UBC Theses and Dissertations

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UBC Theses and Dissertations

A category of topological spaces and sheaves Fraga, Robert Joseph 1963

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A CATEGORY OF TOPOLOGICAL SPACES AND SHEAVES by Robert Joseph Fraga B.A., Pomona C o l l e g e , 1961 A t h e s i s s u b m i t t e d i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r the degree o f MASTER OF ARTS i n the Department of Mathematics We accept t h i s t h e s i s as conforming t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA August, 1963 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r -m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . , I t i s understood t h a t copying, or p u b l i -c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department of M a t h e m a t i c s  The U n i v e r s i t y of B r i t i s h Columbia,. Vancouver 8, Canada. Date September 5. 1963 ABSTRACT The problem of t h i s paper i s to def i n e a category of t o p o l o g i c a l spaces and sheaves i n t o p o l o g i c a l and a l g e b r a i c terms. I t i s then necessary to show that i f the t o p o l o g i c a l space i s , i n p a r t i c u l a r , a f f i n e space and the sheaf over i t the sheaf of germs of r e g u l a r f u n c t i o n s * then the p a i r , c o n s i s t i n g of the t o p o l o g i c a l space and the sheaf of germs of r e g u l a r f u n c t i o n s , c a l l e d an a f f i n e v a r i e t y , i s an element of the category. . We then g e n e r a l i z e t h i s r e s u l t by showing that a l g e b r a i c v a r i e t i e s belong to the category. Before d e f i n i n g the category, i t i s necessary to e s t a b l i s h i n Section 1 some elementary r e s u l t s of general topology which are used i n proving p r o p e r t i e s of the category. I t i s al s o necessary to def i n e the Z a r i s k i topology i n purely t o p o l o g i c a l terms. This i s done i n Section 2. ACKNOWLEDGEMENT I w o u l d l i k e t o e x p r e s s my t h a n k s t o Dr. M a r i o B o r e l l i who s u g g e s t e d t h e t o p i c o f t h i s t h e s i s and g u i d e d my work t h r o u g h o u t i t . I w o u l d a l s o l i k e t o t h a n k D r . Roy W e s t w i c k f o r h i s c o n s c i e n -t i o u s r e a d i n g o f t h e m a n u s c r i p t . My t h a n k s a r e due t o t h e N a t i o n a l R e s e a r c h C o u n c i l o f Canada and t h e D e p a r t m e n t o f M a t h e m a t i c s a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a f o r t h e f i n a n c i a l s u p p o r t t h a t t h e y gave me d u r i n g t h e t i m e t h a t I was w o r k i n g on t h i s t h e s i s . INTRODUCTION The p u r p o s e o f t h i s t h e s i s w i l l be t o c o n s t r u c t a x i o m a t i c a l l y a 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 s h e a v e s (X , O^y) w h i c h w i l l be shown t o c o n t a i n a l l p r e - a l g e b r a i c v a r i e t i e s . The t h e s i s i s d i v i d e d i n t o , f o u r s e c t i o n s as f o l l o w s : S e c t i o n 1 c o n c e r n s i t s e l f w i t h an a r b i t r a r y t o p o l o g i c a l s p a c e and t h e c o a r s e s t t o p o l o g y t h a t any s u c h s p a c e can have f o r c e r t a i n t o p o l o g i c a l p r o p e r t i e s t o h o l d . S e c t i o n 2 i s an a p p l i c a t i o n o f .the i d e a s d e v e l o p e d i n S e c t i o n 1 t o t h e a l g e b r a i c c a s e . The aim o f t h i s s e c t i o n w i l l be t o show t h a t t h e . u s u a l Z a r i s k i t o p o l o g y may be d e f i n e d i n p u r e l y t o p o l o g i c a l t e r m s . We t a k e o u r t o p o l o g i c a l s p a c e f i r s t t o be an a l g e b r a i c a l l y c l o s e d f i e l d , k. Then we g e n e r a l i z e o u r r e s u l t s t o a f f i n e s p a c e , k x . . .xk, n t i m e s . The s e c t i o n c o n c l u d e s w i t h a d i s c u s s i o n o f n e c e s s a r y and s u f f i -c i e n t c o n d i t i o n s t h a t a s e t o f f u n c t i o n s f r o m a f f i n e s p a c e i n t o k be q u o t i e n t s o f p o l y n o m i a l s . I n S e c t i o n 3, we d e f i n e a 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 s h e a v e s and deduce t h e e s s e n t i a l p r o p e r t i e s , o f t h e c a t e g o r y . We c o n c l u d e i n S e c t i o n 4 by g i v i n g e x a m p l e s o f s p e c i f i c t o p o l o g i c a l s p a c e s and s h e a v e s w h i c h b e l o n g t o t h e c a t e g o r y . The n o t a t i o n i n t h e l a s t two s e c t i o n s i s e s s e n t i a l l y t h a t o f J e a n - P i e r r e S e r r e i n h i s p a p e r , F a i s c e a u x A l q e b r i q u e s C o h e r e n t s , 2 [ l j . We assume at the s t a r t a rudimentary knowledge of general topology, a b s t r a c t algebra, and sheaf theory. Throughout the t h e s i s , the f o l l o w i n g n o t a t i o n w i l l be c o n s i s t e n t : ^ ( X ) and denote r e s p e c t i v e l y the set of a l l open sets and the set of a l l c l o s e d sets of the t o p o l o g i c a l space X. k w i l l denote an a l g e b r a i c a l l y c l o s e d f i e l d . A t o p o l o g i c a l space X with a sheaf over i t , w i l l be denoted by (X, • The .stalk i n the sheaf over the p o i n t x w i l l be denoted by & x . SECTION 1 We begin with a few d e f i n i t i o n s . Defn. 1.1 By property, we s h a l l mean a f a m i l y of t o p o l o g i e s on a space X. Defn. 1.2 A topology on a space 0/ (X) w i l l be s a i d to be the A-coarsest f o r a property S i f ^ f(X) S and whenever ^'(X) i s another topology s . t . ^f'(X)<p ^ " ( X ) , then Defn. 1.3 A topology on a space (X) w i l l be s a i d to be B-coarsest f o r a property S i f ^ ( * ) and whenever £^'(X) i s another topology i n S, then ^ ( X ) c ^ ' ( X ) . Remark Note that i f 3 a topology on X, 0^ (X);, which i s B-coarsest f o r 5, then i t i s unique. For suppose 3 a topology ^ ' ( x ) * $ ^ which i s B-coarsest f o r S. Then, since ^ ( x ) i s a topology on X i n 5, ^ ' ( X ) « | ^ ( X ) . T h i s , however, c o n t r a d i c t s the hypothesis that ^ ( * ) 1 S the B-coarsest topology f o r S. D e f n . 