Algebraic Geometry 3. Exercises sheet 5 Carlos de Vera Piquero, [email protected]

We will extend the notion of divisor from the one given in the previous Exercises sheet. First we recall the definition of sheaf of total quotient rings of (the structure sheaf of) a scheme. Let X be an arbitrary scheme. If U = Spec(A) is an open affine subset of X, let S be the set of elements of A which are not zero divisors, and K(U) = S−1A be the localization of A by the multiplicative system S. K(U) is called the total quotient ring of A. For an arbitrary open set U, define

S(U) := {s ∈ Γ(U, OX ): s is not a zero divisor in OX,x, ∀x ∈ U}. −1 Then the association U 7→ S(U) Γ(U, OX ) defines a presheaf, whose associated sheaf of rings KX is the so-called sheaf of total quotient rings of OX . On an arbitrary scheme, KX extends the concept of function × × field of an integral scheme. Write KX , resp. OX , for the sheaf (of multiplicative groups) of invertible elements in K, resp. OX . × × A Cartier divisor on a scheme X is by definition a global section of the sheaf KX /OX . A Cartier divisor on X can therefore be described by giving an open cover {Ui} of X, and for each i an element × × fi ∈ Γ(Ui, KX ) such that for each pair i, j one has fi/fj ∈ Γ(Ui ∩ Uj, OX ). We write CaDiv(X) for the group of Cartier divisors on X. A Cartier divisor is called principal if it lies in the image of the natural × × × map Γ(X, KX ) → Γ(X, KX /OX ), which we will denote f 7→ div(f). Two Cartier divisors C1,C2 are 1 linearly equivalent if their difference C1 − C2 is principal. Writing CaPrin(X) for the group of principal Cartier divisors, we denote by CaCl(X) = CaDiv(X)/CaPrin(X) the class group of Cartier divisors. A Cartier divisor is said to be effective, or positive, if it belongs to the image of the canonical map × × × × Γ(X, (OX ∩ KX )/OX ) → Γ(X, KX /OX ). Let now X be a Noetherian, integral scheme, regular in codimension 1, and C ∈ CaDiv(X) be a Cartier divisor on X, represented by some system {(Ui, fi)} as explained above. Since X is integral, we have × × fi ∈ Γ(Ui, KX ) = K(X) . Given a prime divisor Y = {y} in X, one can consider the valuation vY (fi) for each i such that y ∈ Ui (that is, Y ∩ Ui = ∅). If j is another index with y ∈ Uj, then one can check vY (fi) = vY (fj), hence one can define the multiplicity of C in Y as the integer vY (C) := vY (fi). This does not depend on the system {(Ui, fi)} representing C. Being X Noetherian, and so quasi-compact, n n there is a finite covering {Ui}i=1 of X. Then C is represented by a system {(Ui, fi)}i=1. For each i, there are only finitely many prime divisors Y on X such that vY (fi) 6= 0 (cf. Exercise 1 of the previous sheet). In particular, vY (C) 6= 0 only for a finite number of prime divisors of X. One then defines the cycle associated with C as X cyc(C) := vY (C) · Y ∈ Div(X), Y with the sum being over all prime divisors of X. Then cyc : CaDiv(X) → Div(X) is a morphism.

Exercise 1: Let X be a Noetherian, integral, normal scheme. Prove that cyc is an injective morphism, whose image is the subgroup of Weil divisors on X which are locally principal. Moreover, show that cyc(C) is a principal Weil divisor (resp. an effective Weil divisor) if and only if C is principal (resp. effective) in CaDiv(X). In particular, cyc induces an injective morphism cyc : CaCl(X) → Cl(X).

A scheme X is said to be locally factorial if for every x ∈ X the local ring OX,x is a UFD. Observe that since every UFD is normal, every locally factorial scheme is normal. A Weil divisor D on X is called locally principal if for every x ∈ X there exists an open neighborhood Ux of x such that D|Ux is principal. Exercise 2: Let X be a Noetherian, integral, normal scheme. Prove that the following statements are equivalent. i) X is locally factorial. ii) Every Weil divisor on X is locally principal. iii) The map cyc : CaDiv(X) → Div(X) is an isomorphism. Hint: for the equivalence ii) ⇐⇒ iii), use Exercise 1. For the equivalence i) ⇐⇒ ii), one can proceed along the following steps: (1) Let X be a scheme and y ∈ X. Let U = Spec(A) be an affine neighborhood of y, put y = [p] for some prime p of A, and let iy : Spec(OX,y) → X be the canonical morphism obtained by composing the morphism of schemes Spec(OX,y) → U induced by A → Ap with the inclusion U ⊂ X. Show that iy gives a bijection between Spec(OX,y) and the set {z ∈ X : y ∈ {z}}.

1 × × Although KX and OX are shaves of multiplicative groups, we use additive notation for Cartier divisors. 1 2

Further, for each prime p ∈ Spec(OX,y), there is an isomorphism OX,z ' (OX,y)p, where z = iy(p). (2) From (1), there is a correspondence between the irreducible closed subsets of Ty := Spec(OX,y) and the irreducible closed subsets of X containing y. Indeed, if Y is an irreducible closed subset of X with y ∈ Y , then Y has a generic point, say z. Then y ∈ {z} and from (1) we have z = iy(p) for some prime p ∈ Ty. The map Y 7→ V (p) gives the desired correspondence. If Y ⊆ X is an irreducible closed subset containing y, we put Y ∩ Ty := V (p). (3) Suppose now that X is a Noetherian, integral, and normal scheme, and let y ∈ X. Then Ty is again a Noetherian, integral, and normal scheme. By the preceding points, there is a correspondence between prime divisors of X passing through y and prime divisors of Ty. Given a Weil divisor P D = Y nY · Y ∈ Div(X), the restriction of D to y is defined as the Weil divisor X D|y := nY · (Y ∩ Ty) ∈ Div(Ty), Y

with the sum being over prime divisors Y such that y ∈ Y . Using (1), show that div(f)|y = × divy(f) for all f ∈ K(X) , where divy(f) stands for the divisor associated with f in Div(Ty) (which makes sense since Frac(OX,y) = K(X)). (4) If X is a Noetherian, integral, and normal scheme, and D ∈ Div(X) is a Weil divisor, prove that D|y is principal in Div(Ty) for all y ∈ X if and only if D is locally principal. (5) Use Exercise 3 of the Exercises sheet 4 to deduce that X is locally factorial if and only if Cl(Spec(OX,x)) = 0 for all x ∈ X. Then apply (4).