1.1. Aﬃne varieties 11 Then dom(f) = {P ∈ V : h(P ) = 0 for some h ∈ Jf} hence its comple- ment V \ dom(f) = V(Jf) is closed. As dom(f) is an open subset of the irreducible closed subset V , it is dense in V . b) dom(f) = V ⇔ V(Jf) = ∅. By Hilbert’s Nullstellensatz 1.1.6, this last equality implies 1 ∈ Jf and so f ∈ C[V ]. c) We have dom(f) ⊃ Vh if and only if h vanishes on V(Jf). By Hilbert’s Nullstellensatz, this is equivalent to hr ∈ Jf for some r ≥ 1. This means that f = g/hr ∈ C[V ][1/h]. Part b) of Proposition 1.1.18 says that the polynomial functions are precisely the rational functions that are “everywhere regular”. If U is an open subset of an irreducible variety V , a rational function f in C(V ) is regular on U if it is regular at each point of U. The set of rational functions of C(V ) which are regular on U is a subring of C(V ). We denote it by O(U). The local ring of V at a point P ∈ V is the ring OP := {f ∈ C(V ) : f is regular at P }. It is isomorphic to the ring C[V ]M P obtained by localizing the ring C[V ] at the maximal ideal MP = {f ∈ C[V ] : f(P ) = 0}. This is indeed a local ring, i.e. it has a unique maximal ideal, namely MP C[V ]M P . Remark 1.1.19. If V is an irreducible aﬃne variety, then C[V ] = P ∈V OP . Indeed, as C[V ] is an integral domain, we can apply [Ma] Lemma 2, p.8. If V is an arbitrary aﬃne variety, U an open subset of V , a function f : U → C is regular at a point x in V if there exists g, h ∈ C[V ] and an open subset U0 of U containing x such that for all y ∈ U0, h(y) = 0 and f(y) = g(y)/h(y). A function f : U → C is regular in an open subset U of U if it is regular at each point of U . Let us observe that a principal open set can be seen as an aﬃne variety. If Vf = {x ∈ An C : f(x) = 0}, for some f ∈ C[X1,...,Xn], the points of Vf are in 1-1 correspondence with the points of the closed set of An+1 C {(x1,...,xn,xn+1) : f(x1,...,xn) xn+1 − 1 = 0}. Hence Vf has an aﬃne

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