1.4 We s h a l l s a y t h a t a p r o p e r t y S i s p r e s e r v e d u n d e r i n t e r s e c t i o n s i f i t s a t i s f i e s t h e f o l l o w i n g c o n d i t i o n : I f and Oj- a r e two t o p o l o g i e s on t h e same s p a c e X i n 5, t h e n 0^(\<3j-L'?. We s h a l l now show t h a t f o r p r o p e r t i e s p r e s e r v e d u n d e r i n t e r s e c -t i o n t h e d e f i n i t i o n s f o r A - c o a r s e s t and B - c o a r s e s t t o p o l o g i e s a r e e q u i v a l e n t . Theorem 1.5 L e t S be a p r o p e r t y p r e s e r v e d u n d e r i n t e r s e c t i o n s . Then a t o p o l o g y on a s p a c e X i s A - c o a r s e s t f o r S i f , and o n l y i f , i t i s B - c o a r s e s t . P r o o f : Assume t h a t i s B - c o a r s e s t f o r S b u t n o t A - c o a r s e s t . Then 3 (^ • , a t o p o l o g y on X, i n 5 s . t . m-^Of. The e x i s t e n c e o f s u c h a t o p o l o g y c o n t r a d i c t s t h e a s s u m p t i o n t h a t i s B - c o a r s e s t . Hence ^ must be A - c o a r s e s t . Assume now t h a t i s t h e A - c o a r s e s t t o p o l o g y f o r 5 and l e t (jj ' be a t o p o l o g y on X a l s o i n S. C o n s i d e r t h e i n t e r s e c t i o n t o p o l o g y f\ ^* . S i n c e S i s a p r o p e r t y p r e s e r v e d u n d e r i n t e r s e c t i o n s , ^ f) S 5. Now ^ f l ^ ' - ^ 1 and i s t h e A - c o a r s e s t t o p o l o g y f o r S. Hence and t h i s i m p l i e s t h a t OJC 1 . T h e r e f o r e i s t h e B-5 c o a r s e s t topology f o r 5. We s h a l l now show that the two p r o p e r t i e s that w i l l be of i n t e r e s t to us l a t e r are preserved under i n t e r s e c t i o n . Theorem 1.6 Let X be a set and Y a t o p o l o g i c a l space. Consider an a r b i t r a r y set of f u n c t i o n s y = { f | f : X — > - Y } . I f 5 denotes the set of t o p o l o g i e s on X f o r which a l l f u n c t i o n s fe'Y are continuous, then 5 i s a property preserved under i n t e r s e c t i o n s . Proof: Let and be two t o p o l o g i e s on X f o r which a l l fe"V are continuous. Consider f ~ x ( G ) = M where f and G f ^ ( Y ) . Then since a l l f t l r are continuous fo r and M £ ^ * s i n c e a l l ftlr are continuous f o r • Hence M t Theorem 1.7 Let X be a set and ¥ = {f ( f: X—± x \ an a r b i t r a r y set of functions.. I f S denotes the set of t o p o l o g i e s fo r which a l l f u n c t i o n s f * l r are continuous, then 5 i s a property preserv/ed under i n t e r s e c t i o n s . 6 P r o o f : L e t ^  and 0[7 ' be two t o p o l o g i e s on X f o r w h i c h a l l f u n c t i o n s f t 3r a r e c o n t i n u o u s . L e t f "*"(G) = M where ff, 3r and G« . Then f ~ ^ " ( G ) f ^ s i n c e a l l f u n c t i o n s f t ? a r e c o n t i n u o u s f o r $ . S i m i l a r l y f ~ ^ ( G ) g ' . Hence V\tOJ (\(7j-. T h e r e f o r e 0^f\(^- i s a t o p o l o g y f o r w h i c h a l l f u n c t i o n s fC 3r a r e c o n t i n u o u s , and S i s p r e s e r v e d u n d e r i n t e r s e c t i o n s . I n v i e w o f t h e p r e v i o u s r e s u l t s , we can r e p l a c e t h e words ' A - c o a r s e s t ' and ' B - c o a r s e s t ' i n what f o l l o w s by t h e s i n g l e w o r d , ' c o a r s e s t ' , s i n c e t h e p r o p e r t i e s t h a t w i l l be o f i n t e r e s t t o us a r e t h o s e d e f i n e d i n Theorems 1.6 and 1.7; however, we s h a l l r e t a i n t h e n o t a t i o n A- and B - c o a r s e s t i n o r d e r t o e l u c i d a t e t h e n a t u r e o f t h e p r o o f s . Theorem 1.8 L e t X be a s e t and Y a t o p o l o g i c a l s p a c e . L e t y= {f | f : X — * Y $ be an a r b i t r a r y s e t o f f u n c t i o n s . Then 3 a t o p o l o g y on X w h i c h i s B - c o a r s e s t f o r w h i c h a l l f u n c t i o n s f £ ? a r e c o n t i n u o u s . P r o o f : We d e f i n e a c l a s s o f s e t s & i n X as f o l l o w s : 5 £ % 4 S = f _ 1 ( G ) where f t ¥ and G f Of {Y) . 7 L e t be t h e t o p o l o g y g e n e r a t e d on X by ^ . By c o n -s t r u c t i o n , i s a t o p o l o g y f o r w h i c h a l l f u n c t i o n s ffc3r a r e c o n t i n u o u s . F u r t h e r m o r e , assume t h a t 1 i s a n o t h e r t o p o l o g y f o r w h i c h a l l f u n c t i o n s f f i Y a r e c o n -t i n u o u s . T h e n ^ c : ^ ' and s i n c e i s a t o p o l o g y • Hence 0 ^ i s t h e B - c o a r s e s t t o p o l o g y f o r w h i c h a l l f u n c t i o n s f fc 3r a r e c o n t i n u o u s . SECTION 2 The purpose of t h i s s e c t i o n w i l l be to show that the usual Z a r i s k i topology may be defined i n purely t o p o l o g i c a l terms. Theorem 2.1 Let k be an a l g e b r a i c a l l y c l o s e d f i e l d . Then every n o n - t r i v i a l topology on k f o r which a l l polynomials i n one v a r i a b l e are continuous i s T^. Proof: Let ^ b e a topology on k, not the t r i v i a l topology, f o r which a l l polynomials i n one v a r i a b l e are continuous. Let K % k be a non-empty c l o s e d set f o r the topology Let k' £ K be f i x e d . Consider the set ^ which c o n s i s t s of those polynomials f s . t . f(p) = k 1 where p i s an a r b i t r a r y p o i n t of k. I t i s c l e a r t h a t ^ i s not empty since f'fclP where f* = k* + m'(X - p). We a s s e r t that O f'^K) . { p}. sty I t i s c l e a r from the d e f i n i t i o n of ¥ that n f- x(K)=>jp} Now assume that 3 a p o i n t ql k s . t . q 4 p and 0 f - ^ K ) 3 f q f . Since K ^k,3 a p o i n t k # t k but k * i K . Consider the polynomial, f*(X) = (k + - k'^X + k'q - k #p q - p /k* \ ' i \ q - P / 9 C l e a r l y f * £ Tr , but since