Algebraic Geometry Fall 2013 & Spring 2014 Eduard Looijenga Rings are always supposed to be commutative and to possess a unit 1. A ring homomorphism will always take unit to unit. We allow that 1 = 0, but in that case we get of course the zero ring f0g. If R is a ring, then we denote the multiplicative group of invertible elements (units) of R by R×. We say that R is a domain if R has no zero divisors, equivalently, if (0) is a prime ideal. An R-algebra is a ring A endowed with a ring homomorphism φ : R ! A, but if φ is understood, then for every r 2 R and a 2 A, the product φ(r)a is often denoted by ra. We say that A is finitely generated as an R-algebra if we can find a1; : : : ; an in A such that every element of A can be written as a polynomial in these elements with coefficients in R; in other words, the R-algebra homomorphism R[x1; : : : ; xn] ! A which sends the variable xi to ai is onto. This is not to be confused with the notion of finite generation of an R-module M resp. which merely means the existence of a surjective homomorphism of R-modules Rn ! M. Similarly, a field K is said to be finitely generated as a field over a subfield k if we can find elements b1; : : : ; bn in K such that every element of K can be written as a fraction of two polynomials in these elements with coefficients in k. Contents Chapter 1. Affine varieties5 1. The Zariski topology5 2. Irreducibility and decomposition8 3. Finiteness properties and the Hilbert theorems 14 4. The affine category 18 5. The sheaf of regular functions 23 6. The product 25 7. Function fields and rational maps 27 8. Finite morphisms 31 9. Dimension 36 10. Nonsingular points 39 11. The notion of a variety 47 Chapter 2. Projective varieties 51 1. Projective spaces 51 2. The Zariski topology on a projective space 53 3. The Segre embeddings 56 4. Projections 57 5. Elimination theory and closed projections 58 6. Constructible sets 62 7. The Veronese embedding 63 8. Grassmannians 65 9. Multiplicities of modules 70 10. Hilbert functions and Hilbert polynomials 74 Chapter 3. Schemes 79 1. Presheaves and sheaves 79 2. The spectrum of a ring as a locally ringed space 84 3. The notion of a scheme 88 4. Formation of products 94 5. Elementary properties of schemes 95 6. Separated and proper morphisms 99 7. Quasi-coherent sheaves 109 8. When is a scheme affine? 113 9. Divisors and invertible sheaves 116 10. The projective setting 124 11. Cohomology of coherent modules on a projective scheme 127 12. Ample and very ample sheaves 131 13. Blowing up 134 14. Flatness 138 3 4 CONTENTS 15. Serre duality 141 Chapter 4. Appendix 145 1. The basics of category theory 145 2. Abelian categories and derived functors 151 3. Sheaf cohomology 160 Bibliography 173 CHAPTER 1 Affine varieties Throughout these notes k stands for an algebraically closed field k. Recall that this means that every polynomial f 2 k[x] of positive degree has a root x1 2 k: f(x1) = 0. This implies that we can split off the factor x − x1 from f with quotient a polynomial of degree one less than f. Continuing in this manner we then find × that f decomposes simply as f(x) = c(x − x1) ··· (x − xd) with c 2 k = k − f0g, d = deg(f) and x1; : : : ; xd 2 k. Since an algebraic extension of k is obtained by adjoining to k roots of polynomials in k[x], this also shows that the property in question is equivalent to: every algebraic extension of k is equal to k. A first example you may think of is the field of complex numbers C, but as we proceed you should be increasingly be aware of the fact that there are many others: it is shown in a standard algebra course that for any field F an algebraic closure1 F¯ is obtained by adjoining to F the roots of every polynomial f 2 F [x]. So we could take for k an algebraic closure of the field of rational numbers Q, of the finite field 2 Fq, where q is a prime power or even of the field of fractions of any domain such as C[x1; : : : ; xr]. 