==== Week 2 Graded rings, Spec R and Proj R ==== Week 1 worked with the informal notion of variety, and introduced the notion of a space with locally defined regular functions. This is important background motivation for Grothendieck's more formal and much more general theory of scheme. A graded ring is R = sum R_d taken over integers d >= 0. Here I almost always assume in addition that R_0 is a field k, and R is generated by finitely elements of weight d_i > 0. Week 2 discussed one or two technical points in the general construction of X = Spec R and its structure sheaf OX (the discussion gets fairly gristly), and the notion of the graded spectrum Proj R. The latter generalises the construction of projective space PP^n = (k^{n+1} \ 0) / k^x as the quotient by the multiplicative group GGm. As in the quotient of an affine variety by a finite group, the idea is to take GGm-invariant open sets _not contained in the vertex_ and pass to their rings of GGm-invariant functions and the affine varieties or affine schemes given by their Spec. ==== A projective variety X in PP^n corresponds to a graded ring. Work first with the case of irreducible variety over k = kbar. affine scheme: Spec R and its structure sheaf. Spec of a ring X = Spec R = set of prime ideals (includes maximal ideals) Zariski topology. The closed sets if X are subsets V(I) = set of prime ideals P such that P > I P contains f means f |-> 0 mod P, f vanishes in field k_P = Frac(R/P) Sheaf of rings: open set U |-> ring Ga(U, OX) restriction maps to open subset U1 in U. Particular case: Xf = principal open set X \ V(f) = Spec R[1/f] These Xf form a basis for the topology: any open is union U = union Xf union taken over f s.t. U, V(f) disjoint. Need a few more words on definition of sheaf, and the stalk of a sheaf at a point: for F a sheaf on X, the stalk of F at a point P in X is the direct limit lim_{->} Ga(U, F) or the set of germs of sections near P. If the restriction maps are injective, the direct lim means union. Otherwise, the set of pairs U,s_U with U a nbd of P and s_U a section of F over U modulo _germ equivalence_ which means identify s_U and s1_U1 if they restrict to equal elements on some smaller neighbourhood. Roughly, a germ of function near P. An element s in FP of the stalk can be defined as a section on an open nbd, and any two such are equal on a smaller nbd ===== The construction of the structure sheaf OX of an affine scheme X = Spec R is a complicated array of \forall \forall \exists \exists \suchthat etc. The crucial logical step is that with this definition, Ga(X, OX) = R and for the principal open sets Xf, also Ga(Xf, OX) = R[1/f]. In other words, the definition is consistent with the starting point. At the end, we will have a sheaf of rings OX on X = Spec R with two basic properties: (i) for every f in R, the principal open set Xf = X \ V(f) has sections Ga(Xf, OX) = R[1/f] (ii) for every prime P in X, the stalk of OX at P is the local ring OX,P = RP. Reminder concerning the definitions: (1) the closed sets of the Zariski topology on X are V(I) = {primes containing I}. The principal open sets form a basis for the open sets. [If you have not seen this before, you need to show that V(I) = intersect V(f) taken over f in I, and that any open set X \ V(I) is the union of Xf.] (2) The localisation RP = S^-1R = directlim R[1/f]. The stalk of OX at P is directlim Ga(U,OX) for U open nbd of P = directlim Ga(Xf,OX) for f notin P [If you have not seen this before: the Xf with f notin P form a basis for the open nbds of P. Thus we can refine a limiting sequence {g_U in Ga(U,OX)} for an arbitrary sequence of open nbds to {g_{Xf} in Ga(Xf,OX)} over basic opens.] Statement: Ga(X,OX) = R [Harts, Ch2, Prop 2.2, or same for modules Lemma 5.3] There is a map R[1/f] -> Ga(X,OX) that localises g/f in R[1/f] to g/f in RP for every P in Xf. First, this is INJECTIVE: In fact, if g/f^n <> 0 in R[1/f], its annihilator is a nontrivial ideal, so there is a prime ideal P of R[1/f] contained in it (existence of primes). Then P in Xf, and g/f^n maps to a