i s 1-1 and since f#~ "^(k#) = q, k^K, i t f o l l o w s that f ( K ) p { q ] . Hence fi f _ 1 ( K ) £ } q ] , and our a s s e r t i o n i s proved. Since a l l polynomials f are continuous f o r , i t f o l l o w s that f (K) i s a closed s e t . Hence (If (K) i s a c l o s e d se t . Therefore ^p| = f\f ^"(K) i s a c l o s e d set, and i s T 1. Defn. 2.2 Again l e t k be an a l g e b r a i c a l l y c l o s e d f i e l d . The topology of f i n i t e complements i s that topology i n which open sets are the complements of f i n i t e subsets of k. We s h a l l denote t h i s topology by 0£p^. Theorem 2.3 Excluding the t r i v i a l topology, ^pc ^ s a n A-coarsest topology f o r which a l l polynomials i n one v a r i a b l e are continuous. Proof: I t i s e a s i l y checked that a l l polynomials i n one v a r i a b l e are continuous f o r ^ p £ * ^ e s h a l l now show that $^p£ i s an A-coarsest topology f o r which t h i s i s the case. Assume ,to the contrary that 3 a topology ^ ^prj ^ o r which a l l polynomials f £ k [ x ] are continuous. I f a l l 10 s i n g l e t s of the form {x\ are cl o s e d f o r then ^ = ^p-Q« Hence 3 at l e a s t one s i n g l e t which i s not c l o s e d f o r &J. Theorem 2.1, however, gives us that every topology on k, not the t r i v i a l topology, f o r which a l l polynomials i n one v a r i a b l e are continuous i s T^. Hence OJ cannot be a topology f o r which a l l polynomials f£ k £ x j are c o n t i n u -ous. I t f o l l o w s that i s a n A-coarsest topology f o r which t h i s i s the case. Theorem 2.4 jVZ 1 S *'"ie ^ - c o a r s e s t n o n - t r i v i a l topology f o r which a l l polynomials i n one v a r i a b l e are continuous and hence i s unique. Proof: 5uppose ^ i s a topology, not the t r i v i a l topology, f a r which a l l polynomials i n one v a r i a b l e are continuous. Then by Theorem 2.1 we have that {p\ i s a c l o s e d set f o r 0j where p i s an a r b i t r a r y p o i n t of k. Hence fl^pj, i s not the t r i v i a l topology since f p\ i s a c l o s e d set f o r both t o p o l o g i e s and the complement of f p~\ i s an open set f o r both t o p o l o g i e s . We see, then, i n view of Theorem 1.7, that the set of t o p o l o g i e s on k, excluding the t r i v i a l topology, f o r which a l l polynomials i n one 11 v a r i a b l e are c o n t i n u o u s i s a p r o p e r t y p r e s e r v e d under i n t e r s e c t i o n s . That i s the B - c o a r s e s t t o p o l o g y f o l l o w s i m m e d i a t e l y from Theorems 2.3 and 1.5. That i t i s unique f o l l o w s from the remark a f t e r Defn. 1.3. We c o n s i d e r now a f f i n e space k° = kx. . .xk, n t i m e s . L e t f: k n- i-^k be a p o l y n o m i a l i n n v a r i a b l e s . We denote such a p o l y n o m i a l by ffX^» . . '»X n] o r more s i m p l y by f [ x ] . The t o p o l o g y o f the range space w i l l be ^pr;' Defn. 2.5 The i - t h p r o i e c t i o n map : k n f-k, i = l , . . . , n , o p e r a t e s as f o l l o w s : P^(x) = x^ where x = ( x , , . .,x., . ,,x ). I f i s n s 1, P. i s the 1 i n x i d e n t i t y map. C o n s i d e r a f i n i t e s e t i n k, [k^, . . . , k m \ . The n - t u p l e s i n k n t h a t are mapped by a p o l y n o m i a l f£x] onto any one o f these m v a l u e s are a l l the z e r o s o f the f o l l o w i n g p o l y n o m i a l s : i f X ] " k i f W " k m • Defn. 2.6 C o n s i d e r any f i n i t e s e t [k^, . . . , k m ^ o f k and the p o l y n o m i a l s : f [ x ] - k r . . ., f [ x j - k m . The union o f a l l the z e r o s o f such a s e t o f p o l y n o m i a l s w i l l be a c l o s e d s e t i n k n. L e t f k , , . . ., k \ L 1 m J 12 range aver a l l f i n i t e subsets of k and l e t ffxj range over a l l polynomials i n n v a r i a b l e s . The set of cl o s e d sets that they generate as a sub-base w i l l be denoted by ^ . We s h a l l denote the correspond-i n g set of open sets i n t h i s topology by . It i s c l e a r from i t s c o n s t r u c t i o n that, excluding the t r i v i a l t o -pology, i s the B-coarsest topology on k° f o r which a l l polyno-mials i n n v a r i a b l e s are continuous. We can show that i s an A-coarsest topology f o r which t h i s i s the.case, again excluding the t r i v i a l topology, i n the same way as i n Theorems 2.1 and 2.3. Defn. 2.7 The Z a r i s k i topology on k n i s that topology i n which cl o s e d sets are the common zeros of an i d e a l of poly-nomials i n n v a r i a b l e s . We denote such a topology b y ^ . Theorem 2.8 Let k n be a f f i n e space of dimension n. Let be the Z a r i s k i topology and l e t °e the topology defined i n Defn. 2.6. Then J^k") = f^^^' Proof: The proof w i l l be by complementation. Let 3^ and 3^ denote the sets of cl o s e d subsets of k n which are the complements of the open subsets of ^  and r e s p e c t i v e l y , Let V l^r^ . Then F c o n s i s t s of the common zeros of f ^ ( X ) , 13 i e l , a l l f \ ( X ) b e l o n g i n g t o an i d e a l o f p o l y n o m i a l s i n n v a r i a b l e s . The z e r o s o f f . ( X ) form a c l o s e d s e t F . e \o x x s i n c e t h e y a r e t h e image u n d e r f ^ 1 o f t h e c l o s e d s i n g l e t to). f l F. i s a c l o s e d s e t i n s i n c e a r b i t r a r y i n t e r s e c t i o n s i£l 1 U o f c l o s e d s e t s a r e c l o s e d . However F = f l F. . Hence F t V - , i C l 1 ° C and ^ - ^ j j • We now s u p p o s e t h a t where d e n o t e s t h e s u b - b a s e o f d e f i n e d i n D e f n . 