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CHAPTER 0 PRELIMINARY MATERIAL Paul Vojta University Of CHAPTER 0 PRELIMINARY MATERIAL Paul Vojta University of California, Berkeley 13 February 2021 This chapter gives some preliminary material on number theory and algebraic geometry. Section 1 gives basic preliminary notation, both mathematical and logistical. Sec- tion 2 describes what algebraic geometry is assumed of the reader, and gives a few conventions that will be assumed here. Sections 3 and 4 give a few more details on schematic denseness, associated points, rational maps, rational sections of line sheaves, and how the above relate to the relation between Cartier divisors and line sheaves. Sections 5 and 6 discuss the field of definition of a closed subscheme and of a variety, respectively. Section 7 sets the basic notation and gives some fundamental results in number theory. The remaining sections of this chapter describe some more specialized topics in algebraic geometry that will prove useful later: Kodaira's lemma in Section 8, and descent in Section 9. x1. General notation The symbols Z , Q , R , and C stand for the ring of rational integers and the fields of rational numbers, real numbers, and complex numbers, respectively. The symbol N sig- nifies the natural numbers, which in this book start at zero: N = f0; 1; 2; 3;::: g . When it is necessary to refer to the positive integers, we use subscripts: Z>0 = f1; 2; 3;::: g . Similarly, R≥0 stands for the set of nonnegative real numbers, etc. ` ` The set of extended real numbers is the set R := {−∞} R f1g . It carries the obvious ordering. ¯ ¯ If k is a field, then k denotes an algebraic closure of k . If α 2 k , then Irrα,k(X) is the (unique) monic irreducible polynomial f 2 k[X] for which f(α) = 0 . Unless otherwise specified, the wording almost all will mean all but finitely many. Numbers (such as Section 2, Theorem 2.3, or (2.3.5)) refer to the chapter in which they occur, unless they are preceded by a number or letter and a colon in bold-face type (e.g., Section 3:2, Theorem A:2.5, or (7:2.3.5)), in which case they refer to the chapter or appendix indicated by the bold-face number or letter, respectively. 1 2 PAUL VOJTA x2. Conventions and basic results in algebraic geometry It is assumed that the reader is familiar with the basics of algebraic geometry as given, e.g., in the first three chapters of [H 2], especially the first two. Note, however, that some conventions are different here. This book will primarily use the language of schemes, rather than of varieties. The reader who prefers the more elementary approach of varieties, however, will often be able to mentally substitute the word variety for scheme without much loss, especially in the first few chapters. With the exception of Appendix B, all schemes are assumed to be separated. Usu- ally, schemes will also be assumed to be of finite type over some base scheme. We often omit Spec when it is clear from the context; e.g., X(A) means X(Spec A) n n when A is a ring, PA means PSpec A , and X ×A B means X ×Spec A Spec B when A and B are rings. The following definition gives slightly different names for some standard objects. Definition 2.1. Let X be a scheme. Then a vector sheaf on X is a locally free sheaf of OX -modules on X of constant finite rank. Its rank is the value of that constant rank (if X 6= ; ). A morphism of vector sheaves is a morphism of OX -modules on X .A line sheaf is a vector sheaf of rank 1 . A vector sheaf F of rank r on X can be described concretely by giving an open ∼ r cover fUigi2I of X and isomorphisms φi : F ! O for all i 2 I such that for all Ui Ui i; j 2 I , the automorphism −1 r r φi ◦ φ : O ! O Ui\Uj j Ui\Uj Ui\Uj Ui\Uj is given by an element of GLr(Γ(Ui \ Uj; OX )) . If G is another vector sheaf on X , ∼ s with corresponding open cover fVjgj2J and isomorphisms j : G ! O , then a Vj Vj morphism of vector sheaves F ! G is a morphism ρ: F ! G of sheaves such that, for all i 2 I and all j 2 J , the morphism Or ! Os corresponding to ρ Ui\Vj Ui\Vj Ui\Vj via the isomorphisms φi and j is a linear homomorphism of OUi\Vj -modules. A line sheaf is also called an invertible sheaf by many authors. Varieties Not all authors use the same definition