Cartier divisor

On a scheme $X$ , a Cartier divisor is a global section of the sheaf $\\mathcal{K}^{*}/\\mathcal{O}^{*}$ , where $\\mathcal{K}^{*}$ is the multiplicative sheaf of meromorphic functions, and $\\mathcal{O}^{*}$ the multiplicative sheaf of invertible regular functions (the units of the structure sheaf).

More explicitly, a Cartier divisor is a choice of open cover $U_{i}$ of $X$ , and meromorphic functions $f_{i}\\in\\mathcal{K}^{*}(U_{i})$ , such that $f_{i}/f_{j}\\in\\mathcal{O}^{*}(U_{i}\\cap U_{j})$ , along with two Cartier divisors being the same if the open cover of one is a refinement of the other, with the same functions attached to open sets, or if $f_{i}$ is replaced by $gf_{i}$ with $g\\in\\mathcal{O}_{*}$ .

Intuitively, the only carried by Cartier divisor is where it vanishes, and the order it does there. Thus, a Cartier divisor should give us a Weil divisor, and vice versa. On “nice” (for example, nonsingular over an algebraically closed field) schemes, it does.

单词	CartierDivisor
释义	Cartier divisor On a scheme $X$ , a Cartier divisor is a global section of the sheaf $\\mathcal{K}^{}/\\mathcal{O}^{}$ , where $\\mathcal{K}^{}$ is the multiplicative sheaf of meromorphic functions, and $\\mathcal{O}^{}$ the multiplicative sheaf of invertible regular functions (the units of the structure sheaf). More explicitly, a Cartier divisor is a choice of open cover $U_{i}$ of $X$ , and meromorphic functions $f_{i}\\in\\mathcal{K}^{}(U_{i})$ , such that $f_{i}/f_{j}\\in\\mathcal{O}^{}(U_{i}\\cap U_{j})$ , along with two Cartier divisors being the same if the open cover of one is a refinement of the other, with the same functions attached to open sets, or if $f_{i}$ is replaced by $gf_{i}$ with $g\\in\\mathcal{O}_{*}$ . Intuitively, the only carried by Cartier divisor is where it vanishes, and the order it does there. Thus, a Cartier divisor should give us a Weil divisor, and vice versa. On “nice” (for example, nonsingular over an algebraically closed field) schemes, it does.
随便看	Laplace transform of logarithm LaplaceTransformOfPeriodicFunctions Laplace transform of periodic functions LaplaceTransformOfPowerFunction Laplace transform of power function LaplaceTransformOfSineIntegral Laplace transform of sine integral LaplaceTransformOfTnft Laplace transform of t^n⁢f⁢(t) LaplaceTransformsOfDerivatives Laplace transforms of derivatives Laplacian Laplacian Laplacian from Rectangular to Spherical Coordinates LaplacianFromRectangularToSphericalCoordinates1 LaplacianMatrixOfAGraph Laplacian matrix of a graph LargeIdeal large ideal LargeIntegersThatAreOrMightBeTheSmallestOfTheirKind large integers that are or might be the smallest of their kind LarsAhlfors Lars Ahlfors LarsHormander Lars Hörmander