By Smirnov V.I.

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**Example text**

J have the same restriction to YJl Yk (i, kEI). j (iEI) match to define an element of k[Y], hence f Ek[X] (8) k[ Y]. If now X is not affine, argue similarly using a finite open cover of X. , by definition, isomorphic to a k-open sub-set of an affine k-variety X. Note 26 Background Material from Algebraic Geometry AG that if X is quasi-affine and irreducible, then k(X) is the quotient field of k[X], (since k(X) = k(X)). § 13. 1 The functor of points. Let V be a k-variety. sch(specK(B), V). If B has a k-structure Bk we also write V(B k) = V(B)k for the set of morphisms as above which are defined over k.

6) characterizes separability of K(V) over K(W). 4 Corollary. If (Xi: Vi ...... Wi (i = 1,2) are two separable morphisms then x a 2 : VI x V2 ...... WI X W2 is separable. This follows easily from criterion (2). (Xl §18. Normal Varieties This section contains the main results needed in Chapter II, §6 for the construction of homogeneous spaces. 1 Definition. e. if 0 x is an integral domain integrally closed in its field of fractions. e. it has an irreducible open neighborhood. Consequently most questions involving normality can be easily reduced to the case of irreducible varieties.

This corresponds to the map Derk(B, M) .... Derk(A, M) defined by: Xf---+XOf, for each B-module (and hence also A-module) M. In this way D A is functorial in A. 2 Polynomial rings. If A = k[T1, ... , Tn] is a polynomial ring, then D is a free A-module with basis dT1, ... ,dTw Moreover d:A .... D is given by 8f df=L~dT 8Ti I for f EA. These assertions translate the fact that a derivation X: A -+ M is determined by the XTi , which can be arbitrarily prescribed. 3 Residue class rings. Let A' = AjJ for some ideal J, and let M be an 34 Background Material from Algebraic Geometry AG A'-module (or A-module annihilated by J).