AN INTRODUCTION TO AFFINE SCHEMES BROOKE ULLERY Abstract. This paper gives a basic introduction to modern algebraic geom- etry. The goal of this paper is to present the basic concepts of algebraic geometry, in particular affine schemes and sheaf theory, in such a way that they are more accessible to a student with a background in commutative al- gebra and basic algebraic curves or classical algebraic geometry. This paper is based on introductions to the subject by Robin Hartshorne, Qing Liu, and David Eisenbud and Joe Harris, but provides more rudimentary explanations as well as original proofs and numerous original examples. Contents 1. Sheaves in General 1 2. The Structure Sheaf and Affine Schemes 3 3. Affine n-Space Over Algebraically Closed Fields 5 4. Affine n-space Over Non-Algebraically Closed Fields 8 5. The Gluing Construction 9 6. Conclusion 11 7. Acknowledgements 11 References 11 1. Sheaves in General Before we discuss schemes, we must introduce the notion of a sheaf, without which we could not even define a scheme. Definitions 1.1. Let X be a topological space. A presheaf F of commutative rings on X has the following properties: (1) For each open set U ⊆ X, F (U) is a commutative ring whose elements are called the sections of F over U, (2) F (;) is the zero ring, and (3) for every inclusion U ⊆ V ⊆ X such that U and V are open in X, there is a restriction map resV;U : F (V ) ! F (U) such that (a) resV;U is a homomorphism of rings, (b) resU;U is the identity map, and (c) for all open U ⊆ V ⊆ W ⊆ X; resV;U ◦ resW;V = resW;U : Date: July 26, 2009. 1 2 BROOKE ULLERY In order to simplify notation, we will refer to resV;U (f) for f 2 F (V ) as the restriction of f to U, or simply as fjU , and we refer to the elements of F (V ) as the sections over V . We can similarly define presheaves of modules, abelian groups, or even just sets. However, for our purposes, we will only be dealing with presheaves of commutative rings, and from now on, we will assume that all rings mentioned are commutative and have a multiplicative identity 1. In order for a presheaf to be a sheaf, it needs to satisfy an additional condition, called the sheaf axiom. We state this as a definition: Definition 1.2. Let F be a presheaf on a topological space X, and let U ⊆ X be open. Then F is a sheaf if it satisfies the following condition, known as the sheaf axiom: S If U = i2J Ui is an open covering of U and ffigi2J is a set of elements with fi 2 F (Ui) for all i 2 J such that fijUi\Uj = fjjUi\Uj for each pair i; j 2 J, then there exists a unique element f 2 F (U) such that fjUi = fi for all i 2 J. Now we look at a simple example of sheaves over discrete spaces. Example 1.3. Consider the set X = f0; 1g given the discrete topology, and let F be a sheaf over X. The two single-element open sets only contain themselves in their respective open coverings, so they give us no information about F . Instead, let's consider the covering ff0g; f1gg of X. Let f0 2 F (0) and f1 2 F (1). Then, since there is only one section over the empty set, we have f0jf0g\f1g = f0j; = f1j; = f1jf0g\f1g: Thus by the sheaf axiom, there is a unique section g over X such that gjf0g = f0 and gjf1g = f1. That is, F (X) is set theoretically equal to F (f0g) × F (f1g), and the restriction maps are simply the projection maps. More generally, if Y is any space given the discrete topology, it is clear that Q F (Y ) = y2Y F (fyg). Another important feature of a sheaf is its stalks. The stalks describe the space and its sheaf locally, near a given point. We give a more precise definition. Definition 1.4. Let X be a topological space and F a presheaf on X. Let x 2 X. Then the stalk of F at x, denoted Fx, is defined to be 0 1 , G = lim (U) = B (U)C ∼; Fx −! F B F C x2U⊆X @x2U⊆X A U open U open where ∼ is an equivalence relation such that a ∼ b if a 2 F (U); b 2 F (V ) and there is an open neighborhood W ⊆ U \ V such that ajW = bjW : We illustrate this concept with a few