NOTES ON CARTIER AND WEIL DIVISORS AKHIL MATHEW Abstract. These are notes on divisors from Ravi Vakil's book [2] on scheme theory that I prepared for the Foundations of Algebraic Geometry seminar at Harvard. Most of it is a rewrite of chapter 15 in Vakil's book, and the originality of these notes lies in the mistakes. I learned some of this from [1] though. Recall: Definition 0.1. A line bundle on a ringed space X (e.g. a scheme) is a locally free sheaf of rank one. The group of isomorphism classes of line bundles is called the Picard group and is denoted Pic(X). Here is a standard source of line bundles. 1. The twisting sheaf 1.1. Twisting in general. Let R be a graded ring, R = R0 ⊕ R1 ⊕ ::: . We have discussed the construction of the scheme ProjR. Let us now briefly explain the following additional construction (which will be covered in more detail tomorrow). L Let M = Mn be a graded R-module. Definition 1.1. We define the sheaf Mf on ProjR as follows. On the basic open set D(f) = SpecR(f) ⊂ ProjR, we consider the sheaf associated to the R(f)-module M(f). It can be checked easily that these sheaves glue on D(f) \ D(g) = D(fg) and become a quasi-coherent sheaf Mf on ProjR. Clearly, the association M ! Mf is a functor from graded R-modules to quasi- coherent sheaves on ProjR. (For R reasonable, it is in fact essentially an equiva- lence, though we shall not need this.) We now set a bit of notation. Definition 1.2. If M is a graded R-module, we let M(k) be the graded R-module defined via M(k)n = Mn+k, with the R-multiplication defined in the obvious man- ner. (This is isomorphic to M as an R-module, but generally not as a graded R-module.) Definition 1.3. Let R be a graded ring as above. The sheaf R](k) on ProjR associated to the graded R-module R(k) is denoted O(k). The sheaf O(1) is called the twisting sheaf of Serre. Example. O(0) is the structure sheaf. This is really most interesting only when the ring R satisfies a reasonable con- dition. Date: December 2, 2010. 1 2 AKHIL MATHEW Proposition 1.4. If R is generated by R1 as R0-algebra, then the O(k) are free. Further, O(k1) ⊗ O(k2) 'O(k1 + k2) for any k1; k2 2 Z. Proof. Indeed, let f 2 R1. Then we have an isomorphism of R(f)-modules R(k)(f) ' R(f) k given by dividing by f . So R](k) = O(k) is free on each SpecR(f). But these cover ProjR as R is generated by R1. The second part follows from the next result applied to M = R(k1);N = R(k2): Proposition 1.5. Let R be a graded ring generated by R1 as R0-algebra. Then if M; N are graded R-modules, the canonical homomorphism Mf ⊗ Ne ! M^⊗ N is bijective. N 1.2. The case of projective space. It is useful to look at this twisting procedure n in case R = A[x0; : : : ; xn] for A a ring, in which case ProjR = PA. In this case, the elements of degree one generate the ring, and the twisting sheaves O(k) are line bundles. We can in fact compute their sections. n Proposition 1.6. Let X = PA for a ring A. Then Γ(X; O(k)) is the space of homogeneous polynomials in x0; : : : ; xn of degree k. (In particular, it is zero for k < 0.) Proof. We begin with a general observation. Let M be a graded R-module over some graded ring R. There is a natural (functorial) map M0 ! Γ(ProjR; Mf) because M0 maps into M(f) for any f 2 R in a natural way. In general, this is not an isomorphism. Here M = R(k), R = A[x0; : : : ; xn]. The elements of degree zero are the elements of degree k in R, i.e. the homogeneous polynomials of degree k. So this is the natural isomorphism, but we should check that it is in fact an isomorphism. n We can do this as follows. PA is covered by open sets D(xi); 0 ≤ i ≤ n. Over D(xi), we have that the global sections are rational functions with xi's in the denom- inator and of homogeneous degree k. But the intersection of any two A[x1; : : : ; xn]xi inside the full ring A[x1; : : : ; xn]x1:::xn is just the polynomial ring. This implies the result. N 0 0 n Corollary 1.7. If m 6= m , then O(m) 6' O(m ) on PA. In particular, there is an injection