Comparing the Cartier and Weil divisor sheaves on a curve #
On a Noetherian integral curve with discrete valuation rings at codimension-one points, a
Cartier divisor and its associated Weil divisor define the same subsheaf of rational functions.
Locally, a Cartier equation f makes a rational function q a section precisely when f q is
regular; the Weil condition says that the order of f q is nonnegative at every closed point.
The one-dimensional regularity criterion identifies these conditions.
The resulting isomorphism 𝒪_X(D) ≅ 𝒪_X(D_Weil) commutes with both inclusions into the rational
function sheaf. It lets degree and cohomology computations on either presentation be used with the
other.
References #
- R. Hartshorne, Algebraic Geometry, Chapter II, Proposition 6.11.
The Cartier and Weil conditions define the same rational-function sections on a curve.
The line bundle of a Cartier divisor is canonically the line bundle of its Weil divisor. The isomorphism is induced by equality of their rational-function sections.
Equations
Instances For
The inverse comparison map preserves the inclusion into rational functions.
The inverse comparison map preserves the inclusion into rational functions.
The comparison map preserves the inclusion into rational functions.
The comparison map preserves the inclusion into rational functions.
The Cartier and Weil divisor presentations give the same line-bundle class.