2.6. Then 5 may be t a k e n t o be t h e s e t o f a l l z e r o s o f t h e p o l y n o m i a l s f ( X ) - k^, i = l , . . . ,n , f a p o l y n o m i a l i n n v a r i a b l e s . C o n s i d e r h t h e i d e a l g e n e r a t e d by t h e p o l y n o m i a l ~| f ( f ( X ) - ^ . i = ( I t s s e t o f common z e r o s i s p r e c i s e l y t h e s e t 5. Hence 5 E 3-j and - | ^ c ^ . ^ C ^ i m p l i e s t h a t £ s i n c e 3^  , as t h e s e t o f a l l c l o s e d s u b s e t s : o f k n c o r r e s p o n d i n g t o 2^ » l s c l o s e d u n d e r f i n i t e u n i o n s and a r b i t r a r y i n t e r -s e c t i o n s . T h e r e f o r e 1^ c 3^  . C o m b i n i n g b o t h p a r t s o f t h i s a r g u m e n t , we see t h a t = ^ and, by c o m p l e m e n t a t i o n , 14 Remark The H i l b e r t Base Theorem g i v e s us t h a t i s o b t a i n e d f r o m t h e s e t u n d e r t h e o p e r a t i o n o f t a k i n g f i n i t e i n t e r s e c t i o n s . We now t r y t o c h a r a c t e r i z e i n f u n c t i o n a l t h e o r e t i c a l manner t h e r i n g k£x]]. We n o t e t h a t we have an a d d i t i o n and a m u l t i p l i c a t i o n d e f i n e d f o r k [ x]as f o l l o w s : ( f + g ) ( x ) = f ( x ) + g ( x ) ( f g ) ( x ) = f ( x ) g ( x ) where f and g b e l o n g t o k[x^ and x c k . Theorem 2.9 L e t IA be a f a m i l y o f f u n c t i o n s f r o m k n i n t o k. A n e c e s s a r y and s u f f i c i e n t c o n d i t i o n t h a t J{ be t h e r i n g o f p o l y n o m i a l s i n n v a r i a b l e s , (P , i s t h a t i t s a t i s f y t h e f o l l o w i n g t h r e e c o n d i t i o n s : 1) J{ i s c l o s e d u n d e r a d d i t i o n and m u l t i p l i c a t i o n , 2) J{ c o n t a i n s a l l c o n s t a n t f u n c t i o n s and p r o j e c t i o n s , 3) A l l f u n c t i o n s i n <R may be g e n e r a t e d by t h e f u n c t i o n s i n 2 ) . P r o o f : N e c e s s i t y i s o b v i o u s . We p r o c e e d t o s u f f i c i e n c y . S u p p o s e d s a t i s f i e s 1 ) , 2 ) , and 3 ) . We s h a l l show f i r s t t h a t Q £</?. 15 x, 1 L e t P ( X ) = ^ k. . X, . . ..X i , . . i 61 11» * " x n 1 n 1 n I f i n i t e k. £k . 1 * • • Now X^e t/l s i n c e <^  c o n t a i n s a l l p r o j e c t i o n s . X^ X = X^. . . X^ s i n c e i s c l o s e d u n d e r m u l t i p l i c a t i o n . 'I i , i Hence X, \ . .X n eA . 1 n k. £t/? since*/? c o n t a i n s a l l c o n s t a n t maps. x l ' ' ' f X n k. X, x . . .X n f ( / s i n c e Jl i s c l o s e d u n d e r xl» • -'^-p, 1 n m u l t i p l i c a t i o n . . X l 1 fl /7 3 > k. . X. . . .X M s i n c e ''I i s c l o s e d . ^ — . < T i , , . . , I 1 n i , . , i t i 1' ' n 1 n I f i n i t e u n d e r a d d i t i o n . T h e r e f o r e P ( X ) f c < ^ and we have p r o v e d t h a t C StA • Now s u p p o s e t h a t F ( X ) £ J\ . We s h a l l show t h a t F ( X ) must be a p o l y n o m i a l i n n v a r i a b l e s , and hence We o b s e r v e f i r s t t h a t s i n c e k i s a f i e l d , X.X. = X.X. 1 J J 1 and k^Xj = ^ j ^ i * Hence, a f t e r t h e a p p r o p r i a t e r e a r r a n g e -ment o f f a c t o r s * we have by 3) t h a t 16 Remark We note t h a t the f o l l o w i n g i s e a s i l y proved: L e t k nQ =£(x) [ (x)£k n and Q(x) £ 0^ be the open s e t f o r the t o p o l o g y ^ on which the p o l y n o m i a l Q f a i l s to v a n i s h . C o n s i d e r the s e t o f a l l r a t i o n a l f u n c t i o n s , P : k"^—*-k. L e t the t o p o l o g y on both k n and k be j Q The t o p o l o g y on k nQ w i l l be t h a t induced by Then a l l r a t i o n a l f u n c t i o n s P : k n,=—^ k are c o n t i n u o u s . Q 1 4 SECTION 3 We a r e now p r e p a r e d t o c o n s t r u c t a 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 s h e a v e s and t o i n v e s t i g a t e some o f t h e p r o p e r t i e s o f t h a t c a t e g o r y . L e t X d e n o t e an a r b i t r a r y t o p o l o g i c a l s p a c e and k an a l g e b r a i c a l l y c l o s e d f i e l d w i t h t h e t o p o l o g y o f f i n i t e c o m p l e m e n t s ^ p £ * ^-e^ £ (X) d e n o t e t h e s e t o f c o n t i n u o u s f u n c t i o n s f r o m X t o k. D e f n . 3.1 G i v e n X, l e t £ B ^ ^ ^£j be a c o l l e c t i o n o f c l o s e d s u b -s e t s o f X s . t . fl. = f . (0) f o r some f.«£(X). Then B.'B. = ( f . f . ) _ 1 ( 0 ) . 1 J 1 J I t i s e a s i l y s e e n t h a t t h i s d e f i n i t i o n i s i n d e p e n d e n t o f t h e f u n c t i o n s u s e d s i n c e B.«B. = B.VJB.. Note a l s o t h a t B. = f . - 1 ( 0 ) i J - i j x i i m p l i e s t h a t B^ = ( c f ^ ) ***(0), f o r a l l c£k, c 4= 0« We n e x t make t h e f o l l o w i n g d e f i n i t i o n . D e f n . 3.2 (B. + B . ) , _ . »» ( f . + f . ) ~ 1 ( 0 ) where B. = f . ~ 1 ( 0 ) and Bj = f j " 1 ( 0 ) . N ote t h a t t h i s d e f i n i t i o n o f a d d i t i o n and s u b t r a c t i o n depends IB on the c h o i c e o f the index s e t of f u n c t i o n s , f. . We extend both d e f i n i t i o n s above to f i n i t e p r o d u c t s , sums, and d i f f e r e n c e s . Defn. 3.3 (P ^ = j B ^ w i l l be c a l l e d a pre-base f o r 3r (X) i f the f o l l o w i n g c o n d i t i o n s are s a t i s f i e d : a) 3 a s e t o f f u n c t i o n s % s . t . f l % i m p l i e s : i ) f*£(X) i i ) c f f c * ^ , f o r every c£k i i i ) f - 1 ( 0 ) = B f o r some B e f ^ . b) I f B t ^ x , 3 f t % s . t . B = f _ 1 ( D ) . c) (ft = ^ ^B ±^ , ° , ± ( f f j ^ i s a base f o r £ (X) . Theorem 3.4 L e t Y c X where X i s a t o p o l o g i c a l space w i t h a pre-base /B.