Zariskis Main Theorem

This is completely false if tr.d.K/k 1, essentially because a local ring ox at a closed point of a variety X can be a valuation ring only when dim X = 1. 

Example M. Let x Pn be a closed point, and let B = Bx (Pn). Let p B - Pn be the canonical map. Then (a)both B and Pn are non-singular (in fact, B and Pn are both covered by open pieces isomorphic to An), is proper and Pn is complete, so B is complete too, and p is birational. In other words, we have 2 non-singular and complete varieties, and a birational map between them, which is not an isomorphism if n 2. 

Among higher dimensional varieties, however, we can put together the ideas of Th. 2, 7, and of this section in  

Proposition 6. Let X be a normal projective n-dimensional variety. Then X is the normalization of a hypersurface H C Pn+1. 

Assume X is a closed subvariety of P. In the proof of Theorem 2 we constructed a projection 

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As XW is smooth, p is an isomorphism over U(i, 2)(X) by Zariski s main theorem [Hartshorne (2), V. 5.2]. Now we show (1). As p is an isomorphism over Ai3i Z(lt2)(X), it is enough to prove the smoothness at the points of E = ni(E). Le Barz has given analytic local coordinates around any point e E and so proved the smoothness of H3(X). To simplify notations we will assume that the dimension of A is 3. The argument for general dimension d is completely analogous, only more difficult to write down. Now let... 

N.B. Au lieu d invoquer 8.5 on peut aussi invoquer le "Main Theorem" de Zariski, qui implique directement que u est tme immersion ouVerte, done un isomorphisme. ... 

En fait nous allons voir que toute A-algebre quasi-finie est de cette forme e est le Main Theorem de Zariski. 

Y = Spec(B ). II resulte immediatement du Main theorem de Zariski (chap.iv) et du fait que A est normal, que le morphisme canonique B - A correspond a une immersion ouverte X - Y. Soient S = Spec(A/l), X = Xx S > Y = Yx S lors x est un sous-schSma ouvert de Y, mais c est aussi un sous-sous-schSma fermS de Y, puisque X est fini sur S... 

Finally, forms (IV.) and (V.) of the Main Theorem are even deeper. (V.) is a much more global statement since the properness of / is involved. There is a cohomological proof, due to Grothendieck (cf. EGA, Ch. Ill, 4.3) and a proof using a combination of projective techniques and completions, due to Zariski. As... 

In this chapter we review and elaborate on—with proofs and/or references— some basic abstract features of Grothendieck Duality for schemes with Zariski topology, a theory initially developed by Grothendieck [Gr ], [H], [C], Dehgne [De ], and Verdier [V ]. The principal actor in this Chapter is the twisted inverse image pseudofunctor, described in the Introduction. The basic facts about this pseudofunctor—which may be seen as the main results in these Notes—are existence and flat base change, Theorems (4.8.1) and (4.8.3). 

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