Appendix A Sheaves and Abstract Algebraic Varieties A.1 Sheaves Let X be a topological space. For each open set U eX, consider the set F(U, q of all ((>valued functions on U. This set naturally forms a C-algebra under pointwise addition and multiplication of functions. Definition: A sheafR ofC-valued functions on X assigns to each open set U C X a subalgebra R(U) C F(U, q in a way that is "compatible with both restriction and gluing," that is, • For any open sets Ul C U2 C X and f E R(U2 ), the restriction of f to U1 is in R(U1 ) • • If {U"'}"'EA is an open cover of an open set U ~ X and f E F(U, q is such that flu", E R(U",) for all a, then f E R(U). The functions f E R(U) are called the sections of the sheaf R over the open set U C X. If R is a sheaf of C-valued functions and if f E R(U1 UU2 ), then note that flul and flu2 both have the same restriction to U1 n U2 , namely flu1nu2 • Conversely, if hE R(Ud and 9 E R(U2 ) are such that hl u1nU2 = glu1nU2 , then the map f E F(U1 U U2 , q given by f(x) = {h(X) if x E U1 , g(x) if x E U2 , 146 Appendix A. Sheaves and Abstract Algebraic Varieties is well-defined. Clearly, flu! = hand flu2 = g. So by the second property of sheaves, f E R(U1 U U2 ). We say that hand g glue together to give f. In a similar way, we can define a sheaf of JR.-valued functions, or a sheaf of functions with values in any ring, or even in any set. Examples of sheaves of functions: • Regular functions on a quasi-projective variety V form a sheaf Ov of C-valued functions. The sheaf associates to each open set U C V the C-algebra Ov(U) of regular functions on U. • Continuous JR.-valued functions on a topologiCal space form a sheaf of JR.-valued functions. • COO-functions on a smooth manifold form a sheaf of JR.-valued functions. • HolomorphiC functions on a Riemann surface form a sheaf of C-valued functions. Definition: The sheaf Ov of regular functions on a quasi-projective variety V is called the structure sheaf of the variety. The structure sheaf Ov determines V (up to isomorphism), even if we have only limited information about Ov. For example, in Section 4.3, we proved that for an affine variety, the ring of global sections of Ov is the coordinate ring qV], which in turn recovers the affine variety up to iso­ morphism. In other words, an affine variety is determined by the global sections of its structure sheaf. Only slightly more difficult is the fact that every quasi-projective variety is determined by the rings of sections of the structure sheaf on any affine cover, together with the restriction maps Ov (Ui ) -+ Ov (Ui n Uj ), which recover for us the way these affine pieces are glued together. Later in this appendix we will define an abstract variety, whiCh will be determined by partial information about its structure sheaf in a similar way. This situation is unique to algebraic geometry: A manifold is not determined by such limited information about its sheaf of continuous (or differentiable, or complex holomorphic) functions. For each open set U C X, a sheaf R has a natural restriction Rlu to U. The sections of RI u over an open set U' C U are just the sections R(U') of the original sheaf. Some caution is in order: The ring of sections R(U) and the restriction sheaf Rlu are two different objects; R(U) is a ring, while Rlu is a sheaf (assigning rings to open sets of U). For an open subset V of a quasi-projective variety W, the structure sheaf Ov agrees with the restriCtion of the sheaf Ow to the open set V. A topological space, together with a sheaf of C-valued functions on it, is an example of a ringed space. An understanding of ringed spaces is essential A.I. Sheaves 147 for eventually mastering the definition of a scheme, so we introduce the definition here. Definition: A sheaf of rings R on a topological space X assigns to each open set U C X a ring R(U) in such a way that the following axioms are satisfied • If U1 C U2 , then there is a ring homomorphism R(U2 ) --+ R(U1 ). This map is called "the restriction map from U2 to U1 ," and the image of any element f E R(U2 ) under this map is denoted by flu,. • If U1 C U2 C U3 , then the restriction