nonzero element of RP = OXp at that P. (An element is in the kernel of localisation S^-1 means that s*g = 0.) The injective part only involved finding one P at which the localisation of g/f^n is nonzero, so that was easy. The surjective part involves the full definition of OX. For an open subset U in X = Spec R a section s in Ga(U,OX) means I have sP in AP for every P in X satisfying (local property) there is an open cover of U by principal opens Xh and for each Xh an element g/h^n in R[1/h] that represents sP at P for all P in Xh. Suppose that for a given f in R, some set {h in R} give rise to basic opens Xh that between them cover Xf. This means that for every P in Xf, there is h so that P in Xf. The local property in the definition of OX means I can choose {h in R} so that (1) Xh cover Xf, and (2) sP in AP is represented by g/h^n in R[1/h] for every P. Now (1) together with the familiar (existence of primes) result implies f is in the radical of I = ideal {h}. In fact (1) with the definition Xf = X \ V(f) means that f \not P for any P in Xh so that f in every prime of V(I). Hard point: Let Xf = Spec R \ V(f). If s in Ga(Xf, OX) in the definition of OX, then s in R[1/f] -- this means that Ga(Xf, OX) = R[1/f] as in the starting point description of OX. This is quite similar to "uniformly continuous" in ep-de calculus. We require s to be "continuous" (in the local sense of being covered by open) over the whole of Xf, and conclude it is actually defined once and for all by an element of R[1/f] with the single denominator f. Digression. The proof below and in [Harts, Ch2, Prop 2.2, or for modules and their associated sheaf of OX-modules, Lemma 5.3] is slightly complicated logic that takes care to get correct, but remembering the classical case based on the NSS should help. If we work with the old notion of irreducible variety over k = kbar, the statement is that if s in k(X) is regular at every point of a principal open set Xf in X then s is in the affine coordinate ring k[X][1/f] of Xf. Recall the argument: define the ideal of denominators of s. Then the assumption is that f vanishes on V(Denom(s)). So the NSS implies that f is in the radical, or f^r is a denominator of s, so s in k[X][1/f]. Proof of surjective assertion. Saying s in Ga(Xf,OX) means that for every P in Xf, there is expression s = gi/hi in R_P. The contrapositive of this is that any P that contains one possible denominators hi of an expression s = gi/hi also contains f. Therefore f is contained in every prime that contains one of the hi. Thus f in rad( ideal generated by all the hi ). Because rad(I) = intersection of all primes containing I. (the "scheme version of Nullstellensatz", the first result on existence of primes in commutative algebra). Now f^r in (ideal of hi) so it is in the ideal of finitely many hi, and to treat s we can assume there are finitely many hi with Xhi in Xf, Xf = union Xhi, and s = gi/hi^Ni. The inclusions mean hi in rad(f) and f in rad({hi}). We can replace hi -> hi^Ni or f -> f^N, so the rad. Now gi/hi ~ gj/hj on the overlaps X_{hi*hj}, so (hi*hj)^N*(hj*gi-hi*gj) = 0 and f^N*(hj*gi-hi*gj) = 0. Since there are only finitely many i, we can suppose these holds for fixed N and all i,j. Then g = f^N*hi*gj in R, so g in R[1/f] as required. QED From Spec to Proj Proj R = Spec R \ V( ideal R_+ ) = union Spec (R[1/g])^0 taken over all homogeneous g of degree > 0. This means take gi of degree di > 0 (all of them if your prefer, or just finitely many, or just the xi if you prefer). The construction (R[1/g])^0 means the ring { f / g^n | all f in R_{n*di} } -- allow only powers of g in denominator, and adjust f to be homogeneous of deg n*di so that f/g^n is homogeneous of degree 0, so GGm-invariant. In scheme theoretic terms Proj R is the set of homogeneous prime ideals, (not irrelevant, meaning p not R_+ -primary, so that V(P) in Spec R contains nontrivial GGm orbit). Its Zariski topology and structure sheaf are constructed exactly as for Spec, using GGm-invariant open sets and GGm-invariant fractions as above. === Graded ring R and Proj R === R graded ring Definition ring R = sum R_d taken over d in NN = {0,1,2,..