Some lectures on Algebraic Geometry∗ V. Srinivas 1 Affine varieties Let k be an algebraically closed field. We define the affine n-space over k n n to be just the set k of ordered n-tuples in k, and we denote it by Ak (by 0 convention we define Ak to be a point). If x1; : : : ; xn are the n coordinate n functions on Ak , any polynomial f(x1; : : : ; xn) in x1; : : : ; xn with coefficients n in k yields a k-valued function Ak ! k, which we also denote by f. Let n A(Ak ) denote the ring of such functions; we call it the coordinate ring of n Ak . Since k is algebraically closed, and in particular is infinite, we easily see n that x1; : : : ; xn are algebraically independent over k, and A(Ak ) is hence a 0 polynomial ring over k in n variables. The coordinate ring of Ak is just k. n Let S ⊂ A(Ak ) be a collection of polynomials. We can associate to it n an algebraic set (or affine variety) in Ak defined by n V (S) := fx 2 Ak j f(x) = 0 8 f 2 Sg: This is called the variety defined by the collection S. If I =< S >:= ideal in k[x1; : : : ; xn] generated by elements of S, then we clearly have V (S) = V (I). Also, if f is a polynomial such that some power f m vanishes at a point P , then so does f itself; hence if p m I := ff 2 k[x1; : : : ; xn] j f 2 I for some m > 0g p is the radical of I, then V (I) = V ( I). n Conversely, let X ⊂ Ak be an algebraic set. Then we can associate to it (i) the ideal I(X) of polynomials vanishing on X, and ∗Lectures at S. N. Bose Institute, Calcutta, April 1997. 1 ∼ (ii) the ring A(X) = k[x1; : : : ; xn]=I(X) of polynomial functions on X, the coordinate ring of X. Clearly I(X) = pI(X), and A(X) is a reduced ring (has no non-zero nilpo- tents). p Theorem 1 (Hilbert's Nullstellensatz) Let I = I be a radical ideal in the polynomial ring k[x1; : : : ; xn]. Then I(V (I)) = I. Corollary 1 There is a one-one correspondence between three sets of ob- jects: n (i) algebraic sets in Ak (ii) radical ideals in k[x1; : : : ; xn] (iii) pairs (A; (t1; : : : ; tn)) where A is a reduced k-algebra, and t1; : : : ; tn are n elements of A which generate A as a k-algebra. In particular, if m ⊂ k[x1; : : : ; xn] is a maximal ideal, it is a non-trivial n radical ideal, so that V (m) 6= φ; if P = (a1; : : : ; an) 2 Ak lies in V (m), then m ⊂ I(P ) =< x1 − a1; : : : ; xn − an >. Since m is maximal, we must have equality. Thus all maximal ideals in k[x1; : : : ; xn] are of the form < x1 −a1; : : : ; xn −an > for suitable a1; : : : ; an 2 k. If m is a maximal ideal, the natural map k ! S(X)=m is an isomorphism; identifying S(X)=m with k, if Xi 2 S has image ai in S(X)=m, the corresponding point P is clearly (a1; : : : ; an). Another conclusion is that any radical ideal I ⊂ k[x1; : : : ; xn] is the intersection of all maximal ideals containing it; this just restates that a n variety in Ak is determined by its points. The points on an affine variety X are naturally in bijection with the maximal ideals of A(X), i.e., with the maximal ideals of k[x1; : : : ; xn] containing I(X). We can the associate to a point P 2 X the local ring of X at P , as follows: if m ⊂ A(X) is the corresponding maximal ideal, then the local ring is OP;X := A(X)m is defined to be the localization of A(X) at m. n The set of varieties in a given affine space Ak is closed under finite unions and arbitrary intersections, with the following properties: (i) if I ⊂ J are ideals, then V (J) ⊂ V (I), (ii) V (I) [ V (J) = V (IJ) = V (I \ J) for any pair of ideals I;J, and 2 X (iii) \αV (Iα) = V ( Iα) for any family of ideals fIαg of k[x1; : : : ; xn]. α The proofs are easy. To prove the second property, for example, note that V (I) [ V (J) ⊂ V (I \ J) ⊂ V (IJ), since (use the first property) we have n IJ ⊂ I \J and I \J ⊂ I, I \J ⊂ J. If x 2 Ak −V (I)[V (J), then there exist f 2 I, g 2 J with f(x) 6= 0 and g(x) 6= 0; now fg 2 IJ with fg(x) 6= 0. Thus n the varieties in Ak satisfy the axioms for the closed subsets in a topology; n the resulting topology on Ak (and the induced subspace topology on any n affine variety in Ak ) is called its Zariski topology. A closed