Definition 109.49.1. Throughout, let $S$ be any scheme and let $X$ be a Noetherian, integral scheme.

A

*Weil divisor*on $X$ is a formal linear combination $\Sigma n_ i[Z_ i]$ of prime divisors $Z_ i$ with integer coefficients.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 $X$.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 followsIf $f \in {\mathcal O}_{X, Z}^\ast $ then $v_ Z(f) = 0$.

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

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

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

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

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).

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\[ 0 \to {\mathcal O}_ S^\ast \to {\mathcal K}_ S^\ast \to {\mathcal K}_ S^\ast /{\mathcal O}_ S^\ast \to 0. \]A

*Cartier divisor*on a scheme $S$ is a global section of the quotient sheaf ${\mathcal K}_ S^\ast /{\mathcal O}_ S^\ast $.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 recipeIf 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 $v_ Z(\tau ) = 0$.

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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