Schemes

Schemes We next introduce locally ringed spaces, affine schemes, and general schemes. References: Hartshorne II.2, Eisenbud-Harris I.1, EGA 1.1. 1 Ringed and locally ringed spaces A ringed space is a topological space X equipped with a sheaf OX on X with values in Ring (called the structure sheaf ). This definition isn’t so useful because it doesn’t force the topology to have much to do with the ring structure; for instance, any ring can be viewed as a ringed space on a one-element topological space. A more useful notion is that of a locally ringed space. This is a ringed space in which for each x ∪ X, the stalk OX,x of OX at x is a local ring, i.e., a ring with a unique maximal ideal mX,x . (The zero ring is not a local ring!) For example, suppose X is a manifold and let OX be the sheaf of real-valued continuous functions. We check that (X, OX ) forms a locally ringed space. Given x ∪ X, let mX,x be the ideal of OX,x consisting of germs of functions taking the value 0 at x. This is clearly an ideal, and the quotient OX,x /mX,x is certainly contained in R. Since X is a manifold, the quotient is nonzero, so mX,x is indeed a maximal ideal of OX,x . To check that it is the unique maximal ideal, it suffices to check that any f ∪ OX,x not contained in mX,x is a unit in OX,x . For such an f , f (x) is some nonzero real number, so we can find an open subinterval I ∩ R such that f (x) belongs to I but 0 does not. Represent f by a continuous function on some open subset U of X containing x, which I’ll also call f . The key point is that by continuity, V = f −1 (I) is again an open subset of X containing x, and f takes nonzero values everywhere on V . Hence there exists a multiplicative inverse g of f on V , which is necessarily continuous. Similarly, a smooth manifold, complex manifold, or abstract algebraic variety equipped with the obvious sheaf is a locally ringed space. For any x ∪ X, the quotient OX,x /mX,x is a field. We denote it by �(x) and call it the residue field of x. In the aforementioned examples, the residue fields of all of the points of x are the same (either R, C, or a prescribed algebraically closed field), but that will not be the case for schemes! I’ll talk about morphisms of (locally) ringed spaces later. For the moment, let me at least point out that an isomorphism of (locally) ringed spaces is what you think: a homeomor­ phism of topological spaces and corresponding bijections of sections which commute with restriction. 2 The prime spectrum of a ring The notion of a locally ringed space is a sufficiently broad generalization of manifolds that it admits a meaningful functor from the category of arbitrary (commutative unital) rings. 1 This gives rise to the concept of an affine scheme; to define this, we must first recall the construction of the prime spectrum of a ring. See the exercises for lots of examples. Let R be an arbitrary ring. Following Zariski, we define the prime spectrum of R, denoted Spec(R), to be the set of prime ideals of R. (An ideal p of R is prime if R/p is an integral domain. The zero ring is not an integral domain, so the trivial ideal is not prime.) For a general ring, this is a better idea than using only maximal ideals because a ring homomorphism � : R � S induces a map Spec(S) � Spec(R) taking p ∩ S to �−1 (p). (The latter is prime because � induces an injective map R/�−1 (p) � S/p, so the source is an integral domain.) By contrast, � may not carry maximal ideals of S to maximal ideals of R; for instance, consider � : Z � Q. Again following Zariski, we equip the set Spec(R) with the Zariski topology, in which the closed sets have the form V (I) = {p ∪ Spec(R) : I ∩ p} for I an ideal of R. This is indeed a topology because V (I) → V (J) = V (I ∼ J) = V (IJ) � � � � V (Ii ) = V Ii . i i We will use a special basis of open sets for this topology: the distinguished open sets, of the form D(f ) = {p ∪ Spec(R) : f ∪ / p} for f an element of R. Note that this basis is nice in the sense that the intersection of any two distinguished opens D(f ) and D(g) is again a distinguished open, namely D(f g). Note also that for � : R � S a homomorphism, the induced map Spec(S) � Spec(R) is continuous because the inverse image of D(f ) is D(�(f )). Lemma. Any distinguished open D(f ) of Spec(R) is quasicompact for the Zariski topology. In particular, Spec(R) = D(1) itself is quasicompact. Proof. It is enough to prove that any covering of D(f ) by distinguished open subsets admits a finite subcover. If the sets D(fi ) cover D(f ) (for i running over some arbitrary index set), then the radical of (f ) is contained in the radical of the ideal generated by the f i . In particular, some power of f is in the ideal generated by the fi . But that means that we can write f as a finite R-linear combination of the fi , so those D(fi ) already cover D(f ). For example, if k is an algebraically closed field, then Spec k[x] consists of one point of the form (x − a) for each a ∪ k, plus a point corresponding to the prime ideal (0). The latter is an example of a generic point, a point whose closure is equal to the entire space in question. For the analogous picture of Spec k[x, y], see Hartshorne Example 2.3.4. 