G/N. Let QQ denote the total quotient sheaf of G, and let K = T(X, QL). Let = T(X, (QÖ)/Ö ), the group of Cartier divisors, let 5ß =

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 48, Number 2, April 1975 A NOTE ON THE RELATIONSHIP BETWEEN WEIL AND CARTIER DIVISORS JAMES HORNELL 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, 0 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( G is closed. An example is given which shows one difficulty of the nonreduced case. By a prime ideal p of U is meant a subsheaf p of ideals of U such that for every open U C X, r(i7, p) is a prime ideal of V(U, Ü). 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 G is a proper prime ideal of G such that for each open U C X, V{U, N) is either V{U, 0) or consists entirely of zero divisors of r((7, (J). A divisorial prime ideal p of U is a prime ideal p of (J which contains a zero prime ideal N of 0 such that p/n is of height one in G/N. Let QQ denote the total quotient sheaf of G, and let K = T(X, QL). Let = T(X, (QÖ)/Ö ), the group of Cartier divisors, let 5ß = T(X, (ßO)*)/T(X, 0*), and let S = S/5ß. $ is the set of principal Cartier divisors, and =ß defines linear equivalence on 5).) Let (/) denote the principal Cartier divisor defined by / K. Received by the editors January 31, 1973 and, in revised form, May 3, AMS (MOS) subject classifications (1970). Primary 14C20; Secondary 14B05, 14C10. Key words and phrases. Divisors, Cartier divisors, generalized equivalence relation. Copyright 1975, American Mathematical Society 276

2 WEIL AND CARTIER DIVISORS 277 Let i) C X be the set of nondivisorial prime ideals p of G such that if / is a regular element of G then G p is an embedded prime ideal of G f. Let C_ be the union of JJ and the set of nonnormal (nonregular) divisorial prime ideals. If X, 0 is reduced, then -0 is the set of nondivisorial prime ideals of depth (grade) one, and both and C are finite. Let D be the group of Weil divisors of X, the free abelian group generated by the divisorial prime ideals of G. Let p be a divisorial prime ideal of G, let / e K, let f x and f2 be regular elements of 0 such that / = /j//2, and define 0 \ / 0 p \ (I I p k'if)-lbw7v-^w7^, where the sum is over all zero prime ideals N C p with p/n of height one in G//V. Let the principal Weil divisor defined by / e K* be (/) = 2 AAf)p, where the sum is over all divisorial prime ideals p. Let S be the set of all divisorial prime ideals which are contained in some element of C. Let D\S be the free abelian group generated by those divisorial prime ideals not belonging to S. Let / be the subgroup of D of principal divisors (/) where f K is such that / eü for all q C, consider / as a subgroup of D\S, and let C = {D\S)/I. {D/l is the generalized class group of X, G and / defines a generalized equivalence relation. Letting P be the subgroup of D of all principal divisors (/), D/P is the class group of X, Ü, and P defines linear equivalence on D.) To construct the injection from» to C for X, G reduced, only the following approximation lemma for Cartier divisors is needed. It is a variant of the usual approximation [3, Lemma 3, p. 166]. Recall that if p j,, ps and p are prime ideals of G, then p C p. U Up if and only if p C p. for some i = 1,, s. Let <?,,, <?, also be prime ideals of G. The condition below that p C(pl U Ufs)n(?1U- U i7() is equivalent to p C p. r\ q. for some i and /. Lemma. Eef X, G be a subscheme of a projective space over a field. Let p,,'",p be prime ideals of G, let. D be a Cartier divisor on X, and let q,,, q, be prime ideals of G such that D = (1) for every prime ideal p of with p C (p U U ps) H (q U U <? ). TÂew z7;ere is an

3 278 JAMES HORNELL element f of K such that (/) = D for i = 1,, s and (/) = (1) for i i i 'z / = 1,,/. Proof. First assume the Lemma is true for s = 1, and induct on s. Assuming true for s - 1, there is an /" e K with (/ ) = (1) for i = 1, t t and (/") =D for z= l,...,s-l. Then (D - (/")) = U) forz = l,.--, i i P i i s - 1, and (D - if")) = D for i = 1, *, t. It follows that if p is a prime ideal with pc(pt u uí,_i Ufj u--- UfP npa then (D - (/")X = dx. Thus there is an /' K* such that (/') = (D-(/")).,(/'). =(D. for f-1,--., 5-1, and (/') =(1) for i = 1,, fs p -> </ ^ f. Let /=/"//' K*. Now let s = 1 and p = p.. Let A = r((7, G) where U is an affine open subset of X meeting each irreducible component of X and containing the points q., ', q and p which are to be considered as prime ideals of A. Let g. and g be regular elements of A such that D = (g,/g2)fl* Let P., ", P be the associated prime ideals of Ág for which g,/g2 A p. P. is not a zero prime ideal because g2 P.. If P. C p then Dp = z l l ' i igjg2)p (l)p and P. ( (? U U q, and let e. e P;. be a regular element z' i of A such that e i q} U U c^(. Or, if P p, let e{ P. be a regular element of A with e. p. There is an integer n > 1 such that (et e )"g. Ag2. Let j = (ex e^fgjgj1 e A, and let Ä2 = (ßj erf. Then Dp = {h./h ), and no isolated prime ideal of Ah. or A A is contained in p O (?1U U9<). Let Q., *, Q be the associated prime ideals of AO in A. For / = 1, 2, let k. be an element of p contained in all the isolated prime ideals of A h. such that for i = 1,, t, k. q. if and only if h. < q., and such that Pi ' ' ' i i i i for i = 1,, r, k. Q. if Q.t q^> '"Uj. There is an integer n > 1 such that &" e A A. for j = 1,2, for &. is contained in each associated prime i P i l ideal of Ah.. For ; = 1, 2, let /. = h.+ kn + 1. Then /. i q. U» U? /. p j ' All ' 1 1 ' 7 is a regular element of A, 1 + A- ^"+ is a unit of A, and ifaf2\ = ^ö* Let/ = /j//2. Q.E.D. The injection çs: S C is to be constructed. Let a e S. By the Lemma there is a divisor D in a such that > = (IX for all p e C and therefore D = p P P