\ and a s e t o f f u n c t i o n s A = f f l , c a s s o c i a t e d w i t h the pre-base. L e t ^ ( Y ) be the t o p o l o g y induced on Y by t h a t o f X. L e t X y = ^ f (Y Then {B^H Y^ w i t h the a s s o c i a t e d s e t o f f u n c t i o n s ](y i s a pre-base f o r Y. P r o o f : I t i s e a s i l y checked t h a t s a t i s f i e s i ) , i i ) , and i i i ) o f c o n d i t i o n a) i n Defn. 3.3. L e t <R> = ^ B ^ ' ° ' i ) a n d ^ Y = ( f e n Y ^ » ° ' ±) • 19 I t i s c l e a r that y C £(y). We a s s e r t that 6 y = ft ft Y where (ft f\ Y = [ F<= Y | F t £ (Y) s . t . F = BOY where B£&\. Let B ' £ (2> HY. Then B 1 = B Q Y where B € «3 . Then 3 a fu n c t i o n f generated from % by the r i n g o perations s . t . B = f " 1 ^ ) . Furthermore B A Y = ( f | Y) ~ 1 (0) * ^ y . Hence Now l e t B * t 6 y . Then 3 a f u n c t i o n f * s . t . B # = ( f * ) - 1 ( 0 ) and furthermore 3 f * , generated from % , s . t . f # | Y = f # . ( f l ) " 1 ( Q ) = B^t ( 3 . Hence B*f) Y = ( f * | Y ) " 1 ( 0 ) = ( f ^ ) _ 1 ( Q ) = B # , and B* 0 YC^HY. Hence A Y. Combining t h i s r e s u l t with the previous one, we have that « 3 HY = & y. We s h a l l now use the f a c t that a set of close d subsets of Y , (3 y , i s a ba s i s f o r the close d sets of Y i f , and only i f , given Fe3r(Y) and a point P ^ -F, 3 B fC * y s . t . F C B and P {. B. Let Ft y(Y) and P ^ F . Then F = F'OY where F ' t ^ U ) . Since £ B ^ i s a pre-base f o r X, 0 i s a base f o r X, and 3 B'fcfo s . t . F' <=B* and P 4B« . Then P^B' f\Y and F = ( F ' n Y ) e ( B ' n Y ) . Therefore HY i s a base f o r Y. I t f o l l o w s that * 3 y =(3HY i s a base f o r Y and hence ^B fl Y^ i s a pre-base f o r Y. 20 Defn. 3.-5 Suppose f and g belong to 5uppose f u r t h e r that g(x) 4= 0 f ° r some x £ X. Then f i s defined l o c a l l y 9 at x and w i l l be c a l l e d a r e g u l a r quotient at x. Following Jean-Pierre Serre, £l}, p. 199, we s h a l l make the f o l l o w i n g d e f i n i t i o n . Defn. 3.6 We s h a l l mean by a sheaf a t r i p l e (X, ) X c o n s i s t i n g of two t o p o l o g i c a l spaces, X and ^ y , and a map ^ : 0"^—>-X which i s an open map and a l o c a l homeomorphism. We s h a l l now define a category <C of t o p o l o g i c a l spaces and sheaves. Defn. 3.7 The category < C w i l l c o n s i s t of t o p o l o g i c a l spaces and sheaves (X, &y) where i s isomorphic by means of the map I to a subsheaf of the sheaf of germs of contin-uous f u n c t i o n s from X i n t o k. See, f o r example, £l}, p. 201, f o r the d e f i n i t i o n of a sheaf of germs of con-tinuous f u n c t i o n s . A p a i r (X, $~y) i s an element of £ i f the f o l l o w i n g c o n d i t i o n s are s a t i s f i e d : 1) i s a sheaf of r i n g s , 2) contains the germs of a l l constant f u n c t i o n s , 21 3) Each s t a l k Y i s c l o s e d under the o p e r a t i o n X , A o f t a k i n g r e g u l a r q u o t i e n t s , 4) 3 a f i n i t e open c o v e r i n g o f X and a s e t o f c l o s e d s e t s fE~\ o f X s . t . each U. tH, c o n s i d e r e d as a t o p o l o g i c a l space w i t h the t o p o l o g y i n -duced by t h a t o f X, has a pre-base fEL,f\U.\ w i t h an a s s o c i a t e d s e t o f f u n c t i o n s A • = if- \ Furthermore ^ x j ^ i 9 e n e r a " t e d under the r i n g o p e r a t i o n s and the o p e r a t i o n o f t a k i n g r e g u l a r q u o t i e n t s by the germs of c o n s t a n t f u n c t i o n s and the f u n c t i o n s f.„ f^.. Suppose i s a t o p o l o g i c a l space and a sheaf over i t and Yc X. L e t ^(Y) be the t o p o l o g y induced on Y by t h a t o f X. We c o n s t r u c t a sheaf over Y as f o l l o w s : We d e f i n e a map <f> : & ^ by ^ ( f ) = f | Y . Note t h a t ji i s onto a l t h o u g h not n e c e s s a r i l y 1-1. Theorem 3.8 ( X . ^ J t C and Y <=• X i m p l i e s t h a t ( Y , ^ y ) c C where i s d e f i n e d as above. 22 P r o o f : I t i s e a s i l y checked t h a t (Y, fry) s a t i s f i e s c o n d i t i o n s 1 ) , 2 ) , and 3) of Defn. 3.7. n S i n c e ( X , f r ^ ) t C , 3 an open c o v e r i n g o f X, *M = IJ U. and a s e t o f c l o s e d s e t s IB .\ _ s . t . each U. has a pre-base / B. 0 U . s „ w i t h an a s s o c i a t e d s e t o f f u n c t i o n s = • Now *U y = tj (IL 0 Y) i s a f i n i t e open c o v e r -i s l i n g o f Y and ^ B ^ O Y } i s a s e t of c l o s e d s e t s o f Y. We s h a l l show t h a t these two s e t s s a t i s f y the requ i r e m e n t s of c o n d i t i o n 4) o f Defn. 3.7. We d e f i n e ( P , . . _ Y , = } ( B . n U . J O Y ] and ( U ± n Y ) 1 i * i ^ ( U . O Y ) = l f l Y w h e r e f t * T h a t P ( u . n y ) i s a pre-base f o r U^ O Y w i t h the a s s o c i a t e d s e t o f f u n c t i o n s ^ (U A Y ) f° i-'- o w s ^ r o r n Theorem 3.4 where t L ( \ Y p l a y s the i r o l e o f Y and IL p l a y s the r o l e of X. That % ^y ^ Y j generates fry i s c l e a r s i n c e X y generates fr^ and the i map ^  : ^"^V*^, i s 0 , r t o ' Theorem 3.9 Suppose t h a t (X, ) i C a n d t h a t IX = U U. i s the X , s | i f i n i t e open c o v e r i n g o f X r e q u i r e d by c o n d i t i o n 4) of Defn. 3.7. Then the top o l o g y on X i s A - c o a r s e s t 23 f o r w h i c h t h e f o l l o w i n g two p r o p e r t i e s h o l d : i ) ( X , Q'y) I s a s u b s h e a f o f t h e s h e a f o f germs o f c o n t i n u o u s f u n c t i o n s , and i i ) A l l IL a r e open. P r o o f : ' L e t \B*\ C £ (X) and B = B' 0 U-. be s u c h t h a t £B } FI * f fl *• l o t ' «C « * l i s a p r e - b a s e f o r IL w i t h t h e a s s o c i a t e d s e t o f f u n c t i o n s = f f . i We a s s e r t f i r s t t h a t (Q (X) i n d u c e s on LL an A - c o a r s e s t t o p o l o g y f o r w h i c h a l l f ' ^ x l ^ i a r e germs o f c o n t i n u o u s f u n c t i o n s . The p r o o f w i l l be by c o m p l e m e n t a t i o n . L e t "3? (U. ) be t h e s e t o f c l o s e d s u b s e t s o f U. and l e t ^ ' ( L L ) be t h e s e t o f c l o s e d s u b s e t s f o r an A - c o a r s e s t t o p o l o g y f o r w h i c h a l l ^ c ^ x | ^ i a r e 9 E R M S °^ c o n t i n u o u s f u n c t i o n s . Then B. „ = f . ~X (0) £ X.' (U . ) where f *X j. IK J X 1* S i m i l a r l y < B., ± B ) ( f f , = (f.^ ± f . > - 1 ( 0 ) £ * « ( U . ) , and B.