map R(U3 ) --+ R(Ud is the composition of the restriction maps R(U3 ) --+ R(U2 ) --+ R(U1 ). • If {UoJ"'EA is an open cover of an open set U C X and {g"'}"'EA is a collection of elements g", E R(Uc,) such that for all indices 00, /3, g",lu"nu!3 = g,elua nu!3' then there exists a unique g E R(U) such that glua = g", for all 00. A topological space X, together with a sheaf of rings on X, is called a ringed space. The rings R(U) in the definition above are just abstract rings: They need not be rings of functions on the set U. In particular, the word "restriction" above should not be interpreted literally as the restriction of functions. Our sheaves of C-valued functions are examples of sheaves of rings on the corresponding spaces. In fact, they are all sheaves of C-algebras, since each ring R(U) is actually a C-algebra. Although an abstract sheaf of rings is not necessarily a sheaf of functions, one should think of every sheaf of rings as very much like a sheaf of functions. The third axiom in the definition of a sheaf of rings-also called the sheaf axiom-guarantees that the elements of R(U) really behave like functions: They are uniquely defined by their values on any open cover of U. It is easy to check that every sheaf of C-valued functions is a sheaf of rings. We could also define a sheaf of Abelian groups, a sheaf of sets, a sheaf of algebras, or even a sheaf of objects in almost any category. Just strike out all occurrences of the word "ring" in the preceding definition and replace it by the word "Abelian group," "set," or "algebra." A topological space may come equipped with several different sheaves of rings or algebras. For example, on cn with its usual Euclidean topology, we have not only the sheaf of continuous functions, but also the sheaf of holomorphic functions. We can also equip the set cn with the Zariski topology, where we have the sheaf of regular functions. Definition: Let Rand S be two sheaves of rings on a space X. A map of sheaves of rings 148 Appendix A. Sheaves and Abstract Algebraic Varieties consists of a ring map R(U) GJ!!) S(U) for each open set U C X such that whenever U1 C U2, the following diagram commutes: G(U2) R(U2) • S(U2) j j R(U1 ) • S(Ud G(Ud where the vertical maps are the restriction maps. If the sheaves of rings are actually sheaves of (C>algebras, the maps are furthermore required to preserve the ([:::-algebra structure, that is, each R(U) GJ!!) S(U) must be I[>linear. It does not make sense to speak of maps of sheaves when the sheaves are on two different topological spaces. However, given a continuous map X ----* Y of topological spaces, there is a way to define a sheaf of rings on Y from any sheaf of rings on X. Definition: Given a sheaf R on a topological space X and a continuous map X ~ Y of topological spaces, the push-forward f* R of R is the sheaf on Y defined as follows. For each open set U C Y, If R is a sheaf of rings on X, then f* R is a sheaf of rings on Y. Definition: A map of ringed spaces (X, Ox) ----* (Y, Oy) is a pair (F, F#) consisting of a continuous map of topological spaces X ~ Y and a map F# of sheaves of rings on Y, Oy ---+ F*Ox. Example: Let V ~ W be a map of quasi-projective algebraic varieties. Then there is a naturally induced map of ringed spaces (V, Ov) ----* (W, Ow) where Ov (respectively Ow) is the sheaf of regular functions on V (respec­ tively W). The map of topological spaces is F, and the map Ow ----* F*Ov A.I. Sheaves 149 of sheaves of rings is defined by the pullback: For each open set U C W, Ow(U) ---+ Ov(F-1 (U)), 9 r------+ F# (g) = 9 0 F. This idea works more generally, as the next example shows. Example: If X ~ Y is any continuous map of topological spaces, there is always a morphism of ringed spaces (X,Fx) ----t (Y,Fy), where Fx is the sheaf of CC-valued functions on X and Fy is the sheaf of CC-valued functions on Y. Indeed, the map of sheaves Fy ----t F*Fx is defined using the pullback, Fy(U) ----t F x (F-1 (U)), 9 r------+ goF. If Fx and Fy instead denote the sheaves of continuous CC-valued functions on X and Y, then (X, F x) ----t (Y, Fy) will be a morphism of these ringed spaces.