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The functor O : AV −→ FRA given by X → O(X), ϕ → ϕ∗ defines an anti-equivalence of categories. Proof. 4 we know that every algebra A ∈ FRA is isomorphic to the regular function algebra O(X) of some affine variety X. 5. Let us now describe a functor Sp : FRA −→ AV inverse to O : AV −→ FRA. For a reduced affine k-algebra A we set Sp(A) := {m → A, max. ideal} , the “(maximal) spectrum” of the ring A. , Tn ] −→ A, denote a → k[T ] its kernel and X := N (a) → k n its zero set. Then X −→ Sp(A), x → mx is a bijection and hence can be used to define a topology and regular functions on Sp(A).

E. O(X) = k: A regular function can be regarded as a morphism f : X −→ P1 = k ∪ {∞} avoiding the value ∞. Thus the closed set f (X) is finite. 6. An affine variety X is complete iff it is finite. 7. A complex algebraic variety X is complete iff Xh is compact. 12. Projective varieties are complete. Proof. 2 it suffices to show that Pn is complete. g. affine variety and Y → Pn × Z a closed set. With B := prZ (Y ) we have the commutative diagram Y ↓ → Pn × Z ↓ ? B → Z 61 and want to see, why we are allowed to remove the question mark.

F (q) ∈ I(Z). 7. , fr ) := {[t]; f1 (t) = ... , Tn ]. Proof. Apply Prop. 6 to the affine cone C(X) → k n+1 . Note that homogeneous polynomials are functions only on k n+1 and not on Pn , but since such a polynomial either vanishes identically on a punctured line k ∗ · t (t = 0) or has no zeros there, the above description of a projective variety makes sense nevertheless. 8. , Tn ]. , Tn ] f ∈a satisfies N (a) = C(Y ). Proof. t. π : k n+1 \ {0} −→ Pn , we have C(Y ) = k ∗ · ({1} × Y ). , t0 t0 .