« B. = ( f • f , F T ) " 1 ( 0 ) t ¥ * Now t h i s i m p l i e s t h a t 6 = ^ { 8 ^ , ° , ± ( f > f ^ 1 ( U ± ) , F u r t h e r m o r e (3 i s a base f o r ^ ( U ± ) . Then J ( I L ) £ (LL) , 2 4 but s i n c e ^ ' ( I L ) i s an A - c o a r s e s t t o p o l o g y f o r which a l l f*£?J U. are germs o f c o n t i n u o u s f u n c t i o n s , we con-c l u d e t h a t Y 1 = "I? , thus p r o v i n g our a s s e r t i o n . I t i s c l e a r t h a t the t o p o l o g y on X i s such t h a t a l l f i are germs o f c o n t i n u o u s f u n c t i o n s . Assume t h a t i t i s not an A - c o a r s e s t t o p o l o g y f o r which t h i s i s the case. Then 3 a t o p o l o g y on X, tfjr'(X), f o r which: a) a l l are germs o f c o n t i n u o u s f u n c t i o n s , b) I L f i ^ ' ( X ) , and c) such t h a t ' ( X ) ^ ^ ( X ) . S ince the i n c l u s i o n i s pro p e r , l e t G t ^ ( X ) but G ^ ^ ' U ) . C o n s i d e r GQU. , i = l , . ,,n. For some i, G f\ LL ^ Oj' (LL) , f o r o t h e r w i s e n (GO U ± ) £ ^ ' ( X ) , i = l , . . ,n and G = \J (G n IL) & <p » (X) c o n t r a r y t o h y p o t h e s i s . Hence f o r some i , ^ , ^ i ) % ^ ^ ± ^ ' We have proved, however, t h a t ^ ( b \ ) 1 S a n A - c o a r s e s t t o p o l o g y f o r which a l l f i ^ ^ j l L are germs o f c o n t i n u o u s f u n c t i o n s . Hence 1 S * D O c o a r s e a t o p o l o g y f o r a l l ft&A U. to be germs o f c o n t i n u o u s f u n c t i o n s , thus c o n t r a d i c t i n g the d e f i n i t i o n of ^ f ' ( X ) . I t f o l l o w s t h a t ^ ( X ) i s an A - c o a r s e s t t o p o l o g y f o r which a l l f I ^ are c o n t i n u o u s . 25 Theorem 3.10 Suppose X = U x . i s a t o p o l o g i c a l space, X . t ^ C X ) , i=l 1 1 0 and suppose f u r t h e r t h a t J 3 sheaves s . t . i ( X . , ^ x )£ C , i = l , . .,n. Then 3 a sheaf ^ x i on X s . t . (X, ^ Y ) « < C a n d (X. , # Y ) g (X. , ^ J x , ) A X A . X A 1 X i f the f o l l o w i n g two c o n d i t i o n s are s a t i s f i e d : 1) L e t X. . = X . A X.. 3 an isomorphism f . . : & y l x . : — y 0"Y l x . . s . t . f. . f . . = f. . . x j X J x j X J x j k j j x kx T h i s c o n d i t i o n guarantees the e x i s t e n c e o f a sheaf 0x as w e l l as o f the isomorphism *±« < V * x . » - » - < x i ' * x l V • x 2) I f I . : ^ v ^ S Y where S x i s the sheaf o f 1 i i i germs o f c o n t i n u o u s f u n c t i o n s i s the i n c l u s i o n map r e q u i r e d by Defn. 3.7, then T h i s c o n d i t i o n a l l o w s us to c o n c e i v e of ^ x as a subsheaf o f the sheaf of germs of c o n t i n -uous f u n c t i o n s . P r o o f : Since <p . p r e s e r v e s the r i n g s t r u c t u r e and r e g u l a r 26 q u o t i e n t s o f t h e s h e a v e s frs^ , we need o n l y a s c e r t a i n i t h a t 3 = j ^ j u j » a n open c o v e r i n g o f X, and a s e t J f i n i t e o f c l o s e d s e t s " L ^ i ^ j ^ s . t . L L , w i t h t h e t o p o l o g y i n -duced by t h a t o f X , has a p r e - b a s e P, = | B ! flU.\ J ' r U ^  •-- c oc x £ R i n o r d e r t o p r o v e t h a t (X, ^y) * ^» We d e f i n e a s e t Q = [ f | ft ft and f ( x ) = • } . I t i s e a s i l y s e e n ' i t h a t Q i s an i d e a l . F u r t h e r m o r e 1 Q . Suppose h 4 Q. Then 1 i Q . Let. H be an i d e a l i n y and s u p p o s e h c H b u t h ^ Q. As we have seen a b o v e , t h i s i m p l i e s t h a t h has an i n v e r s e 1 and hence H = T T x ' x i Thus Q i s t h e u n i q u e m a x i m a l i d e a l o f S i n c e X , A . ^ i s an i s o m o r p h i s m , J*5^(Q) i s t h e m a x i m a l i d e a l o f ^ f X _ . Thus 0.(Q) = {f*.\\ 1 °~X,X a n d \ l") = ° i • s u p p o s e X . = U U i , u j£^ (X . ) , and f u r t h e r m o r e t h e s e t o f c l o s e d s e t s ^ B ' ^ ^ i n X^ i s t h e s e t r e q u i r e d by D e f n . 3.7 s . t . { \ i s a p r e - b a s e f o r Ll"*" where B^^ = B ' ^ f l U ^ a n d ^ y i = f \ ^ s ^' n e s e * °^ f u n c t i o n s a s s o c i a t e d w i t h t h e p r e - b a s e . We a s s e r t t h a t i s an open c o v e r i n g o f X t h a t w i t h t h e s e t o f c l o s e d x Now 27 se t s to be defined l a t e r s a t i s f i e s c o n d i t i o n 4) of Defn. 3.7. That LJU i s a union of open sets of X f o l l o w s from i ot the f a c t that U ^ e ^ f U J and X±€^(X) i m p l i e s that U X E ^ ( X ) . That UU U 1 , covers X f o l l o w s immediately from the f a c t that X. = U U 1 and X = U X. . w i l l be defined to be the unique extension ot an X of B ^ P ^ U 1 ) . That B 1 ? can be extended and that the extension, F^^ , i s unique w i l l be the concern of the l a t t e r h a l f of t h i s proof. Let P , , i = f f\ U 1 I. . I t i s c l e a r that (° , . i i n c l u d e s a l l those sets which c o n s t i t u t e d a pre-base f o r as one of the open sets covering X^ , i . e . Furthermore, suppose ( f . „) X(Q) = B^^ and f . ^ - * ^ . , ! . let p O l I B f t Then i t f o l l o w s from the argument about i d e a l s that ( t . f . ^ f l ) " 1 ( Q ) = B 1^. Furthermore, That the set of f u n c t i o n s ^ ^ ( ' X y i ) = C ^ i ^ i ^ g ^ R ZD-generates the sheaf ^ U1. f o l l o w s from the f a c t that OK 28 X y i generates # x 1 U * and ^ ± : (X ±, ^ x ) • ( X ^ &^ \ X ±) ' t i i i s an isomorphism. Now assume that U 1^ = U^fl ={= 0, and assume f u r t h e r that B j ^ f f - . i i s s . t . B 1^ fl f 0. Now f.^„is defined on U 1^ as a consequence of c o n d i t i o n 2) i n the hypothesis, and furthermore can be obtained as a quotient of polynomials i n f v^-, say P . where Q f 0 on Then l e t B * j ^ » P" 1(G). We a s s e r t that B* ^ n U1J' = B 1^ 0 U l j . This f o l l o w s s i n c e B ^ C f l U 1 ] = P ~ 1 ( 0 ) n U 1 i P V ( 0 ) A U 1^ si n c e Q 4= 0 on U 1^ = B i ( i n u i J . We a s s e r t furthermore that a coherency c o n d i t i o n i s s a t i s f i e d : 5uppose U 1 ^ k = 0 UJ' 0 £ 0. Suppose a l s o that B * j P n U x i = B J P n u H and that B* k^ 0 U i k = B ^ f ) Uik.. Then B * ^ n uj ^  = B * ^ A U^<. This f o l l o w s from 1) i n the i i k hypothesis where we set X^^^ = U^^^ and conside r that i f f . . : f .