of variety. Here we use the following definition. Definition 2.2. Let k be a field. A variety over k , also called a k-variety, is an integral scheme, of finite type (and separated) over Spec k . If it is also proper over k , then we say it is complete.A curve is a variety of dimension 1 . A morphism of k-varieties is a morphism of schemes over Spec k .A subvariety (resp. open subvariety, closed subvariety) of a given variety over k is an integral subscheme (resp. open integral subscheme, closed integral subscheme) of that variety (with induced map to Spec k ). Note that, since k is not assumed to be algebraically closed, the set of closed ¯ ¯ points of a variety X over k is the set X(k) , modulo the action of Autk(k) . Also, the residue field k(P ) for a closed point P will in general be a finite extension of k . PRELIMINARY MATERIAL 3 Remark 2.3. We have not assumed a variety to be geometrically integral. An advantage of this choice is that every irreducible closed subset of a variety will again be a variety, so that there is a natural one-to-one correspondence between the set of points of a variety and its set of subvarieties. This choice also agrees with the general philosophy that definitions should be weak. Since integrality is an absolute definition, if X is a variety over a number field k , then it is also a variety over any smaller number field k0 , since one can compose X ! Spec k with the map Spec k ! Spec k0 to get X to be a scheme over k0 . The latter is discussed more in Section 6. In addition, we have (not just for regular field extensions): Proposition 2.4. The association X 7! K(X) induces an arrow-reversing equivalence of categories between the category of varieties and dominant rational maps over k , and the category of finitely-generated field extensions of k . Proof. The proof of ([H 2], I Thm. 4.4) extends directly to the present case; see also ([EGA], I 7.1.16). Remark 2.5. Let k ⊆ L be fields. Then, for varieties (or schemes) X over k , the sets X(L) satisfy the following basic properties: (a). An(L) = Ln in the obvious way. A similar statement holds for Pn(L). (b). A morphism f : X1 ! X2 of schemes over k induces a natural map fL : X1(L) ! X2(L) : (c). If f is an immersion, then fL is injective. (d). If f is surjective and L is algebraically closed, then fL is surjective. (e). Since the field L has no nilpotents, Xred(L) = X(L). (f). An inclusion L1 ⊆ L2 of fields induces a natural injection X(L1) ,! X(L2). (g). If k0 is an extension field of k such that both k and L are contained in some larger field, then for schemes X over k there is a natural injection 0 0 X(L) ,! (X ×k k )(k L). Note, however, that if L is not algebraically closed, then fL in part (d) need not be surjective; consider for example f : 1 ! 1 defined by z 7! z2 . AQ AQ We note that cohomology is geometric in nature: Let X be a scheme over k , let F be a quasi-coherent sheaf on X , let L be a field containing k , let XL = X ×k L , ∗ let f : XL ! X be the projection morphism, and let FL = f F . Then, by the fact that cohomology commutes with flat base extension ([H 2], III Prop. 9.3), i ∼ i H (XL; FL) = H (X; F ) ⊗k L for all i ≥ 0 . Later chapters of the book will work extensively with schemes of finite type over the ring of integers of a number field. Such schemes have many of the same properties as varieties over a field. We describe one set of such properties here, generalizing a basic result on dimension in ([H], I Thm. 1.8A). 4 PAUL VOJTA Proposition 2.6. Let f : X ! Y be a morphism of finite type, where X and Y are noetherian integral schemes and Y is regular of dimension 1 . Assume that (i) Y has infinitely many points, or (ii) f is proper. Then: (a). For all closed points x of X , the point y := f(x) is a closed point of Y , and the induced extension k(x)=k(y) of residue fields is a finite extension. (b). dim X = tr: deg K(X)=K(Y ) + 1 ; and (c). X is catenary and equicodimensional; in particular, (2.6.1) dim fxg + codim(x; X) = dim X for all points x 2 X . (Here equicodimensional means that all closed points of X have the same codimension in X .) Proof. After passing to an open affine, we may assume that Y is affine, say Y = Spec A.
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