subset of a variety is referred to as a subvariety; subvarieties of X are clearly in bijection with radical ideals in A(X)(i.e., with radical ideals in k[x1; : : : ; xn] which contain I(X)). We can define a suitable notion of morphisms between affine varieties. n m First consider the case of morphisms f = (f1; : : : ; fm): Ak ! Ak ; we define these to be mappings whose component functions fi are polynomial n m functions. Let x1; : : : ; xn and y1; : : : ; ym be the coordinates on Ak and Ak respectively. For any polynomial function g(y1; : : : ; ym), the function f ∗(g) := g ◦ f n is then a polynomial function on Ak . If yi ◦ f = fi(x1; : : : ; xn) are the component functions of the mapping f, then clearly ∗ f (g) = g(f1(x1; : : : ; xn); f2(x1; : : : ; xn); : : : ; fm(x1; : : : ; xn)): ∗ Thus f : k[y1; : : : ; ym] ! k[x1; : : : ; xn] is a homomorphism of k-algebras. Conversely, given a k-algebra homomorphism ' : k[y1; : : : ; ym] ! k[x1; : : : ; xn], n m let '(yi) = fi(x1; : : : ; xn); then f = (f1; : : : ; fm): Ak ! Ak is a morphism, and f ∗ = '. Note that a morphism is clearly continuous for the Zariski topology. n m If X ⊂ Ak , Y ⊂ Ak are affine varieties, a mapping f : X ! Y is n m called a morphism if it is the restriction of a morphism fe : Ak ! Ak (the morphism fe need not be unique). Since fe extends f, we have fe(X) ⊂ Y , and so fe∗I(Y ) ⊂ I(X), and we have an induced k-algebra homomorphism f ∗ : A(Y ) ! A(X). Conversely, let ' : A(Y ) ! A(X) be a k-algebra homo- morphism. Let a1; : : : ; an 2 A(X), b1; : : : ; bm 2 A(Y ) be (the restrictions to X and Y ) of the coordinate functions. Then '(bi) = fi(a1; : : : ; an) for some polynomials fi in n variables over k. If n m fe = (f1; : : : ; fm): Ak ! Ak ; 3 and f is its restriction to X, then one checks that f(X) ⊂ Y , and f ∗ = '. Thus morphisms between affine varieties correspond precisely to k-algebra homomorphisms (in the opposite direction) between their coordinate rings. Since we have defined a notion of morphisms, in particular, we have the notion of an isomorphism. 1 Morphisms f : X ! Ak are also called regular functions on X. These correspond to k-algebra homomorphisms k[x1] ! A(X), i.e., to elements of A(X) (a homomorphism k[x1] ! A(X) is determined by the image of x1). Some of the subtleties in the definition of a morphism become apparent from the following examples. 2 1 2 Example 1.1: Let x; y be coordinates on Ak. Let X = Ak, Y = V (y − 3 2 2 3 1 x ) ⊂ Ak, and let f : X ! Y , f(a) = (a ; a ) for all a 2 Ak. If t is 1 ∗ 2 ∗ 3 the coordinate on Ak, then f (x) = t , f (y) = t . Hence the component functions of f are polynomials, and so f is a morphism. One sees easily that f is bijective, which amounts to the statement that Y = f(a2; a3) j a 2 2 ∗ kg ⊂ Ak. Now A(X) = k[t] is the polynomial ring, and f A(Y ) is the k- subalgebra generated by t2; t3. In particular the subalgebra does not contain t, so that f is not an isomorphism, though it induces an isomorphism of the quotient fields of A(Y ) and A(X). This situation occurs because the variety Y has a singularity at the origin. 2 2 Example 1.2: Let f : Ak ! Ak be given by f(a; b) = (a; ab). If x; y are the coordinates on the target space, then (i) the fibres of f over the Zariski ∗ 2 2 open set fx 6= 0g consist of exactly 1 point (ii) the map f : A(Ak) ! A(Ak) induces an isomorphism on quotient fields (iii) the only point of the y-axis fx = 0g in the image of f is the origin (0; 0), whose preimage is the y-axis. This is called an affine blow up of the origin, since we have `replaced' the origin of the target A2 by its preimage, a line, while (more or less) leaving 2 the rest of Ak unchanged. Note that the image of f contains a non-empty (dense) open subset of 2 Ak, but is neither open nor closed. Thus the open mapping theorem is not valid here. n n Example 1.3: Let k have characteristic p > 0. Then F : Ak ! Ak given p p p by F (a1; a2; : : : ; an) = (a1; a2; : : : ; an) is a morphism which is bijective, but ∗ n not an isomorphism (the map F identifies A(Ak ) with its subring consisting th n of p powers).