Algebraic Geometry

The following sections are included:

• Preface

• Table of Contents

• Information for the Reader

In these notes, the letter k will be reserved for a fixed algebraically closed field. The term ring will always mean a commutative ring with unity.

Let x₁,⋯,x_n be indeterminates. Then k[x₁,⋯,xn] will denote the polynomial ring over k in n variables, and k(x₁,⋯,x_n) its field of fractions. The elements in k[x₁,⋯,x_n] may be regarded as functions on affine ra-space Aⁿ = kⁿ. By an algebraic set we will mean the set V(Σ) of zeros of some subset Σ of k[x₁,⋯ ,x_n]. It is clear that if a is the ideal generated by Σ then V(a) = VCE). Consequently every algebraic set has the form V(a), where a is an ideal. On the other hand, according to the Hilbert Basis Theorem (Exercise 1.4), every ideal in k[x₁, ⋯ ,x_n] is finitely generated, and so every algebraic set also has the form V(Σ), where Σ is a finite set.…

The purpose of this chapter is to develop some useful commutative algebra. In particular, this will allow us to prove the Nullstellensatz. References for this material are Lang [18], Atiyah and Macdonald [3], and Zariski and Samuel [36] Chapter IV, Section 4.

Let R be a ring. (Recall that the term ring always means a commutative ring with unit throughout these notes.) R is called a local ring if it has a R is called a local ring if it has a unique maximal ideal. Let F be a field. A subring R ⊆ F called a valuation ring of F if 0 ≠ x ∈ Fimplies that either x ∈ R or x^-1 ∈ R. Clearly then F is the field of fractions of R.

Let X ⊆ Aⁿ be an affine variety, and let p = I(X), so that p is a prime ideal of the polynomial ring k [x₁, …, x_n]. In the proof of the weak Nullstellensatz, we have already encountered the quotient ring A = k[x₁, … , x_n]/p. This is the ring generated by the coordinate functions on Aⁿ, where two functions are identified if their difference is in p, i.e. if they agree on X. We may think of A as the ring of functions on X generated by the coordinate functions in the ambient affine space, so it is called the coordinate ring of X.

We recall the notion of transcendence degree. Let K be a field, F/K an extension which we may assume for our purposes to be finitely generated. If ξ₁, … , ξ_n is any set of generators, we may reorder them so that ξ₁, … , ξ_d are algebraically independent, and ξ_d+1, … , ξ_n are algebraic over the subfield K(ξ₁, … , ξ_d). The set ξ₁, … , ξ_d is called a transcendence basis. A transcendence basis is simply a maximal set of algebraically independent elements…

Let us begin with an observation. Let X be an affine variety, and x ∈ X. We consider the local ring . This ring contains all elements of the function field F of X which are regular in some neighborhood of x. Thus, it is particularly adapted to the study of the variety near x. On the other hand, one should not think that the ring contains only local data. For example, F is the field of fractions of , and so the birational equivalence class of the variety is determined by the local ring . If X and Y are affine varieties, and if x ∈ X and y ∈ Y, we expect local rings and to be isomorphic if and only if X and Y are birationally equivalent, and some neighborhood U of x is biregularly isomorphic to some neighborhood V of y. That is, there is a rational map ϕ from X to Y such that ϕ is regular on U and ϕ⁻¹ is regular on V. Thus we may say that the local ring contains a mixture of local and global data, but the global data is birational in nature, while the local data is more precise. Of course we could make a similar observation about the local ring along a closed subvariety Z of X…

We begin with a discussion of the fibers of a morphism. This chapter is strongly influenced by Mumford [21].