1. The Zariski topology n Any f 2 k[x1; : : : ; xn] determines in an evident manner a function k ! k. In such cases we prefer to think of kn not as vector space—its origin and vector addition will be irrelevant to us—but as a set with a weaker structure. We shall make this precise later, but it basically amounts to only remembering that elements of k[x1; : : : ; xn] can be understood as k-valued functions on it. For that reason it n n is convenient to denote this set differently, namely as A (or as Ak , if we feel that we should not forget about the field k). We refer to An as the affine n-space over k.A k-valued function on An is then said to be regular if it is defined by some f 2 k[x1; : : : ; xn]. We denote the zero set of such a function by Z(f) and the n n complement (the nonzero set) by Af ⊂ A . If f is not a constant polynomial (that is, f2 = k), then we call Z(f) a hypersurface of An. EXERCISE 1. Prove that f 2 k[x1; : : : ; xn] is completely determined by the reg- ular function it defines. (Hint: do first the case n = 1.) So the ring k[x1; : : : ; xn] can be regarded as a ring of functions on An under pointwise addition and multiplica- tion. (This would fail to be so had we not assumed that k is algebraically closed: q for instance the function on the finite field Fq defined by x − x is identically zero.) 1This can not be done in one step and involves an infinite process which involves in general many choices. This is reflected by the fact that the final result is not canonical, although it is unique up to a (in general nonunique) isomorphism; whence the use of the indefinite article in ‘an algebraic closure’. 2 (qn) Since the elements of any algebraic extension of Fq of degree n ≥ 2 are roots of x − x, we only need to adjoin roots of such polynomials. 5 6 1. AFFINE VARIETIES EXERCISE 2. Prove that a hypersurface is nonempty. It is perhaps somewhat surprising that in this rather algebraic context, the lan- guage of topology proves to be quite effective: algebraic subsets of An shall appear as the closed sets of a topology, albeit a rather peculiar one. n LEMMA-DEFINITION 1.1. The collection fAf : f 2 k[x1; : : : ; xn]g is a basis of a 3 topology on An, called the Zariski topology . A subset of An is closed for this topology if and only if it is an intersection of zero sets of regular functions. PROOF. We recall that a collection fUαgα of subsets of a set X is a basis for a topology if and only if its union is all of X and any intersection Uα1 \ Uα2 is a union of members of fUαgα. This is here certainly the case, for we have U(0) = X n and U(f1) \ U(f2) = U(f1f2). Since an open subset of A is by definition a union n of subsets of the form Af , a closed subset must be an intersection of subsets of the form Z(f). EXAMPLE 1.2. The Zariski topology on A1 is the cofinite topology: its closed subsets 6= A1 are the finite subsets. EXERCISE 3. Show that the diagonal in A2 is closed for the Zariski topology, but not for the product topology (where each factor A1 is equipped with the Zariski topology). So A2 does not have the product topology. We will explore the mutual relationship between the following two basic maps: n I fsubsets of A g −−−−! fideals of k[x1; : : : ; xn]g [\ n Z fclosed subsets of A g −−−− fsubsets of k[x1; : : : ; xn]g: n where for a subset X ⊂ A , I(X) is the ideal of f 2 k[x1; : : : ; xn] with fjX = 0 and n for a subset J ⊂ k[x1; : : : ; xn], Z(J) is the closed subset of A defined by \f2J Z(f). Observe that I(X1 [ X2) = I(X1) \ I(X2) and Z(J1 [ J2) = Z(J1) \ Z(J2): In particular, both I and Z are inclusion reversing. Furthermore, the restriction of I to closed subsets defines a section of Z: if Y ⊂ An is closed, then Z(I(Y )) = Y . We also note that by Exercise1 I(An) = (0), and that any singleton fp = n (p1; : : : ; pn)g ⊂ A is closed, as it is the common zero set of the degree one polyno- mials x1 − p1; : : : ; xn − pn. EXERCISE 4. Prove that I(fpg) is equal to the ideal generated by these degree one polynomials and that this ideal is maximal. EXERCISE 5. Prove that the (Zariski) closure of a subset Y of An is equal to Z(I(Y )). n Given Y ⊂ A , then f; g 2 k[x1; : : : ; xn] have the same restriction to Y if and only if f − g 2 I(Y ). So the quotient ring k[x1; : : : ; xn]=I(Y ) (a k-algebra) can be regarded as a ring of k-valued functions on Y .