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The Stacks project

Definition 111.49.1. Throughout, let S be any scheme and let X be a Noetherian, integral scheme.

  1. A Weil divisor on X is a formal linear combination \Sigma n_ i[Z_ i] of prime divisors Z_ i with integer coefficients.

  2. A prime divisor is a closed subscheme Z \subset X, which is integral with generic point \xi \in Z such that {\mathcal O}_{X, \xi } has dimension 1. We will use the notation {\mathcal O}_{X, Z} = {\mathcal O}_{X, \xi } when \xi \in Z \subset X is as above. Note that {\mathcal O}_{X, Z} \subset K(X) is a subring of the function field of X.

  3. The Weil divisor associated to a rational function f \in K(X)^\ast is the sum \Sigma v_ Z(f)[Z]. Here v_ Z(f) is defined as follows

    1. If f \in {\mathcal O}_{X, Z}^\ast then v_ Z(f) = 0.

    2. If f \in {\mathcal O}_{X, Z} then

      v_ Z(f) = \text{length}_{{\mathcal O}_{X, Z}}({\mathcal O}_{X, Z}/(f)).
    3. If f = \frac{a}{b} with a, b \in {\mathcal O}_{X, Z} then

      v_ Z(f) = \text{length}_{{\mathcal O}_{X, Z}}({\mathcal O}_{X, Z}/(a)) - \text{length}_{{\mathcal O}_{X, Z}}({\mathcal O}_{X, Z}/(b)).
  4. An effective Cartier divisor on a scheme S is a closed subscheme D \subset S such that every point d\in D has an affine open neighbourhood \mathop{\mathrm{Spec}}(A) = U \subset S in S so that D \cap U = \mathop{\mathrm{Spec}}(A/(f)) with f \in A a nonzerodivisor.

  5. The Weil divisor [D] associated to an effective Cartier divisor D \subset X of our Noetherian integral scheme X is defined as the sum \Sigma v_ Z(D)[Z] where v_ Z(D) is defined as follows

    1. If the generic point \xi of Z is not in D then v_ Z(D) = 0.

    2. If the generic point \xi of Z is in D then

      v_ Z(D) = \text{length}_{{\mathcal O}_{X, Z}}({\mathcal O}_{X, Z}/(f))

      where f \in {\mathcal O}_{X, Z} = {\mathcal O}_{X, \xi } is the nonzerodivisor which defines D in an affine neighbourhood of \xi (as in (4) above).

  6. Let S be a scheme. The sheaf of total quotient rings {\mathcal K}_ S is the sheaf of {\mathcal O}_ S-algebras which is the sheafification of the pre-sheaf {\mathcal K}' defined as follows. For U \subset S open we set {\mathcal K}'(U) = S_ U^{-1}{\mathcal O}_ S(U) where S_ U \subset {\mathcal O}_ S(U) is the multiplicative subset consisting of sections f \in {\mathcal O}_ S(U) such that the germ of f in {\mathcal O}_{S, u} is a nonzerodivisor for every u\in U. In particular the elements of S_ U are all nonzerodivisors. Thus {\mathcal O}_ S is a subsheaf of {\mathcal K}_ S, and we get a short exact sequence

    0 \to {\mathcal O}_ S^\ast \to {\mathcal K}_ S^\ast \to {\mathcal K}_ S^\ast /{\mathcal O}_ S^\ast \to 0.
  7. A Cartier divisor on a scheme S is a global section of the quotient sheaf {\mathcal K}_ S^\ast /{\mathcal O}_ S^\ast .

  8. The Weil divisor associated to a Cartier divisor \tau \in \Gamma (X, {\mathcal K}_ X^\ast /{\mathcal O}_ X^\ast ) over our Noetherian integral scheme X is the sum \Sigma v_ Z(\tau )[Z] where v_ Z(\tau ) is defined as by the following recipe

    1. If the germ of \tau at the generic point \xi of Z is zero – in other words the image of \tau in the stalk ({\mathcal K}^\ast /{\mathcal O}^\ast )_\xi is “zero” – then v_ Z(\tau ) = 0.

    2. Find an affine open neighbourhood \mathop{\mathrm{Spec}}(A) = U \subset X so that \tau |_ U is the image of a section f \in {\mathcal K}(U) and moreover f = a/b with a, b \in A. Then we set

      v_ Z(f) = \text{length}_{{\mathcal O}_{X, Z}}({\mathcal O}_{X, Z}/(a)) - \text{length}_{{\mathcal O}_{X, Z}}({\mathcal O}_{X, Z}/(b)).


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