Math 216A. A gluing construction of Proj(S)

Math 216A. A gluing construction o Proj(S) 1. Some basic deinitions Let S = n 0 S n be an N-graded ring (we ollows French terminology here, even though outside o France it is commonly accepted that N does not include 0). Morphisms between N-graded rings are understood to respect the grading. The irrelevant ideal is S + = n>0 S n ; keep in mind that we allow S 0 to have a nontrivial ideal theory (that is, it need not be a ield). An element S is homogeneous i S d or some d, and then d is unique i 0; we call d the degree o (when 0), and we consider 0 as having arbitrary degree. Note that the equation deg(g) = deg() + deg g is valid even i one o, g, or g vanishes, using the convention that 0 may be considered to have arbitrary degree. For example, S + is exactly the set o elements (including 0) with positive degree. For a general element S, the homogeneous parts o are the projections d o into each S d (so d = 0 or all but initely many d). An ideal I in S is homogeneous i an element = n 0 n o S lies in I i and only i each homogeneous part n lies in I. It is a simple exercise (inducting on degrees) to check that an ideal generated by homogeneous elements is a homogeneous ideal, and that homogeneous ideal I in S is prime i and only i it is a proper ideal and g I I or g I or homogeneous, g S. It is also clear that the kernel o a morphism o N-graded rings is a homogeneous ideal, and that or any homogeneous ideal I o S there is a natural N-grading on S/I. Deinition 1.1. Let S be an N-graded ring. The topological space Proj(S) has underlying set Proj(S) = {p a homogeneous prime such that S + p}, and the closed sets are the loci V (I) = {p Proj(S) I p} or homogeneous ideals I o S (context will prevent conusion with the analogous V (I) notation or aine schemes). It is easy to check that the V (I) s do satisy the axioms to deine the closed sets or a topology on Proj(S) (the empty set is V (S) and Proj(S) = V (0)). A homogeneous prime p ails to contain I i and only i there exists a homogeneous element I that does not lie in p (here we use crucially that I is homogeneous). Thus, a base o open sets or the topology on Proj(S) is given by loci D + () = {p Proj(S) p} = Proj(S) V (S) or homogeneous S. A crucial act is that it is even enough to take with positive degree: Lemma 1.2. A base o open sets or the topology on Proj(S) is given by loci D + () or homogeneous S +. Proo. Consider a homogeneous S and a point p D + (). We need to ind a homogeneous element g S + such that p D + (g) D + (). Since p does not contain S + (by the deinition o Proj(S)), there exists h S + not in p. Thus, the condition p implies h p, and h is homogenous with positive degree (possibly even h = 0) since both and h are homogeneous and deg h > 0. We conclude that p D + (h), and clearly D + (h) D + (). Beware that Proj(S) is generally not quasi-compact! For example, Proj(k[x 1, x 2,... ]) with ininitely many indeterminates o degree 1 is not quasi-compact, as it is covered by opens D + (x i ) and there is evidently no inite subcover. This issue is best understood as ollows: 1

2 Theorem 1.3. For an N-graded ring S, Proj(S) is empty i and only i all elements o S + are nilpotent. More generally, or positive-degree homogeneous elements and { i } i I in S, D + () D + ( i ) i and only i some power o lies in the homogeneous ideal generated by the i s. In particular, a collection o D + ( i ) s with all deg i > 0 covers Proj(S) i and only i every element o S + has some power lying in the homogeneous ideal generated by the i s. The contrast with Spec is o course that Spec(A i ) s cover Spec A i and only i the i s generate the unit ideal. The intererence o S + in the analogous covering criterion or Proj, coupled with the possibility that S + might not be initely generated, is the reason why Proj(S) can ail to be quasi-compact. On the other hand, in most interesting situations the ideal S + is initely generated and hence Proj(S) is quasi-compact. However, we note that some undamental constructions o Mumord in the study o moduli o abelian varieties rest crucially on the use o non-quasi-compact Proj s. Proo. Let I be the homogeneous ideal generated by the i s, so the complement o D + ( i ) is the set o p Proj(S) that contain I. Hence, we need to determine when D + () is disjoint rom the set o such p s, or equivalently when every p Proj(S) that contains I also contains ; we want to show that this condition is exactly the condition that a power o lies in I. Passing to the N-graded S/I, we are reduced to proving that a homogeneous S + lies in p or all p Proj(S) i and only i is nilpotent; keep in mind that has positive degree. One direction is obvious, and conversely we must prove that i S + is homogeneous o degree d > 0 and is not nilpotent, then there exists a homogeneous prime p such that p (and so S + p too, so p Proj(S)). We will make use o an auxiliary construction that will play an important role later. Let S (d) = n 0 S dn (so S (d) = S i d = 1). This is naturally an N-graded ring with vanishing graded pieces in degrees not divisible by d. Consider the localized ring (S (d) ) ; since (S (d) ) = S (d) [T ]/(1 T ), by assigning T degree d we see that (S (d) ) naturally has a Z-grading (with vanishing terms away rom degrees divisible by d). For example, s/ n is assigned degree deg(s) nd or homogeneous elements s S (d). Let (S (d) ) () (S (d) ) denote the direct summand o degree-0 elements in the Z-graded (S (d) ). This is a ring, and i is not nilpotent in S then it is not nilpotent in S (d), so then (S (d) ) 0 and hence the subring (S (d) ) () is nonzero. It then ollows that there exists a prime ideal q in (S (d) ) (). We will use this to construct a homogeneous prime p in S (d) that does not contain (and so in particular does not contain S (d) + since deg > 0); the ideal generated by the homogeneous a S such that a d S (d) lies in p is then readily checked to be a homogeneous prime ideal o S that does not contain (this rests crucially on the act that membership in the homogeneous p may be checked on component-parts). Let p be the contraction o q(s (d) ) under S (d) (S (d) ). The ideal p o S (d) does not contain, since otherwise q(s (d) ) would contain the degree-0 element 1, which is absurd since (q(s (d) ) ) (S (d) ) () = q is a proper ideal. To check that p is homogeneous prime, irst observe that (by contstruction) q(s (d) ) is a homogeneous ideal o the Z-graded (S (d) ), so p is a homogeneous ideal o the N-graded S (d). Hence, to veriy primality it is suicient to work with homogeneous elements. That is, we consider homogeneous a, a S (d) with respective degrees dn and dn and we assume aa p. Our goal is to prove a p or a p. Since aa p, the homogenous image o aa in (S (d) ) is contained in q(s (d) ), so aa = (x/ e ) r with r Z, x S kd, and x/ e q (S (d) ) (). Thus, by comparing degrees we get dn + dn = dr, so n + n = r. Hence, aa / r = (a/ n )(a / n ) (S (d) ) is a product o terms with degree 0. However, a a n = x n e (S(d) ) () (q(s (d) ) ) = q, so by primality o q in (S (d) ) () we conclude that at least o the degree-0 elements a/ n or a / n lies in q! Hence, either a or a in S (d) map into q(s (d) ) upon inverting, so by deinition either a or a lie in p.

3 2. First steps towards a scheme structure For homogeneous S +, we get an open set D + () Proj(S) consisting o those p Proj(S) that do not contain. These are a base o open sets, and we claim that D + () is naturally homeomorphic to Spec S (), where S () S is the degree-0 part o the Z-graded localization o S at the homogeneous. To deine a homeomorphism ϕ : D + () Spec S (), to each p D + () we associate the prime ideal p () = (ps ) S () Spec S () ; this is prime because it is the contraction o the prime ps = p o S under the ring map S () S (note that p is prime since p is a prime o S not containing ). Theorem 2.1. The map ϕ : D + () Spec(S () ) is a homeomorphism. Proo. For any homogeneous ideal a o S, we generalize the above operation on homogeneous prime ideals by deining ϕ(a) = (as ) S (). For any p D + (), we claim (1) ϕ(a) ϕ(p) a p. Once this is proved, it will ollow that ϕ is at least injective. The ( ) implication is obvious, and or the converse it suices to prove that i a a is a homogeneous element then a p. Let n = deg a 0 and let d = deg > 0. It ollows that a d n as S () = ϕ(a) ϕ(p) = ps S (), so there exists a homogeneous x p such that a d / n = x/ m in S with md = deg(x). Thus, or some e 0 we have e ( m a d n x) = 0 in S, and since p we must have m a d n x p. However, x p, so m a d p. Since p is prime, p, and d is positive, we conclude a