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A Algebraic Varieties A Algebraic Varieties A.1 Basic definitions Let k[X1,X2,...,Xn] be a polynomial algebra over an algebraically closed field k with n indeterminates X1,...,Xn. We sometimes abbreviate it as k[X]= k[X1,X2,...,Xn]. Let us associate to each polynomial f(X)∈ k[X] its zero set n V(f):= {x = (x1,x2,...,xn) ∈ k | f(x)= f(x1,x2,...,xn) = 0} n 2in the n-fold product set k of k. For any subset S ⊂ k[X] we also set V(S) = f ∈S V(f). Then we have the following properties: n (i) 2V(1) =∅,V(0) =k . = (ii) i∈I V(Si) V( i∈I Si). (iii) V(S1) ∪ V(S2) = V(S1S2), where S1S2 := {fg | f ∈ S1,g∈ S2}. The inclusion ⊂ of (iii) is clear. We will prove only the inclusion ⊃. For x ∈ V(S1S2) \ V(S2) there is an element g ∈ S2 such that g(x) = 0. On the other hand, it ∀ ∀ follows from x ∈ V(S1S2) that f(x)g(x) = 0( f ∈ S1). Hence f(x)= 0( f ∈ S1) and x ∈ V(S1). So the part ⊃ was also proved. By (i), (ii), (iii) the set kn is endowed with the structure of a topological space by taking {V(S) | S ⊂ k[X]} to be its closed subsets. We call this topology of kn the Zariski topology. The closed subsets V(S) of kn with respect to it are called algebraic sets in kn. Note that V(S)= V(S), where S denotes the ideal of k[X] generated by S. Hence we may assume from the beginning that S is an ideal of k[X]. Conversely, for a subset W ⊂ kn the set = > ∀ I(W):= f ∈ k[X]|f(x)= 0 ( x ∈ W) is an ideal of k[X]. When W is a (Zariski) closed subset of kn, we have clearly V(I(W))= W. Namely, in the diagram V ideals in k[X] closed subsets in kn I 322 A Algebraic Varieties we have V ◦I = Id. However, for an ideal J ⊂ k[X] the equality I(V(J))= J does not hold in general. We have only I(V(J)) ⊃ J . The difference will be clarified later by Hilbert’s Nullstellensatz. Let V be a Zariski closed subset of kn (i.e., an algebraic set in kn). We regard it as a topological space by the relative topology induced from the Zariski topology of kn. We denote by k[V ] the k-algebra of k-valued functions on V obtained by restricting polynomial functions to V . It is called the coordinate ring of V . The restriction map ρV : k[X]→k[V ] given by ρV (f ) := f |V is a surjective homomorphism of k-algebras with Ker ρV = I(V), and hence we have k[V ]k[X]/I (V ). For each n point x = (x1,x2,...,xn) ∈ k of V define a homomorphism ex : k[V ]→k of k-algebras by ex(f ) = f(x). Then we get a map e : V −→ Homk-alg(k[V ],k) (x−→ ex), where Homk-alg(k[V ],k)denotes the set of the k-algebra homomorphisms from k[V ] to k. Conversely, for a k-algebra homomorphism φ : k[V ]→k define n x = (x1,x2,...,xn) ∈ k by xi = φρV (Xi) (1 ≤ i ≤ n). Then we have x ∈ V and ex = φ. Hence we have an identification V = Homk-alg(k[V ],k)as a set. Moreover, −1 ={ ∈ | = }⊂ the closed subsets of V are of the form V(ρV (J )) x V ex(J ) 0 V for ideals J of k[V ]. Therefore, the topological space V is recovered from the k-algebra k[V ]. It indicates the possibility of defining the notion of algebraic sets starting from certain k-algebras without using the embedding into kn. Note that the coordinate ring A = k[V ] is finitely generated over k, and reduced (i.e., does not contain non-zero nilpotent elements) because k[V ] is a subring of the ring of functions on V with values in the field k. In this chapter we give an account of the classical theory of “algebraic varieties’’ based on reduced finitely generated (commutative) algebras over algebraically closed fields (in the modern language of schemes one allows general commutative rings as “coordinate algebras’’). The following two theorems are fundamental. Theorem A.1.1 (Aweak form of Hilbert’Nullstellensatz). Any maximal ideal of the polynomial ring k[X1,X2,...,Xn] is generated by the elements Xi −xi (1 ≤ i ≤ n) n for a point x = (x1,x2,...,xn) ∈ k . √ √ Theorem A.1.2 (Hilbert’s Nullstellensatz). We have