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By Donald Knutson

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3 58 union of the images of the c o r r e s p o n d i n g Let U x be the d i s j o i n t Uvl'''''Uvn cover Vx. union of Ux', U v l ' ' ' ' ' U v n Now let the index set I be the c o l l e c t i o n of p o i n t s and for each i e I, X. and Y. the U. and V. 11: above. that i-iv are satisfied. • The class of maps of schemes w h i c h are flat and locally of finite p r e s e n t a t i o n c a t e g o r y of the c a t e g o r y of schemes is a closed sub- and satisfies axioms S 1 and S 2 . 12: The flat t o p o l o g y is the topology a s s o c i a t e d with on the c a t e g o r y of the set of maps, flat and locally of finite presentation.

11: and d i s j o i n t The c a t e g o r y sums. 6). we d e f i n e has fiber functor faithful, and p r e - of g:Z ~ Y (Schemes). A. the term necessary map for the n o t i o n s if f is injec- One open the induced open m a p s and closed open (in if for f':X X Z ~ Z is Y is then a c l o s e d universally closed. ) when it comes something, the t o p o l o g i c a l Luckily of u n i v e r s a l l y maps map and u n i v e r g a l l y to check note subcategories we d e f i n e is used g:Z + Y. should say in the case of open Similarly radiciel a given m a p for every p o s s i b l e it is s u f f i c i e n t this, bijective The above d e f i n i t i o n s it is a p p a r e n t l y spaces.

W e note products, and given by the tensor finite disjoint But o b s e r v e ducts that that of rings ite p r o d u c t unions, the By e x t e n d i n g the c a t e g o r y of schemes, spaces, of a f f i n e takes one gets schemes. 5: An open be true, subspace Y of X with (Note Y need not itself be an affine scheme of the form Z = Spec R/I w h e r e pro- schemes of to of alge- of d i s j o i n t however, union that Spec limits. X is an o p e n Z of an affine of rings. of the u n i o n of affine notion rings, (Spec of an infin- to the c a t e g o r y subscheme scheme associated take i n f i n i t e compactification the correct It w i l l not fiber products unions.

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