4 WEIL AND CARTIER DIVISORS 279 (1) for all p S. For a divisorial prime ideal p, let / and / be regular elements of 0,. with D = (/,//,X, and define P p ' i ' i p %\epfl + N p \ \ Qp\opf2 + N ' where the sum is over all zero prime ideals N of G such that G fi//v is of p p height one in G /N. Let pe^ where the sum is over all divisorial prime ideals p not contained in S. If / e K*and if) = (1) for all p C, then A(/)= (/) e /. Letting cßa be the image of AD in C = {D\S)/I, cß: ' C is a well-defined homomorphisr Theorem. Eei X, G be a reduced subscheme of a projective space over a field. The length homomorphism < : S ' C is infective. The image of cß is the subgroup of locally principal elements of C, which is a subquotient of D. Furthermore cß: (a > C is surjective if and only if G is factorial for all prime ideals p of G which are contained in no element of (_. Proof. The usual argument follows ([4, Propositions, pp. 65, 66]). Let S)\C be the subgroup of 5) of divisors D such that D = (IX for all p C. Let D 5)\G with AD = (1). Let p be a prime ideal of C, and let fx and /, be regular elements of G such that D = (/,//, ). If q C p is a prime ideal contained in C, then D = (1) and G / = G /_. If z? C p is a divisorial q q q ' 1 (J ' 2 r prime ideal not contained in G, then IL (G /(- / ) =? (G /G / ) and, Ü " «a-i O ^ ^ because C is normal of Krull dimension one, G / = G /. Thus G / = q q'1 q'2 q'1 G / for all depth one (grade one) prime ideals q of G, and Vi - Q Vt - Q o?/2-0p/2- Hence G /j = Gp/2 for ail prime ideals p of C, /j//2 e G*, D = (1), and A: \C D \S is injective. Now, to show that cß: E C is injective, let a e S with cßa = 0 e C, and let D a be such that D = (1) for all zj in G. Let /ek ;m be such that / e G for all q in G and AD = (/). Then by the injectivity of A above, D = (/), and a = 0. cß: E C is surjective if and only if A: 5) \C > DXS is surjective. This is true if and only if the group of Cartier divisors is equal to the group

5 280 JAMES HORNELL of Weil divisors for each local ring G where p is a prime ideal of G which is contained in no element of G, which is in turn equivalent to each G being factorial for all these prime ideals p. Q.E.D. If G is not reduced, G may no longer be finite, and the construction used to define <ß may not be applicable. For example, let R = k[x, y, z]/{x, xy) where k is a field, The reduction of R, R/Rx = k[y, z], is the polynomial ring in two variables over k. Let p be a prime element of k[z], Rp has as associated prime ideals (p, x) and (p, x, y), for {x, y) is an embedded component of (0) in U[z]/(p))U y]/(*2, xy) as R/Rp. Thus (p, x, y) e J), and Aü is infinite. The similar projective example given by the homogeneous ring k[w, X, Y, Z]/{X, XY) is such that every homogeneous height one prime ideal is contained in an element of. REFERENCES 1. Séminaire C. Chevalley, 3ieme année: 1958/59, Variétés de Picard, Ecole Normal Supérieure, Paris, I960. MR 28 # J. Hornell, Divisorial complete intersections, Pacific J. Math. 45 (1973), MR 47 # S. Lang, Abelian varieties, Interscience Tracts in Pure and Appl. Math., no. 7, Interscience, New York, MR 21 # D. Mumford, Lectures on curves on an algebraic surface, Ann. of Math. Studies, no. 59, Princeton Univ. Press, Princeton, N. J., MR 35 # M. Rosenlicht, Equivalence relations on algebraic curves, Ann. of Math. (2) 56 (1952), MR 14, 80. DEPARTMENT OF MATHEMATICS, UNIVERSITY OF KANSAS, LAWRENCE. KANSAS Current address: 2017 North 6th Street Terrace, Blue Springs, Missouri 64015