of groups n Z ! Pic(PA): Proof. If one of m; m0 is nonnegative, this is clear by taking the module of global sections. If both are negative, dualization sends O(m) to O(−m), so we are done in this case too. N NOTES ON CARTIER AND WEIL DIVISORS 3 Example. Sheafification is necessary when tensoring sheaves. Consider F = n O(1); G = O(−1) on X = PA. Then G(X) = 0, so F(X) ⊗O(X) G(X) = 0, but (F ⊗ G)(X) = A 6= 0. n n+m Example (Essential exercise, 15.1C in Vakil). dimA Γ(PA; O(m)) = n . This follows from some easy combinatorics. Indeed, this is the space of homogeneous polynomials in n + 1 variables of degree m. If one has m + n slots, and chooses n of them to be separators, then one can get a unique monomial of degree m in n + 1 n variables. These form a basis for Γ(PA; O(m)), so the claim is proved. Finally, we discuss the explicit construction via transition functions of O(m). n Proposition 1.8. For X = PA, O(m) can be described as follows. The sheaf is free on D(xi); 0 ≤ i ≤ n. The transition functions between D(xi) and D(xj) are m xi given by m (which is a regular function on D(xi) \ D(xj)). xj This explicit description makes it clear that O(m1) ⊗ O(m2) = O(m1 + m2), since tensoring line bundles corresponds to multiplying the changes-of-coordinates. Proof. Follows easily from the explicit construction of the local freeness. The map m O(m)jD(xi) ! OjD(xi) was given by division by xi . The map O(m)jD(xj ) !OD(xj ) m was division by xj . N Clearly, these transition functions satisfy the cocycle condition. n Example. Let us now consider the case of complex projective space CP , and look at topological (as opposed to simply algebraic) line bundles, and see what the O(m) look like. n The tautological line bundle on CP (corresponding to pairs (`; x) where n ` 2 CP and x 2 Cn+1 with x 2 `) corresponds to O(−1). Indeed, on each open set D(xi) we can trivialize this bundle by sending a number t 2 C and a line n n+1 ` = [`0 : ··· : `n] 2 CP to x = t(`0=`i; : : : ; `n=`i) 2 C . So given a line ` = [`0 : ··· : `n] and a point x = (x0; : : : ; xn), we get a number from this by finding t such that x = t(`0=`i; : : : ; `n=`i) or equivalently, t = xi: This is the trivialization on the set D(xi). To from D(xi) to D(xj), one has to go from xi to xj, so the multiplication factor is xj=xi = `j=`i which is the inverse of what it is for O(1). 2. Weil divisors 2.1. First definition. Let X be a noetherian scheme. Definition 2.1. A Weil divisor on X is a formal integer combination of integral codimension one closed subvarieties of X. So the group of Weil divisors is free abelian on the codimension one closed subvarieties. A Weil divisor is called effective if it is a nonnegative linear combination of subvarieties. The support of a Weil divisor is the union of the subvarieties that occur. 4 AKHIL MATHEW Suppose now that X is integral and regular in codimension one. The latter means that for any codimension one point x, the local ring Ox is a discrete valuation ring. Let f 2 k(X) be a nonzero rational function, i.e. a section defined over a dense open subset of X. Definition 2.2. If y 2 X is a codimension one point, then there is a discrete valuation on Oy, and thus on the quotient field k(X). The valuation of f is denoted vy(f) or vY (f) if y is the generic point of the subscheme Y . Remark. This can also be defined for a reduced (not necessarily irreducible) scheme by splitting into irreducible components. We remark that the divisors can be restricted to open sets. Proposition 2.3. vy(f) = 0 for almost all codimension one points y 2 Y . Proof. Assume U ⊂ X is an open set on which f is a unit. Then vy(f) = 0 if y 2 U. But X − U is a finite union of subvarieties of proper codimension. In particular, there are only finitely many codimension one points in X − U. This proves the result. N In view of this, we can make: Definition 2.4. Given f 2 k(X)∗, we define the divisor of f, denoted div(f), to be X vy(f)(Y ); y codim 1 where y is the generic point of Y .