Schemes

Exercises 6

Schemes – Divisors

Feel free to quote results from Hartshorne or EGA.

Definitions. Throughout, let X be a Noetherian, integral and separated scheme.

(a) A Weil divisor is a formal linear combination Σn_{i}[Z_{i}] of prime divisors Z_{i} with integer coefficients.

(b) A prime divisor is a closed subscheme Z ⊂ X, which is integral with generic point ξ ∈ Z such that O_{X,ξ} has dimension 1. We will use the notation O_{X,Z} = O_{X,ξ} when ξ ∈ Z ⊂ X is as above. Note that O_{X,Z} ⊂ K(X) is a subring of the function field of X.

(c) The Weil divisor associated to a rational function f ∈ K(X)^{∗} is the sum Σv_{Z}(f)[Z]. Here v_{Z}(f) is defined as follows (c1) If f ∈ O_{X,Z}^{∗} then v_{Z}(f) = 0. (c2) If f ∈ O_{X,Z} then

v_{Z}(f) = length_{O_{X,Z}} (O_{X,Z}/(f)).

(c3) If f = \frac{a}{b} with a, b ∈ O_{X,Z} then

v_{Z}(f) = length_{O_{X,Z}} (O_{X,Z}/(a)) − length_{O_{X,Z}} (O_{X,Z}/(b)).

(d) An effective Cartier divisor on any scheme S is a closed subscheme D ⊂ S such that every point d ∈ D has an affine open neighbourhood Spec A = U ⊂ S in S so that D ∩ U = Spec A/(f) with f ∈ A a nonzero divisor.

(e) The Weil divisor [D] associated to an effective Cartier divisor D ⊂ X of our Noetherian integral separated scheme X is defined as the sum Σv_{Z}(D)[Z] where v_{Z}(D) is defined as follows (e1) If the generic point ξ of Z is not in D then v_{Z}(D) = 0. (e1) If the generic point ξ of Z is in D then

v_{Z}(D) = length_{O_{X,Z}} (O_{X,Z}/(f))

where f ∈ O_{X,Z} = O_{X,ξ} is the nonzero divisor which defines D in an affine neighbourhood of ξ (as in definition (d) above).

(f) Let S be any scheme. The sheaf of total quotient rings K_{S} is the sheaf of O_{S}-algebras which is the sheafification of the pre-sheaf K′ defined as follows. For U ⊂ S open we set K′(U) = S_{U}^{−1}O_{S}(U) where S_{U} ⊂ O_{S}(U) is the multiplicative subset consisting of sections f ∈ O_{S}(U) such that the germ of f in O_{S,u} is a nonzero divisor for every u ∈ U. In particular the elements of S_{U} are all nonzero divisors. Thus O_{S} is a subsheaf of K_{S}, and we get a short exact sequence

0 → O_{S}^{∗} → K_{S}^{∗} → K_{S}^{∗}/O_{S}^{∗} → 0.

(g) A Cartier divisor on any scheme S is a global section of the quotient sheaf K_{S}^{∗}/O_{S}^{∗}.

(h) The Weil divisor associated to a Cartier divisor τ ∈ Γ(X, K_{X}^{∗}/O_{X}^{∗}) over our Noetherian integral separated scheme X is the sum Σv_{Z}(τ)[Z] where v_{Z}(τ) is defined as by the following recipe (h1) If the germ of τ at the generic point ξ of Z is zero – in other words the image of τ in the stalk

(K^{∗}/O^{∗})_{ξ} is “zero” – then v_{Z}(τ) = 0. (h2) Find an affine open neighbourhood Spec A = U ⊂ X so that τ|_{U} is the image of a section f ∈ K(U)

and moreover f = a/b with a, b ∈ A. Then we set

v_{Z}(f) = length_{O_{X,Z}} (O_{X,Z}/(a)) − length_{O_{X,Z}} (O_{X,Z}/(b)).

Remarks. (a) On a Noetherian integral separated scheme X the sheaf K_{X} is constant with value the function field K(X).

(b) To make sense out of the definitions above one needs to show that

length_{O}(O/(ab)) = length_{O}(O/(a)) + length_{O}(O/(b))

for any pair (a, b) of nonzero elements of a Noetherian 1-dimensional local domain O. This will be done in the lectures.

1. Describe how to assign a Cartier divisor to an effective Cartier divisor.

The following questions have some logical dependencies; if you point them out then you won’t have to do all of them.

2. Give an example of a Weil divisor (on a Noetherian integral separated scheme) which is not the Weil divisor associated to a rational function.

3. Give an example of a Weil divisor (on a Noetherian integral separated scheme) which is not the Weil divisor associated to any effective Cartier divisor.

4. Give an example of a Weil divisor (on a Noetherian integral separated scheme) which is not the Weil divisor associated to any Cartier divisor.

5. Give an example of a Weil divisor D (on a Noetherian integral separated scheme) which is not the Weil divisor associated to any Cartier divisor but such that nD is the Weil divisor associate to a Cartier divisor for some n > 1.

6. Give an example of a Weil divisor D (on a Noetherian integral separated scheme) which is not the Weil divisor associated to any Cartier divisor and such that nD is NOT the Weil divisor associate to a Cartier divisor for any n > 1.

7. Give an example of a Cartier divisor which is not the difference of (the Cartier divisors associated to) two effective Cartier divisors.†

† I do not know how to do this one myself, but I think this happens.