Let ϕ : X → Y be a morphism of affine varieties. Let A and B be the respective coordinate rings of X and Y. Recall that ϕ is called dominant if ϕ(X) is dense in Y. It is easy to see that this is equivalent to the induced map ϕ* : B → A being injective. Therefore dim(X) ≥ dim(Y). We are interested in knowing what we can say about the fibers of a dominant map. Thus if y ∈ Y, the fiber over y is ϕ^-1(y), which is a closed subset of Y. As such, it may be decomposed into a finite union of irreducible components, Y₁ ∪⋯∪Y_n. In general, we expect that dim(Y_i) = dim(X) – dim(Y). However, this may not be true without qualification…

Varieties which are not affine arise in a number of ways. The first examples which one encounters are projective spaces. Recall that projective n-space Pⁿ may be constructed as follows . Let V be a finite-dimensional vector space over k. Then P(V) is the set of lines through the origin in V. If dim(V) = n + 1, then we will interpret P(V) as an n-dimensional variety. Since any two n + 1- dimensional vector spaces are isomorphic, we may as well take V = kⁿ⁺¹, and we denote P(V) = Pⁿ…

By a function field of dimension one over our fixed algebraically closed field k, we mean a finitely generated extension field F of transcendence degree one over k. The function field of a curve is such a field; two curves are birationally equivalent if they have isomorphic function fields.

In this chapter, we will establish that every function field of dimension one is the function field of a unique complete nonsingular curve. This is different from the situation for surfaces, where in Chapter 7 we saw that two nonisomorphic complete nonsingular surfaces could be birationally equivalent. We will also show that morphisms of complete nonsingular curves correspond exactly to field homomorphisms, so the study of curves and their mappings is intimately related to the Galois theory of fields…

In this chapter, we will develop some algebra having to do with the behavior of prime ideals when the ground field is extended. This has to do with the fibers of a finite morphism. The algebraic facts which we prove, in addition to the applications to algebraic geometry, are the first theorems which must be proved when one studies algebraic number theory…

In this chapter, we will introduce the powerful technique of completion. In complex analysis, one makes frequent use of the Taylor expansion of a function. Similarly, in algebraic geometry, one may consider expansions in formal power series. In contradistinction to complex analysis, these may not be regarded as giving analytic continuation to a function. Nevertheless, they are a potent tool. The formation of a ring or field of formal power series is an example of the process of completion…

We recall the classical theorem that if M is a compact Riemann surface, and if ω is any meromorphic differential 1-form on M, then the sum of the residues of ω is equal to zero. This may be proved as follows. We start with a canonical dissection of M. For example, if the genus of M is two, we cut it as follows:

The complement of the cut lines is simply connected, and we may now lay the surface out “flat,” on a table of constant negative curvature. It looks like this:

According to Cauchy’s integral formula, the sum of the residues of ω is equal to the integral of (2πi)⁻¹ ω around the perimeter. Observe that each of the cut lines α, β, γ and δ is traversed twice, in opposite directions, so that the integrals cancel. Hence the sum of the residues is zero…

The proof of the Riemann-Roch theorem which we will present is Weil’s adelic proof. This proof was first sketched in 1938 in Weil [29]. Proofs of the Riemann-Roch theorem based on Weil’s adelic approach may be found in Lang [17], Serre [26] and in Weil [33]. (See also [31].) In [33], Weil proved the Riemann-Roch theorem for function fields over a finite field. By contrast, we will prove the Riemann-Roch theorem over an algebraically closed field, and when we need the finite-field version in Chapter 14, we will deduce it from the algebraically closed case…

Elliptic curves are an important topic in number theory. A useful basic reference is Silverman [27]. An elliptic curve is a complete nonsingular curve of genus one. Throughout this chapter, E will be an elliptic curve, and F will be its function field…

Let us recall that the Riemann zeta function is defined by . This series is convergent if re(s) > 1. It has meromorphic continuation to all s. It has a simple pole at s = 1, and is analytic elsewhere. It has the Euler product representation

also valid if re(s) > 1. Finally, there is the functional equation. Let

Then ξ(s) is meromorphic for all s, with simple poles at just s = 0 and s = 1, and ξ(s) = ξ(1 − s)…

The following sections are included:

• References

• Index