p as desired. This completes the proo o injectivity or ϕ. Once we prove ϕ is surjective, and hence is bijective, (1) implies ϕ(v (a) D + ()) = V (ϕ(a)). Hence, or any ideal b o S (), the preimage a o bs in S is a homogeneous ideal satisying ϕ(a) = b. We may thereore conclude that every closed set V (b) in Spec S () corresponds (under the bijection ϕ) to a closed set V (a) D + () in D + (). However, all closed sets in D + () (with the subspace topology rom Proj(S)) have such a orm or some a, so we thereby get that ϕ is a homeomorphism. It remains to check that ϕ is surjective. A key observation is that the natural map (S (d) ) () S () is an isomorphism. The basic idea is that a degree-0 element in S () must have the orm x/ n with homogeneous x o degree deg(x) = nd S nd, so x is in S (d) ; the straightorward details are let to the reader (hint: equality o subrings o S ). Via this identiication, any prime ideal o S () may be considered as a prime ideal in (S (d) ) (). However, in the proo o Theorem 1.3 it was proved (check!) that every prime ideal q o (S (d) ) () has the orm ϕ(p) or some homogeneous prime p o S not containing (that is, or some p D + ()). Let us now write ϕ : D + () Spec(S () ) to denote the homeomorphism constructed above, with S + any positive-degree homogeneous element (so ϕ (p) = ps S () ). We shall use this homeomorphism to endow D + () with a structure o aine scheme, using the structure shea on Spec(S () ). In view o the act that the D + () s orm a base o opens in Proj(S), the key issue is to identiy S () as the ring o sections

4 on the open subset D + () Proj(S), and to this end it is useul to note that S () may be described entirely in terms o the subset D + () Proj(S) and the ring S without mentioning : Theorem 2.2. For homogeneous S +, let T be the multiplicative set o homogeneous elements g S such that g p or all p D + () Proj(S) (despite the notation, T only depends on D + () and not on ). The natural map S () (T 1 S) 0 to the degree-0 part o the Z-graded T 1 S induced by S T 1 S is an isomorphism. Proo. Let d = deg > 0. For injectivity, suppose x S is homogeneous o degree nd and the degree-0 element x/ n S maps to 0 in T 1 S. Hence, there exists g T such that gx = 0 in S. Replacing g with g d i necessary, we can assume deg g = md. Thus, (g/ m )(x/ n ) = 0 in S, and hence this equality holds in S (). By the deinition o T, or all p D + () we have g p, so g/ m is not contained in the prime ideal ϕ (p) = (ps ) S () o S () (as, g p). But ϕ is bijective onto Spec S (), so g/ m S () is not contained in any primes. It ollows that g/ m S () is a unit, so the vanishing o (g/ m )(x/ n ) in S () orces x/ n = 0 in S (). This gives exactly the desired injectivity. Now choose g T and x S with deg(x) = deg(g), so x/g (T 1 S) 0 makes sense. We seek a homogeneous a S o some degree nd (or some n 0) such that a/ n S () maps to x/g in T 1 S. We may replace x with g d 1 x and g with g d to get to the case deg g = md or some m 0. Thus, using the deinition o T and the bijectivity o ϕ we see that g/ m S () is not contained in any prime ideals, so it is a unit. In the degree-0 part o the Z-graded T 1 S we have x g = m g g m x g = m g so (g/ m ) 1 (x/ m ) S () maps to x/g (T 1 S) 0. This proves the desired surjectivity. x m, 3. A scheme structure on Proj(S) By Theorem 2.2, whenever, h S + are homogeneous elements such that D + (h) D + () inside o Proj(S) we have (by the deinitions!) T T h inside S, and so we get a canonical map (2) S () = (T 1 S) 0 (T 1 S) 0 = S (h) on degree-0 parts induced by the map T 1 S T 1 h S o Z-graded localizations. We may thereore consider the diagram o topological spaces D + () ϕ h Spec((T 1 S) 0 ) D + (h) ϕ h Spec((T 1 h S) 0) where the let column is the inclusion within Proj(S). One readily checks (upon reviewing the deinitions o the various maps) that this diagram commutes with the right side an open embedding, ultimately because the canonical equality (S () ) h deg / deg h = S (h) = (S (h) ) deg h /h deg inside o S h (check!) and the act that deg h /h deg S (h) (since T T h ) implies that (2) induces an isomorphism (S () ) h deg / S (h). deg h Clearly D + () D + (g) = D + (g), and by taking h = g above we see that this open subset o D + () is carried by ϕ onto the open subset Spec((S () ) g deg / deg g) Spec(S ()).