} multiplication R_d1 x R_d2 -> R_{d1+d2} additional assumptions (to make a projective scheme over k) R_0 = field k, R generated by finitely many x_i in R_di => R = k[x0,..xn]/IR where IR is homogeneous ideal. GGm = multiplicative group scheme = GL(1) acts on R, with la \in GGm(S) acting on R by multiplication la^d: R_d -> R_d GGm is a reductive group scheme. Its irreducibles are all 1-dimensional, given by la^d, and every representation is a direct sum of representations. Definition of Proj R Proj R = homogeneous prime spectrum Homogeneous ideal means I = sum In -- the condition is that every element f in I can be written as a sum of parts fn in Rn that are homogeneous of degree n in R, and are still in I. If we already know the affine cone Spec R = CX. The irrelevant ideal is M = sum R_d over d > 0. Every homogeneous element of degree > 0 is in M, and M = Ideal(x0,..xn). (Any ideal I containing M^N is also irrelevant.) Then Proj R = (CX \ V(M)) / GGm. Here GGm is the multiplicative group, or the group of invertible 1x1 matrices. For a ring R, GGm(R) = R^x (the units of R), but I consider GGm as a group scheme defined over all rings. You can write k^x or even CC^x if you prefer. (Lots of people write k^* to mean the _set_ of units of k, but the multiplicative x is better. I write GGm or k^x even when I say CC-star in colloquial speech.) Taking the quotient means: take GGm invariant affine open subset CXf = CX \ V(f) for f in R_d (homogeneous of degree > 0). It has affine coordinate ring R[1/f]. The GGm invariants of R[1/f] are (R[1/f])^0 homogeneous elements of degree 0. ==== Weighted projective space I want to get to the wPPs version of nonsingularity, called quasi-smooth. First, an example like PP^2(1,b,c) with b,c > 0 and coprime. PP(1,2,3) or PP(1,3,5) if you want to be specific. Draw picture of (say) PP(1,3,5). It has 2 "horns" P_y and P_z that are quotient points Py = 1/3(1,5) iso 1/3(1,2) and P_z = 1/5(1,3) GGm-action AA^3 / divided by action ( x |-> la*x, y |-> la^3*y, z |-> la^5*z ). At a point P = (x,y,z) with x <> 0, we can choose x = 1 as a "slice" of the equivalence relation given by the GGm-action. Namely, I can pass to (1, y/x^3, z/x^5) as a preferred representative. That does not work near the y-axis or z-axis. Geometrically, if I try to take a preferred representative of the orbit of GGm by setting y = 1, that plane cuts the y-axis in 1 point, but it cuts all the neighbouring orbits with (x,z) <> (0,0) is 3 points, the cube roots of x or z. Algebraically, the ring of fractions R[1/y] allows denominator y or y^n accompanied by x^i*z^k of weight 3*n. I could take x^3/y and x*z/y^2 and z^3/y^5 as generators of the ring (R[1/y])^0, but there is nothing simpler. The affine piece Spec(R[1/y])^0 has a singularity of type 1/3(1,5) at the origin. A way of seeing this that is convenient for computation is: first take a cube root eta = y^{1/3}. Now eta has weight 1, so in a slightly bigger ring x/eta and z/eta^5 are both homogeneous of degree 0. So there is some kind of slice giving preferred representatives of the orbits. But by taking the cube root we have passed to a Galois extension of R with Galois group ZZ/3 (sometimes better viewed as mu_3). The polynomials that are GGm invariant fractions of R are obtained as the invariants of ZZ/3 action. In summary, where the GGm group action has fixed points, the variety is best thought of as locally a quotient variety. The quasismooth condition is that locally, Proj is of the form 1/r(a1,.. ak) where r = deg y, the "slice" coordinate to be set = 1, and the transverse coordinates xi have degree == a1,.. ak mod r. This is called orbifold. The idea that you have to take the quotient, then remember the coordinates xi of the space before taking the quotient is the beginning of the theory of stacks. There are lots of exercises concerned with constructing interesting quasi-smooth weighted hypersurfaces (or with mild singularities). See separate Addendum on some orbifold curves.