more simple examples. Example 1.5. We again consider the space of two elements given the discrete topology. We call it X = f0; 1g: By our definition, we have F0 = F (f0g) t F (f0; 1g)= ∼ : Clearly f0g \ f0; 1g = f0g, and if a 2 F (f0; 1g) then ajf0g 2 F (f0g) so that 0 a ∼ b for some b 2 F (f0g). Thus F0 ⊆ F (f0g). However, if f; f 2 F (f0g) AN INTRODUCTION TO AFFINE SCHEMES 3 0 0 such that f ∼ f , then f = f since resf0g;f0g is the identity on F (f0g). That is, F0 = F (f0g), and similarly F1 = F (f1g). It is again clear that we can generalize to any space Y given the discrete topology: for y 2 Y , Fy = F (fyg). Moreover, if any single point in a space is open, the stalk at the point is simply the sheaf on the set containing only that point. Example 1.6. Now we consider a non-discrete, but still simple, example. Let X = f0; 1g, but this time let the open sets be only ;, f0g, and f0; 1g. From the previous example we see that F0 = F (f0g): Now, it is clear that the stalk at 1 is simply F1 = F (f0; 1g), since f0; 1g is the only open neighborhood of 1. After defining sheaves and stalks, it is natural to define maps between them. Definition 1.7. Let X be a topological space and let F and G be sheaves on X. Then a morphism φ : F ! G of sheaves is a collection of maps φ(U): F (U) ! G (U) where U ⊆ X is open and for every V open in X such that U ⊆ V , the following diagram commutes: φ(V ) F (V ) −−−−! G (V ) ? ? ?resV;U ?resV;U y y φ(U) F (U) −−−−! G (U) In the case where F and G are sheaves of rings, the maps φ(U) are ring homo- morphisms. A morphism of sheaves also induces a morphism of stalks, in this case a ring homomorphism, of the respective sheaves. We denote this morphism φx : Fx ! Gx for x 2 X. 2. The Structure Sheaf and Affine Schemes Now that we have described some of the basics of sheaves, we present a specific sheaf, which we will use throughout the rest of the paper: the structure sheaf. First, we must quickly review a few definitions from commutative algebra: The spectrum of a ring R, denoted Spec R, is a topological space whose under- lying set is the set of prime ideals of R. We give Spec R the Zariski Topology. The closed sets of this topology are of the form V (S) := fp j S ⊆ pg for S an arbitrary subset of R. Given f 2 R, we define the distinguished open set of X = Spec R associated with f to be Xf := Spec R − V (f) = fp 2 Spec R j f2 = pg: The set of distinguished open sets of X, fXf j f 2 Rg, form a basis for the topology on X. Now that we have a topology on X, we can define the structure sheaf OX on X. When the space over which we are dealing is clear, we will denote the structure sheaf simply as O. It turns out that it suffices to define the sections of O over distinguished open sets and the restriction maps between basic open sets. In fact, every sheaf on basis elements of a space Y satisfying the sheaf axiom with respect 4 BROOKE ULLERY to inclusions and coverings can be extended uniquely to a sheaf on Y . For the purposes of this paper, we will omit the proof of this statement. Going back to our space X = Spec R, with basis fXf j f 2 Rg, we set O(Xf ) = p n Rf , the localization of R at f. If Xf ⊆ Xg (that is f 2 (g), or f 2 (g) for some power n), then we define the restriction map resXg ;Xf : Rg ! Rf , as the localization map Rg ! Rgf = Rf . This leads us to the following simple observations: Observation 2.1. If X = Spec R, then O(X) = R. Proof. Consider X1 = Spec R − V (1) = X − ; = X. Then we have O(X) = O(X1) = R1 = R, as desired. Observation 2.2. A point p 2 X = Spec R is closed if and only if p is a maximal ideal of R. Proof. The point p is closed in X if and only if fpg = V (S) for some S ⊆ R if and only if there is no prime ideal properly containing p if and only if p is a maximal ideal of R. Now we are able to construct the stalks of O: Lemma 2.3. Let O be the structure sheaf on Spec R = X, and let p 2 X. Then the stalk at p is Op = Rp, the localization at the prime ideal p.