A note on the relationship between Weil and Cartier divisors

Abstract

Using a generalized equivalence relation, a subquotient of the group of Weil divisors is shown to be isomorphic to the group of Cartier divisors modulo linear equivalence for a reduced subscheme of a projective space over a field. A difficulty of the nonreduced case is discussed. Let X, C be a subscheme of a projective space over a field. A generalized equivalence relation is defined on the group of Weil divisors, and if X, (9 is reduced, a corresponding subquotient is shown to be isomorphic to the group of Cartier divisors modulo linear equivalence. The generalized equivalence for curves appears in [5]. Projectivity may be replaced by the conditions that any finite number of points of X lie in an affine open of X and that the nonregular locus of X, (9 is closed. An example is given which shows one difficulty of the nonreduced case. By a prime ideal p of (9 is meant a subsheaf p of ideals of (9 such that for every open U C X, 1(U, p) is a prime ideal of F'(U, (0). These are the points of X by the usual correspondence. For reducible schemes, the notation for zero-divisors developed in [2] is used. A zero prime ideal N of (9 is a proper prime ideal of (9 such that for each open U C X, F'(U, N) is either ['(U, (9) or consists entirely of zero divisors of ['(u, (9). A divisorial prime ideal p of (9 is a prime ideal p of (9 which contains a zero prime ideal N of (9 such that p/N is of height one in O9/N. 'Let Q(9 denote the total quotient sheaf of (9, and let K = I7(X, QC). 'Let I) = J(X, (Q(9)*/(D ), the group of Cartier divisors, let 83 = F(X, (Q(9)*)/JT(X, (9), and let ( = M)(g. is the set of principal Cartier divisors, and $ defines linear equivalence on M.) Let (/) denote the principal Cartier divisor defined by f c K*. Received by the editors January 31, 1973 and, in revised form, May 3, 1974. AMS (MOS) subject classifications (1970). Primary 14C20; Secondary 14B05, 14C10.