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- Tags: - #AG/basics - Refs: - See Bhatt-Lurie 2022 for Cartier-Witt divisors, generalized Cartier divisors, and their relation to prisms: https://arxiv.org/pdf/2201.06120.pdf#page=1 #resources/papers/2022 - Links: - divisor - Ring theory: - Krull dimension - regular ring - Scheme theory: - Noetherian scheme - integral scheme - separated scheme - closed subscheme
Summary
\(\mathop{\mathrm{Cart}}\operatorname{Div}(X) \subseteq \operatorname{Div}(X)\) and it is interesting to ask when these are equal (i.e. when is an arbitrary divisor locally principal?)
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Idea:
- Weil divisors: formal sums of hypersurfaces (codimension 1 subvarieties). Like homology, essentially \({\operatorname{CH}}_{n-1} = {\operatorname{CH}}^1\).
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Cartier divisors: Weil divisors which are locally given by the divisor of a rational function, i.e. a locally principal divisor. Like cohomology.
- Think of this as a formal \({\mathbf{Z}}{\hbox{-}}\)combinations of embedded integral hypersurfaces.
- Formal sums of arbitrary codimension subvarieties which can still be cut out by a single equation.
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Motivation:
- If \(X \subseteq {\mathbf{P}}^n\) is codim 1 then \(X=V(f)\) is a hypersurface cut out by a single equation. For arbitrary smooth varieties this may not hold, but \(X\) is still locally a hypersurface. For singular varieties, even this can fail, so one needs to distinguish codimenson 1 subvarieties (~Weil) from those locally cut out by a single equation (~Cartier).
Facts
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Generally \(\mathop{\mathrm{Cart}}\operatorname{Div}(X) \subseteq \operatorname{W}\operatorname{Div}(X)\).
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On smoooth varieties, Weil = Cartier.
A Weil divisor that is not Cartier: 
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On smoooth varieties, Weil = Cartier.
Weil divisor
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Related to the divisor class group.
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For Weil divisors, only consider schemes which are noetherian, integral, separated, and regular in codimension one.
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The Weil divisor group of a scheme is the free abelian group \(\operatorname{Div}X\) generated by prime divisors, so finite sums \begin{align*}D \in \mathrm{WDiv}(X) \coloneqq{\mathbf{Z}}[{\mathrm{Sub}}^{n-1}(X)] \implies \quad D = \sum_{Y \in {\mathrm{Sub}}^{n-1}(X)} n_Y [Y].\end{align*}
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There is a map \(\operatorname{Div}: {\mathcal{O}}_X \to \mathrm{WDiv}(X)\)
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Every Weil divisor \(D\) determines a coherent sheaf \({\mathcal{O}}_X(D)\) where \begin{align*}{\mathcal{O}}_X(D)(U) = \left\{{f\in k(x) {~\mathrel{\Big\vert}~}f = 0 \text{ or } \operatorname{Div}(f) + D \geq 0 \text{ on } U}\right\},\end{align*} leading to a SES \begin{align*}1 \to {\mathcal{O}}_X(-D) \to {\mathcal{O}}_X \to {\mathcal{O}}_D \to 1.\end{align*}
Cartier divisor
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A special type of Weil divisor. Related to the Picard group.
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The Cartier divisor group of a variety is defined as \(\mathop{\mathrm{Cart}}\operatorname{Div}(X) \coloneqq{{\Gamma}\qty{X; K_X^{\times}/{\mathcal{O}}_X^{\times}} }\) where \(K_X\) is the sheaf of rational functions on \(X\).
- Equivalently, \(\mathop{\mathrm{Cart}}\operatorname{Div}(X)\) is the group of invertible fractional ideals.
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The Cartier divisor group of a scheme is the free abelian group generated by closed subschemes \(D \subseteq X\) such that the ideal sheaf \({\mathcal{O}}(-D) \subseteq {\mathcal{O}}_X\) is invertible in ${}_{{\mathcal{O}}_X}{\mathsf{Mod}} $.
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If \(f\in K_X^{\times}\) is a rational function on \(X\), then the divisor of \(f\) is \begin{align*}(f) \coloneqq \sum _{Y\in \operatorname{Div}(X) \, \mathrm{prime}} v_Y(f) Y.\end{align*}
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A Cartier divisor \(D\) is principal iff \(D = (f)\), the divisor of a function.
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Generally, \(\operatorname{Pic}(X)=\mathop{\mathrm{Cart}}\operatorname{Div}(X) / \mathop{\mathrm{Prin}}\mathop{\mathrm{Cart}}\operatorname{Div}(X)\), i.e. the quotient of Cartier divisors by the principal divisors.
- Equivalently, \(\operatorname{Pic}(X) = \mathop{\mathrm{Cart}}\operatorname{Div}(X)/K^{\times}\).
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Equivalently, a Cartier divisor is a pair \(({\mathcal{L}}, s \in {{\Gamma}\qty{{\mathcal{L}}} })\) with \(s\) a rational section.
- Effective when \(s\) is everywhere defined, yielding a subscheme as \(Z(s)\).
Associating a line bundle to a Cartier divisor
Results
- If \(X\) is factorial (for instance, when \(X\) is smooth), the Weil and Cartier divisor groups are isomorphic.
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Cartier divisors do not behave well under base change: if \(f\in {\mathsf{Sch}}(Y, X)\) and \(D\subseteq X\), then \(f^{-1}(D) \subseteq Y\) need not be be Cartier divisor.
- For a more general notion, see Cartier-Witt divisors, e.g. in Bhatt-Lurie 2022
Arithmetic definitions of Cartier and Weil divisors
Cartier-Witt divisors