„„ P and f. . : f . . —y R , where P and J 1 ~Q~ k l ~S~ "5" 29 R are quotients of polynomials i n f a n d f ^ o -5 r e s p e c t i v e l y , then f. .: P —>» R , and J Q 5 These c o n s i d e r a t i o n s guarantee that a set B ^ f i ^ U ^ ) can be extended i n a unique way throughout X. We have c a l l e d such an extension F, •P. Then i f ty = (J i s an open covering of X, since we have that FJP«£(X). Hence, i f F ^ O u j # 0 , 3 f't^ J l l j s . t . ( f f ) " 1 ( 0 ) = F ^ f n i j J . This completes the proof ci o of the theorem. Our purpose now w i l l be to show that d i r e c t sums and d i r e c t products of t o p o l o g i c a l spaces and sheaves i n the category <L are themselves i n the category. Theorem 3.11 Let (X , # x ) e C a i j d ( Y , # y ) £ < C . Then (X © Y , # x y)£ C and (X x Y, x y)£ C . Proof: Let U x . and U Y . be the f i n i t e open coverings of X 30 and Y r e s p e c t i v e l y r e q u i r e d by c o n d i t i o n 4) o f D e f n . 3.7. I t i s e a s i l y s een t h a t (X © Y , ^ v ) s a t i s f i e s 1 ) , 2 ) , and 3) o f D e f n . 3.7. To s a t i s f y c o n d i t i o n 4 ) , we sim p l y t a k e ( U X.)U( U Y.) t o be t h e f i n i t e open id 1 j=l J c o v e r i n g o f X @ Y. In t h e c a s e o f t h e d i r e c t p r o d u c t , we p r o c e e d as f o l l o w s : Case 1. We s h a l l assume t h a t t h e open c o v e r i n g o f X r e q u i r e d by c o n d i t i o n 4) o f D e f n . 3.7 i s M = {x], S i m i -l a r l y t h e open c o v e r i n g o f Y w i l l be assumed t o beW = { Y? L e t be t h e p r e — b a s e o f X w i t h t h e a s s o c i a t e d s e t o f f u n c t i o n s and £c^^ t h e p r e - b a s e o f Y w i t h t h e a s s o c i a t e d s e t o f f u n c t i o n s Xy. We n o w c o n s i d e r X x Y. We d e f i n e a s e t o f f u n c t i o n s }6 f r o m X x Y i n t o k as f o l l o w s : I f f C ^ x » ^ f ' e ^ s . t . f * ( x , y ) = f ( x ) . S i m i -l a r l y i f qi%y,3 g 1 * / s . t . g ' ( x , y ) = g ( y ) . ^ i s g e n e r a t e d by t h e f u n c t i o n s £f',g'} u n d e r t h e u s u a l r i n g o p e r a t i o n s . L e t t h e s e t o f c l o s e d s u b s e t s o f X x Y, ) ( X x Y ) , be g e n e r a t e d by t h e s e t £ x Y, X x u n d e r t h e o p e r a t i o n s o f a r b i t r a r y i n t e r s e c t i o n s , f i n i t e u n i o n s , and t h e o p e r a t i o n d e f i n e d i n t e r m s o f t h e f u n c t i o n s { f ' , g ' | . We o b s e r v e t h a t a l l c o n s t a n t maps and t h e f u n c t i o n s b e l o n g i n g t o f> a r e c o n t i n u o u s f o r t h i s t o p o l o g y . 31 We s h a l l d e f i n e t h e p r e s h e a f Py = q u o t i e n t s o f f u n c -t i o n s o f ^ where t h e d e n o m i n a t o r does n o t v a n i s h on 1), U b e i n g an open s e t i n X x Y. We n o t e a l s o t h a t Py i s c l o s e d u n d e r t h e r i n g o p e r a t i o n s . T h i s d e f i n e s t h e s h e a f ^x x Y* S i n c e Py i s g e n e r a t e d by }6 , •-^ ^ y w i l l be s i m i l a r l y g e n e r a t e d . I t i s e a s i l y v e r i f i e d t h a t (X x Y, O" ) s a t i s f i e s A X T 1 ) , 2 ) , and 3) o f D e f n . 3.7. F o r c o n d i t i o n A), we t a k e as t h e open c o v e r i n g o f X x Y,Yv/ = ^X x Y^ . The p r e -b a s e w i l l be f B., x Y, X x Crt"V w i t h t h e a s s o c i a t e d s e t I * ' P oi,Q o f f u n c t i o n s %^ x Y = {^» 9 jj } • *-s e a s i l y c h e c k e d t h a t ^x x Y s a t i s f i e s i ) t i i ) , and i i i ) o f D e f n . 3.3. A l s o t h e g e n e r a t i o n r e q u i r e m e n t o f c o n d i t i o n 4) i n D e f n . 3.7 i s o b v i o u s l y s a t i s f i e d as we o b s e r v e d i n t h e p a r a -g r a p h a b o v e . S i n c e t h e t o p o l o g y on X x Y was c h o s e n t o be t h a t g e n e r a t e d by [_B^ x Y, X x C-|, i t i s c l e a r t h a t ( f B . ^ . n ^ , . , , ) ^ ^ f o r t h e c l ^ d s e t s o f X x Y. T h i s c o m p l e t e s t h e p r o o f f o r Case 1. Case 2 . We now s u p p o s e t h a t t h e open c o v e r i n g o f X i s n I * * V. - U X. and t h a t t h e open c o v e r i n g o f Y i s V/ = U Y. . isl i j:\ J We c o n s i d e r t h e s u b s p a c e X^x Y^ as an open s e t i n X x Y. Vie c o n s t r u c t , j u s t as i n Case 1, a s h e a f o v e r X.x Y. a n d , as b e f o r e , (X.x Y. , ^ Y v ) « C . We a r e now i n a 1 J A . X T . 1 J 32 p o s i t i o n , however, t o apply Theorem 3.10. Both con-d i t i o n s o f the h y p o t h e s i s o f t h a t theorem are s a t i s f i e d as f o l l o w s : I t i s c l e a r t h a t (X, and (Y, 6^) s a t i s -f y a c o n d i t i o n analogous to c o n d i t i o n 1) i n the h y p o t h e s i s of Theorem 3.10. The s e t o f f u n c t i o n s . . = \ f'. . : X.x Y .—>k) generates ^ x y and s i m i l a r l y d s f f : X x Y — k f generates 0 v v . Si n c e *mn ( mn m n ' 3 X x Y m n ^ . . i s generated from ^ x and, ^ v , and s i n c e ^ xj i j m n i s generated from ^ x and ^ y , as i n Case 1, i t i s m n c l e a r t h a t 3 an isomorphism, f U n ) U j r *X.x Y j I ' V V ( V V »~ <?x y | ( X . x Y . ) n ( X m x Y ) m n w i t h the coherency c o n d i t i o n f/ \, \ f, \ / • • \ = f / w • J (pq)(mn) ( m n ) ( i j ) ( p q ) ( i The same g e n e r a t i o n and coherency c o n s i d e r a t i o n s g i v e us t h a t v s a t i s f i e s X x Y Hence (X x Y, # x y ) £ ^ v w s a t i s f i e s c o n d i t i o n 2) o f Theorem 3.10. SECTION 4 We s h a l l now give examples of s p e c i f i c t o p o l o g i c a l spaces and sheaves that are elements of the category C . Theorem 4.1 Let k denote a f f i n e space of dimension r with ( k 1 ) = 2^ » a n d l e t b e t n e sheaf of germs of r e g u l a r f u n c t i o n s . Then ( k r , ) e ^ . Proof: I t i s e a s i l y seen that ( k r , ^ ' j < r ) s a t i s f i e s c o n d i t i o n s li)l, 2), and 3) of Defn. 3.7. For 4) we proceed as f o l l o w s : toe take the open covering of k r, %( , to c o n s i s t only of the e n t i r e space, k . Let l-L