2 3 A presheaf of rings We now specify a presheaf of rings on X = Spec(R), but only on the distinguished open subsets. To do this, we must do a bit of work to clean up their description, to account for the fact that prime ideals don’t see the difference between an element of a ring and a power of that element. Lemma. For f, g ∪ R, we have D(f ) ∩ D(g) if and only if some power of f is a multiple of g. Proof. Note that D(f ) = D(f n ) for any positive integer n. Hence if f n is a multiple of g for some n, then D(f ) = D(f n ) is contained in D(g). Conversely, suppose D(f ) ∩ D(g), or in other words, V (g) ∩ V (f ). Recall that the radical of the ideal (g) is the intersection of the prime ideals containing (g). Since V (g) ∩ V (f ), it follows that the radical of (f ) is contained in the radical of (g), so in particular f belongs to the radical of (g). That is, some power of f is a multiple of g, as desired. A multiplicative subset of R is a subset closed under multiplication. For example, Sf = {1, f, f 2 , f 3 , . . . } is a multiplicative subset. A multiplicative subset S is saturated if for any x ∪ R such that some power of x equals an element of S times a unit, we have in fact x ∪ S. For any multiplicative subset S of R, there is a unique saturated multiplicative subset S̃ containing it, formed in the obvious fashon. By the previous lemma, we now have the following. Corollary. For f, g ∪ R, we have D(f ) = D(g) if and only if S̃f = S̃g . Given any multiplicative subset S of R, there is a unique initial object among the Ralgebras in which each element of S has a multiplicative inverse. It is called the localization of R at S, denoted S −1 R. We can construct it as the polynomial ring in one variable xf for each f ∪ S, modulo the relations xf f − 1. Note that there is a canonical isomorphism S̃ −1 R � = S −1 R since they both satisfy the same universal property. In particular, we can write S̃f−1 R � = R[x]/(xf − 1). From now on, write Rf instead of S̃f−1 R. Let D be the set of distinguished open subsets of X = Spec R. Define a presheaf of rings OX on X specified on D as follows. First put OX (D(f )) = Rf ; this is well-defined by the previous corollary. Then note that given an inclusion D(g) ∩ D(f ), we have Rf ∩ Rg , so the universal property of localization gives a canonical homomorphism Rf � Rg . If you want to write this more concretely (but less canonically), apply the lemma above to write f n = gh for some positive integer n, identify OX (D(f )) = R[x]/(xf − 1) and OX (D(g)) = R[y]/(yg − 1), and take the R-algebra homomorphism R[x]/(xf − 1) � R[y]/(yg − 1), 3 x ⊆� f n−1 hy. 4 The fundamental theorem of affine schemes We are now ready to prove what I call the fundamental theorem of affine schemes. I don’t know whether its appearance in EGA 1 is its first. Theorem 1. The presheaf OX on X = Spec R specified on D satisfies the sheaf axiom for coverings of distinguished opens by other distinguished opens. Consequently, it extends uniquely to a sheaf of rings on Spec R. While we’re at it, though, we may as well prove something stronger which we will need later. This proof is basically the same one used to compute the regular functions on an affine algebraic variety. It may also be thought of as an enhancement of the Chinese remainder theorem; indeed, the latter is an immediate corollary (exercise). Theorem 2. Let M be an R-module. Define a presheaf M̃ of abelian groups on X specified on D by the formula D(f ) ⊆� M �R Rf . Then M̃ satisfies the sheaf axiom for coverings of distinguished opens by other distinguished opens. Consequently, it extends uniquely to a sheaf on Spec R. Proof. By replacing R with Rf , we may reduce to checking the sheaf axiom for a cover of X itself by some distinguished open subsets D(fi ). We first verify that the map M � � i M �R Rfi is injective, as follows. Suppose m ∪ M belongs to the kernel of this map. Then the annihilator of m is an ideal of R which cannot be contained in any prime ideal p of R, or else we would have p ∪ D(fi ) for some i, and the image of m in M �R Rfi would be nonzero. Thus 1 · m = 0, so m = 0. This proves the first half of the sheaf axiom; we must now check the glueing property. For this, we remember that X is quasicompact, so we may reduce to checking for a finite cover. Say D(f1 ), . . . , D(fn ) cover X. Suppose that some D(fi ) cover D(f ), and that we h are given elements mi /fihi ∪ M �R Rfi such that mi /fihi and mj /fj j have the same image in Rfi fj . Since there are only finitely many fi , we may take the nonnegative integers hi to be equal to a common value h. For each i, j, we then have (fi fj )gij (fih mj − fjh mi ) = 0 for some nonnegative integers gij . By rechoosing the mi , we can force gij = 0 for all i, j, that is, we now have literal equalities fih mj = fjh mi . Since D(fih ) = D(fi ), the D(fih ) again cover X, so the fih generate the unit ideal. We may now pick g1 , . . . , gn ∪ R such that g1 f1h + · · · + gn fnh = 1. Put m = g 1 m1 + · · · + g n mn . We then have fih m = � fih gj mj = j � fjh gj mi = mi , j so m is an element of M restricting to mi /fih for each i. This completes the proof of the glueing property, so we are done. 4 5 Schemes From now on, we view X = Spec(R) as a ringed space with structure sheaf OX as defined above. Note that for any prime ideal p of R, the