I(V(J)) = J , where J is the radical { f ∈ k[X]|f N ∈ J for some N 0} of J . For the proofs, see, for example, [Mu]. For a finitely generated k-algebra A denote by Specm A the set of the maximal ideals of A. For an algebraic set V ⊂ kn we have a bijection Homk-alg(k[V ],k) Specm k[V ] (e −→ Ker e) byTheoremA.1.1. Under this correspondence, the closed subsets in Specm k[V ]V are the sets V(I)={m ∈ Specm k[V ]|m ⊃ I }, A.2 Affine varieties 323 where I ranges through the ideals of k[V ]. Theorem A.1.2 implies that there is a one- to-one correspondence√ between the closed subsets in Specm k[V ] and the radical ideals I (I = I)ofk[V ]. A.2 Affine varieties Motivated by the arguments in the previous section, we start from a finitely generated reduced commutative k-algebra A to define an algebraic variety. Namely, we set V = Specm A and define its topology so that the closed subsets are given by = > V(I)={m ∈ Specm A | I ⊂ m}|I: ideals of A . By Hilbert’s Nullstellensatz (its weak form), we get the identification V Homk-alg(A, k). We sometimes write a point x ∈ V as mx ∈ Specm A or ex ∈ Homk-alg(A, k). Under this notation we have f(x)= ex(f ) = (f mod mx) ∈ k for f ∈ A. Here, we used the identification k A/mx obtained by the composite of the morphisms k→ A → A/mx. Hence the ring A is regarded as a k-algebra consisting of certain k-valued functions on V . Recall that any open subset of V is of the form D(I) = V \ V(I), where I is an ideal of A. Since A is a noetherian ring (finitely generated over k), the ideal I is generated by a finite subset {f1,f2,...,fr } of I. Then we have r (r D(I) = V \ V(fi) = D(fi), i=1 i=1 where D(f ) ={x ∈ V | f(x) = 0 }=V \ V(f)for f ∈ A. We call an open subset of the form D(f ) for f ∈ A a principal open subset of V . Principal open subsets form a basis of the open subsets of V . Note that we have the equivalence D(f ) ⊂ D(g) ⇐⇒ (f ) ⊂ (g) ⇐⇒ f ∈ (g) by Hilbert’s Nullstellensatz. Assume that we are given an A-module M. We introduce a sheaf M on the topological space V = Specm A as follows. For a multiplicatively closed subset S of A we denote by S−1M the localization of M with respect to S. It consists of the equivalence classes with respect to the equivalence relation ∼ on the set of pairs (s, m) = m/s (s ∈ S, m ∈ M) given by m/s ∼ m/s ⇐⇒ t(sm − sm) = 0 (∃t ∈ S). By the ordinary operation rule of fractional numbers S−1A is endowed with a ring structure and S−1M turns out to be an S−1A-module. For f ∈ A we set 324 A Algebraic Varieties = −1 := { 2 } Mf Sf M, where Sf 1,f,f ,... Note that for two principal open subsets ⊂ g : → D(f ) D(g) a natural homomorphism rf Mg Mf is defined as follows. We n l have f = hg (h ∈ A, n ∈ N), and then the element m/g ∈ Mg is mapped to l l l l l nl m/g = h m/h g = h m/f ∈ Mf . In the case of D(f ) = D(g) we easily see Mf Mg by considering the inverse. Theorem A.2.1. (i) For an A-module M there exists a unique sheaf M on V = Specm A such that for any principal open subset D(f ) we have M(D(f)) = Mf , and the restriction → ⊂ g homomorphism M(D(f)) M(D(g)) for D(f ) D(g) is given by rf . (ii) The sheaf OV := A is naturally a sheaf of k-algebras on V . (iii) For an A-module M the sheaf M is naturally a sheaf of OV -module. The stalk of M at x ∈ V is given by = −1 = Mmx (A mx) M −lim→ Mf . f(x) =0 O := = −1 This is a module over the local ring V,x Amx (A mx) A. The key point of the proof is the fact that the functor D(f ) → Mf on the category of principal open subsets {D(f ) | f ∈ A} satisfies the “axioms of sheaves (for a basis of open subsets),’’ which is assured by the next lemma. Lemma A.2.2. Assume that the condition f1,f2,...,fr 1 is satisfied in the ring A. Then for an A-module M we have an exact sequence Br B −→ −→α −→β 0 M Mfi Mfi fj , i=1 i,j r − ∈ ≤ ≤ where the last arrow maps (si)i=1 to si sj Mfi fj (1 i, j r).
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