denote the hyperplane of k which c o n s i s t s of a l l those p o i n t s whose i - t h c o o r d i -nate i s 0. These r hyperplanes together with the n u l l set form a pre-base f o r V ( k r ) where X ^ r c o n s i s t s of the p r o j e c t i o n maps X^, i = l , . , , r and the constant maps c, cck. This may be v e r i f i e d e a s i l y as f o l l o w s : Obviously X ^ r s a t i s f i e s i ) , i i ) , and i i i ) of Defn. 3.3. Furthermore, the l o c u s of zeros of any polynomial i n r v a r i a b l e s can be constructed from the given hyperplanes and the n u l l set 0 under the operations s p e c i f i e d f o r 34 the c o n s t r u c t i o n of the set ( 3 i n Defn. 3.3. A l l quot i e n t s of polynomials may be constructed from the p r o j e c t i o n maps and the constant maps under the r i n g o p erations and the operation of ta k i n g r e g u l a r q u o t i e n t s ; hence the generation requirement of 4) i n Defn. 3.3 i s s a t i s f i e d . I t f o l l o w s from the remark a f t e r Theorem 2.9 that (J3 i s a base f o r 3 r ( k r ) since the set of l o c i of zeros of a l l quotients of polynomials i s a base f o r ^ ( k r ) . Defn. 4.2 A q u a s i - a f f i n e v a r i e t y (X, ^y) i s a p a i r c o n s i s t i n g of a t o p o l o g i c a l space X = GflF> where & £ ^ ( k r ) and Yl V ( k r ) , the topology on X i s that induced from k r, and a sheaf 0"^ = £f^ = f(X where f £ \ • C o r o l l a r y 4.3 A l l q u a s i - a f f i n e v a r i e t i e s are i n C . Proof: T h i s i s the immediate consequence of the preceding theorem and Theorem 3.8 i n which X = k and Y = G fl F where G e # ( k r ) and F c T ( k r ) . 35 r s Theorem 4.4 L e t k and k be a f f i n e spaces o f dimension r and s r e s p e c t i v e l y , both s u p p l i e d w i t h the Z a r i s k i t o p o l o g y and sheaves o f germs o f r e g u l a r f u n c t i o n r s We c o n s i d e r the C a r t e s i a n product k x k . L e t £^(krx j ^ S j be the sheaf o f germs o f r e g u l a r func t i o n s o f r+s v a r i a b l e s and l e t (k x k )^ be k x k w i t h the u s u a l Z a r i s k i t o p o l o g y . Then where ( k r x k S, &^x^ ^s ) i s the sheaf c o n s t r u c t e d as i n Theorem 3.11 f o r the c a t e g o r y * t . Pr o o f : We s h a l l d e f i n e a map i : ^ ( ^ r x k s ) "^ ^ k r x k s * L e t f ( X l ' ' "V Y r + 1 ' • • > Y r + s ) & ^ ( x . v ) . ( k r x k s ) R g(x i, . .,x r, Y r + 1,•. • » Y r + s ) where g(x,y) 4= 0» e . e . Now f ='S*z.f.g. where f . = X. J l . . .X. Jv* and *tj— J J J J J| Jv J g. = Y . E J ^ . . . Y . B J ' + 5 . J Jrtl J*** But s i n c e f ^ ^ ^ r and g / ^ ^ a , Vj * ^ <x.y) . k rx k Hence * ( X t y ) ( K * X k * s i m i l a r l y g « ^ ( X f y ) f K r x k s 36 L e t I D : ^ \ , r . 8 , — V S ( k r x k s) and I : . s - > S ( k r x k s) R ( k x k ) R k x k where S ( k r x k S ) i s the sheaf o f germs o f c o n t i n u o u s f u n c t i o n s from k r x k s i n t o k. Then i i s d e f i n e d as f o l l o w s : Ip (-s-H1*-*-)) That i i s onto f o l l o w s from the f a c t t h a t % , r s >^>y'i^ ^ * i s generated from U , r and That i i s 1-1 x, K y, K f o l l o w s from the f a c t t h a t a p o l y n o m i a l w i t h non-zero c o e f f i c i e n t s cannot be the c o n s t a n t z e r o f u n c t i o n . F i n a l l y t h a t i i s a sheaf morphism f o l l o w s from the f a c t t h a t i t p r e s e r v e s the r i n g s t r u c t u r e o f the s t a l k s i n the sheaf. Again f o l l o w i n g J e a n - P i e r r e S e r r e , [ i j , p. 226, we make the f o l l o w i n g d e f i n i t i o n : Defn. 4.5 A p r e - a l q e b r a i c v a r i e t y i s a t o p o l o g i c a l space X w i t h a subsheaf of the sheaf o f germs o f c o n t i n u -ous f u n c t i o n s which s a t i s f i e s the f o l l o w i n g t h r e e c o n d i t i o n s : n i ) X = IJ X ± , X ±£p { X ) . ) X. ^  U. where U. i s an a f f i n e v a r i e t y . 37 i i i ) The maps p ^ u s e d i n i i ) a r e b i - r e g u l a r i s o -m o r p h i s m s . A f u n c t i o n i s s a i d t o be r e g u l a r i f a) f& i s c o n t i n u o u s b) i f x£U and i f ft ^ ( x ) f V » t h e n f'f 1 &XfU • ^ : U — y V i s c a l l e d a b i - r e q u l a r i s o m o r p h i s m i f b o t h P and a r e r e g u l a r . Theorem 4.6 I f ( X , O' ) i s a p r e - a l g e b r a i c v a r i e t y , t h e n ( X , ^ x ) * P r o o f : T h i s f o l l o w s i m m e d i a t e l y f r o m Theorem 3.10 and C o r o l l a r y 4.3. D e f n . 4.7 I f a p r e - a l g e b r a i c v a r i e t y ( X , &y) has t h e p r o p e r t y t h a t t h e d i a g o n a l A o f X x X i s c l o s e d i n X x X, t h e n ( X , ^ Y ) i s c a l l e d an a l g e b r a i c v a r i e t y . C o r o l l a r y 4.8 A l l a l g e b r a i c v a r i e t i e s b e l o n g t o <£ . CONCLUSIONS A l t h o u g h we have p r o v e n s e v e r a l p r o p e r t i e s o f t h e c a t e g o r y ^ and e x h i b i t e d e x a m p l e s o f t o p o l o g i c a l s p a c e s and s h e a v e s t h a t a r e e l e m e n t s o f <C- , s e v e r a l q u e s t i o n s r e m a i n open. The most i m p o r t a n t o f t h e s e i s t h e r e l a t i o n between t h e c a t e g o r y P o f p r e - a l g e b r a i c v a r i e t i e s and t h e c a t e g o r y C . We have shown o n l y thatP c <C . I t seems r e a s o n a b l e to e x p e c t t h a t t h e r e a r e p a i r s o f t o p o l o g i c a l s p a c e s and s h e a v e s w h i c h a r e n o t ' p r e - a l g e b r a i c v a r i e t i e s b u t w h i c h b e l o n g t o <t . BIBLIOGRAPHY S e r r e , J e a n - P i e r r e . " F a i s c e a u x A l g e b r i q u e s C o h e r e n t s " , A n n a l s o f M a t h e m a t i c s , v. 6 1 , 1955, pp. 197-278. L a n g , 5 e r g e . I n t r o d u c t i o n t o A l g e b r a i c Geometry. I n t e r -s c i e n c e P u b l i s h e r s , I n c . , New Y o r k . 

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