5 Likewise, as an open subset o D + (g) it is carried by ϕ g onto the open subset Spec((S (g) ) deg g /g deg ) Spec(S (g)). We now have put three scheme structures on D + (g), namely Spec S (g) and the two as basic opens in Spec S () and in Spec S (g). These three structures are identiied by means o the ring isomorphisms (3) (S () ) g deg / deg g S (g) (S (g) ) deg g /g deg that are really equalities as subrings o S g. Consequently, the cocycle condition or gluing is satisied (it comes down to transitivity or equality among three subrings o S (gh) or any three homogeneous, g, h S + ), so we may glue the structure sheaves O Spec(S() ) over the D + () s via (3). That is, we are gluing the Spec S () s (as ringed spaces) along the Spec S (g) s, where the underlying topological space Proj(S) o the gluing was made at the start. The glued structure shea over P = Proj(S) will be denoted O P, and so the ringed space (P, O P ) is covered by open subspaces (D + (), O P D+()) Spec(S () ) or homogeneous S +. Hence, (P, O P ) is a scheme. Deinition 3.1. Let S be an N-graded ring. The scheme Proj(S) is the topological space denoted Proj(S) above, equipped with the unique shea o rings O P whose restriction to D + () is O Spec(S() ) (using ϕ ) or all homogeneous S +, with the overlap-gluing isomorphism O Spec((S() ) g deg / deg g ) = O Spec(S() ) D+() D +(g) O Spec(S(g) ) D+(g) D +() = O Spec((S(g) ) deg g /g deg ) deined by the isomorphism Spec((S () ) g deg / g) Spec((S (g)) deg deg g /gdeg ) arising rom the canonical ring isomorphism in (3) or homogeneous, g S +. By Theorem 1.3, we obtain a useul alternative description: Corollary 3.2. Let { i } be a collection o homogeneous elements in S + such that every element o S + has some power contained in the ideal generated by the i s. The scheme Proj(S) is obtained by gluing the aine schemes Spec(S (i)) along the open aine overlaps Spec(S (i j)) Spec(S (i)) deined by the isomorphisms S (i j) (S (i)) deg i j / deg j. i Remark 3.3. We emphasize that there is content in this construction, namely that the above ring isomorphisms satisy triple overlap compatibility; this is most painlessly seen in terms o a triple equality o subrings o S (i j k ). Example 3.4. Let S = A[X 0,..., X n ] be an N-graded ring by putting A in degree 0 and declaring each X i to be homogeneous o degree 1. It ollows that Proj(S) is covered by the opens D + (X i ) = Spec S (Xi) = Spec A[X 0 /X i,..., X n /X i ] or 0 i n, and the gluing isomorphism is determined by the isomorphism (S (Xi)) Xj/X i (S (Xj)) Xi/X j deined by X k /X i (X k /X j ) (X i /X j ) 1 or k i. These are exactly the standard ormulas that express projective n-space as the gluing o n + 1 copies o aine n-space along certain open overlaps deined by non-vanishing o various coordinate unctions. Inspired by the above example, or any ring A we deine projective n-space over A to be P n A = Proj(A[X 0,..., X n ]) with the usual grading on A[X 0,..., X n ]. This is naturally a scheme over Spec A since each basic open aine D + () is naturally an A-scheme (as A has degree 0 in the N-grading being used) and the open-aine gluing data is one o A-algebras (more generally, Proj(S) is always naturally a scheme over Spec S 0 ). As a particularly degenerate example, we have P 0 A = Spec A[X 0] (X0) = Spec A.

6 Remark 3.5. I we assign A[X 0,..., X n ] an N-graded structure by putting A in degree 0 and assigning X i some positive degree d i, the resulting N-graded rings are generally not isomorphic as N-graded rings or dierent n-tuples d = (d 0,..., d n ), and their A-scheme Proj s (so-called weighted projective n-spaces over A with weights d) are generally not isomorphic to each other.