t component f and w i l l hence- f o r t h be i d e n t i f i e d w i t h t h e map f from t h e s e t M t o t h e s e t N. C l e a r l y a map f:M -* N i s l o c a l l y s e m i a l g e b r a i c i f and o n l y i f f i s c o n t i n u o u s ( c f . Def. 2 a b o v e ) , and i f f o r e v e r y U € 'f(N) and e v e r y h € #N( U ) t h e
_ i
f u n c t i o n h * f i s l o c a l l y s e m i a l g e b r a i c on f (U). N o t i c e t h a t , i n c a s e M and N a r e s e m i a l g e b r a i c s p a c e s , t h e l o c a l l y s e m i a l g e b r a i c maps from M t o N a r e j u s t the s e m i a l g e b r a i c maps from M t o N as d e f i n e d i n [DK2,§7], I n g e n e r a l t h e f o l l o w i n g P r o p o s i t i o n 1.3 g i v e s a good h o l d on l o c a l l y s e m i a l g e b r a i c maps i n terms o f s e m i a l g e b r a i c maps.
N o t i c e t h a t f o r e v e r y c o n t i n u o u s map f:M -> N and e v e r y a d m i s s i b l e c o v e r i n g (N^|p€J) o f N by open s e m i a l g e b r a i c s e t s t h e r e c e r t a i n l y e x i s t s a c o v e r i n g (M^jaCI) o f M w i t h t h e p r o p e r t i e s needed i n t h e p r o p o s i t i o n .
P r o p o s i t i o n 1.3. L e t f:M N be a ( s e t t h e o r e t i c a l ) map. L e t ( Mal a € I ) and (NplftCJ) be a d m i s s i b l e c o v e r i n g s o f M and N by open s e m i a l g e b r a i c s u b s e t s . Assume t h a t t h e r e i s a map y: I -» J such t h a t f {M^) cNp (a) f o r e v e r y a € I . Then f i s l o c a l l y s e m i a l g e b r a i c i f and o n l y i f t h e
r e s t r i c t i o n f l M :M ->N , , o f f t o M i s a s e m i a l g e b r a i c mao f o r a a y ( a ) a ^ e v e r y a E I .
P r o o f . The " o n l y i f " d i r e c t i o n i s o b v i o u s . So assume t h a t f l M : M_ -»N , * i s s e m i a l g e b r a i c f o r a l l ot€I. L e t U E t?(N) . Then f o r e v e r y aEI
f -1( U ) n Ma = f "1( Ny(a) n u ) n Ma = ( f l M , , ) -1^ ^ n u )
i s an open s e m i a l g e b r a i c s u b s e t o f M . Hence, by axiom ( v i i i ) i n De- f i n i t i o n 1, f ~1( U ) E 7(M). L e t now (U^IAEA) be an a d m i s s i b l e c o v e r i n g o f U E 'f(N) . Then f o r e v e r y a£I
( f ~1 (Ux) fl MaIXEA) = ( (f l Ma) "1 ( Ux fl Ny(a) ) I AEA)
i s an a d m i s s i b l e c o v e r i n g o f f ~1 (U) 0 Ma = (f i MQ) "1 (U fl (q }), i . e . i t p o s s e s s e s a f i n i t e r e f i n e m e n t . We c o n c l u d e t h a t (f 1( U ^ ) I A E A ) i s an
_ i
a d m i s s i b l e c o v e r i n g o f f ( U ) . Thus f i s c o n t i n u o u s . F o r a g i v e n f u n c t i o n h E <^(U) a l l t h e r e s t r i c t i o n s h l U D N ,&€J, a r e s e m i a l g e b r a i c f u n c t i o n s . S i n c e t h e maps f l M a : M a -* Ny (a)a r e s e m i a l g e b r a i c we see t h a t a l l t h e f u n c t i o n s h * f | f ~1( U ) flM ,a€I, a r e s e m i a l g e b r a i c . Thus h - f E e? (f""1u) .
q.e.d.
C o r o l l a r y 1.4. L e t (M,e? ) be a l o c a l l y s e m i a l g e b r a i c space over R and U an open s u b s e t o f M. Then t h e l o c a l l y s e m i a l g e b r a i c maps from U t o t h e s e m i a l g e b r a i c s t a n d a r d space (R,(?_) a r e j u s t t h e f u n c t i o n s f € ©M( U ) .