stalk OX,p is canonically isomorphic to the local ring Rp (the localization of R at the multiplicative set R \ p). Hence Spec(R) is in fact a locally ringed space. At this point, we make schemes out of prime spectra by glueing, just as we would make manifolds out of open subspaces of Rn . We define an affine scheme to be any locally ringed space X isomorphic to Spec(R) for some ring R; note that the ring R is uniquely determined by the fact that �(Spec(R), OSpec(R) ) = R (from the previous theorem). A scheme is a locally ringed space in which each point has an open neighborhood isomorphic to an affine scheme. Warning: if X = Spec(R) is an affine scheme, each distinguished open subset D(f ) is an affine scheme, namely Spec(Rf ) (exercise). By construction, these form a basis of open sets. However, it is possible for there to be an open subset U of X such that (U, O X |U ) is isomorphic to an affine scheme but U is not distinguished. (Counterexample to appear as an exercise.) 6 Schemes by glueing We often specify nonaffine schemes using glueing data. For instance, if X1 and X2 are two schemes admitting open subsets U1 , U2 which are isomorphic as locally ringed spaces, we can glue along this isomorphism to get a third scheme X. For more than two schemes, though, we must add a cocycle condition to keep the glueing maps consistent. Here is how that works. Let us first specify glueing data for sets. Let (Xi )i�I be a collection of sets. For each pair (i, j) ∪ I × I, let Uij be an open subset of Xi , and suppose that Uii = Xi . Let �ij : Uij � Uji be an isomorphism, and suppose that �ii = idXi . Suppose also that for i, j, k ∪ I, �ij restricts to an isomorphism of Uij ∼ Uik with Uji ∼ Ujk , and the cocycle condition �ik = �jk ⊂ �ij holds on Uij ∼ Uik . (In particular, �ji = �−1 ij .) We would like to identify the Xi with subsets of a single set X in such a way that Uij identifies with Xi ∼ Xj and �ij� identifies with the identity map on Xi ∼ Xj . To do this, first form the disjoint union X � = i�I Xi . Then define a binary relation on X � as follows: for xi ∪ Xi and xj ∪ Xj , we say that xi � xj if xi ∪ Uij , xj ∪ Uji , and �ij (xi ) = xj . The glueing conditions guarantee that this is an equivalence relation, so we may form the quotient X of X � by �; this gives the desired glueing. (Exercise: reformulate this definition in terms of a limit construction.) We next specify glueing data for topological spaces. Set notation as above, except that each Uij must be an open subset of Xi , and each �ij must be a homeomorphism. Using the 5 glueing construction for sets, identify the Xi with subsets of a single set X. We may then use the topologies on the Xi as a basis for a topology on X; in particular, Xi is open in X. We must still check, however, that the given topology on Xi coincides with the subspace topology from X (it is only obvious that the subspace topology is finer). Suppose x i ∪ Xi and V is an open neighborhood of xi in X. There then exists some j such that xi ∪ Xj and V contains an open neighborhood of xi for the topology on Xj . Since xi ∪ Xi ∼ Xj = Uji and the latter is open in Xj , V ∼ Uji also contains an open neighborhood of xi for the topology on Xj . Since �ij is a homeomorphism, V ∼ Uji = V ∼ Uij contains an open neighborhood of xi for the topology on Xi . This proves the claim. We next specify glueing data for (locally) ringed spaces. Set notation as above, except that each Xi now carries a structure sheaf OXi , and each �ij is an isomorphism of (locally) ringed spaces. Using the glueing construction for topological spaces, identify the Xi with open subsets of a single topological space X. By the glueing property for sheaves, we now obtain a sheaf of rings OX , so X may be viewed as a ringed space. Moreover, for x ∪ Xi , we have a canonical identification of OX,x with OXi ,x ; hence if each Xi is a locally ringed space, so is X. We finally specify glueing data for schemes. This is the easy part: set notation as above, except that each Xi is a scheme. Then it is evident that X is also locally isomorphic to an affine scheme, so X is a scheme! (This part also works for manifolds and the like.) 7 Examples of glueing Glueing can be a force for both good and evil. Let’s start with good. Start with any ring R. For i = 0, . . . , n, put Xi = Spec R[x0 /xi , . . . , xi−1 /xi , xi+1 /xi , . . . , xn /xi ]. Define the distinguished open subset Uij = D(xj /xi ) ⊗ Xi . Then there is an obvious isomorphism of Uij with Uji given by identifying xk /xi with (xk /xj )(xj /xi ). It is easy to check that the cocycle condition is satisfied, so we get a scheme PnR , the projective space over R. (An alternate construction of projective space uses graded rings. More on this later.) Now for the evil. Let k be an algebraically closed field. Let X1 and X2 be two copies of Spec k[x]. We may glue these on the open sets obtained by removing the point x = 0 (i.e., the distinguished opens D(x)) to get a rather unpleasant object; it is a line with a doubled point. We would like to formulate a condition that rules out such pathologies. In topology, the Hausdorff condition does the job, but that won’t work for schemes. We need a more category-theoretical notion, which will be provided once we define separatedness. 6