T h i s i s e v i d e n t from P r o p o s i t i o n 1.3 and t h e c o r r e s p o n d i n g f a c t f o r s e m i a l g e b r a i c s p a c e s . S i m i l a r l y , t h e l o c a l l y s e m i a l g e b r a i c maps from
(M,C?M) t o t h e s e m i a l g e b r a i c space ( Rn, 0R n) (n > 1) a r e t h e n - t u p l e s (f<l' •••/ fn) o f l o c a l l y s e m i a l g e b r a i c f u n c t i o n s f . , f on M.
C o r o l l a r y 1.5, L e t (M,£>M) be a s e m i a l g e b r a i c space and (N,ct?N) a l o - c a l l y s e m i a l g e b r a i c space o v e r R. L e t (N^I3GJ) be an a d m i s s i b l e c o - v e r i n g o f N by open s e m i a l g e b r a i c s u b s e t s . Then a map f:M -> N i s l o - c a l l y s e m i a l g e b r a i c i f and o n l y i f t h e r e e x i s t s a f i n i t e s u b s e t J ' o f J such t h a t f(M) i s c o n t a i n e d i n t h e u n i o n N1 o f t h e s e t s w i t h
3EJ' and the map f from M t o t h e open s e m i a l g e b r a i c subspace N' o f N i s s e m i a l g e b r a i c .
A g a i n t h i s i s e v i d e n t from P r o p o s i t i o n 1.3. We w i l l o f t e n c a l l t h e l o c a l l y s e m i a l g e b r a i c maps from a s e m i a l g e b r a i c space (M,C^) t o a l o c a l l y s e m i a l g e b r a i c space (N,C>N) t h e " s e m i a l g e b r a i c maps from
(M,C?M) t o (N,fl>N)"
We a r e ready f o r a d i s c u s s i o n o f axiom ( v i i i ) i n t h e d e f i n i t i o n o f a g e n e r a l i z e d t o p o l o g i c a l space (Def. 1 ) . We have seen i n t h e p r o o f o f P r o p o s i t i o n 1.3 t h a t t h i s axiom i s i m p o r t a n t t o o b t a i n a s l i c k d e s c r i p t i o n o f l o c a l l y s e m i a l g e b r a i c maps i n terms o f s e m i a l g e b r a i c maps and spaces. On t h e o t h e r hand one may v e r i f y t h e f o l l o w i n q ob- s e r v a t i o n c o n c e r n i n g o u r d e f i n i t i o n . Assume we had d e f i n e d l o c a l l y s e m i a l g e b r a i c s p a c e s u s i n g axioms ( i ) - ( v i i ) o m i t t i n g ( v i i i ) , and t h a t (M, # ( M ) , C o vM, ® ) were a l o c a l l y s e m i a l g e b r a i c space i n t h i s new s e n s e . Then we c o u l d o b t a i n a l o c a l l y s e m i a l g e b r a i c space
(M, f* (M),Cov^,C^) i n t h e o l d sense as f o l l o w s . L e t f(M) be t h e s e t of a l l U £ f(M) such t h a t e v e r y (U lcc€I) € C o vM( U ) has a f i n i t e r e f i n e - ment. D e f i n e 'f1 (M) as t h e s e t o f a l l U c M w i t h U H W € f(M) f o r e v e r y W 6 j-(M) , and d e f i n e Cov^ as t h e s e t o f a l l f a m i l i e s (Ual<x€I) i n f (M) such t h a t the u n i o n U o f t h e Ua i s an e l e m e n t o f *f' (M), and such t h a t f o r e v e r y W€ r(M) t h e s e t U flW c a n be c o v e r e d by f i n i t e l y many U . Then (M, 5* (M),Cov^) f u l f i l l s a l l t h e axioms ( i ) - ( v i i i ) . Moreover, e v e r y sheaf ^ on t h e s i t e (M, f (M),CovM) e x t e n d s u n i q u e l y t o a s h e a f 7' on t h e new s i t e (M, f ' (M) ,Cov^) . C l e a r l y (M, tf1 (M) , C o v ^ , ^ ) | w
= (M, Cf(M) ,Cov^, O ) |w f o r e v e r y W€ ^(M) , and t h e s e spaces a r e s e m i a l - g e b r a i c . In p a r t i c u l a r (M,'J1 (M) , C o v ^ , ^ ) i s l o c a l l y s e m i a l g e b r a i c i n t h e o l d s e n s e .
Thus, d e s p i t e i t s i m p o r t a n c e , t h e axiom ( v i i i ) s h o u l d be r e g a r d e d as an axiom which does not r e s t r i c t the g e n e r a l i t y o f our c o n c e p t o f l o - c a l l y s e m i a l g e b r a i c s p a c e s .
