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If $D$ is an effective divisor on a curve $Y$, then I can represented $D$ as a zero dimensional quasi-compact subscheme: for a closed point of multiplicity $n \geq 0$, we have a closed immersion from $Spec k[x] / (x^n)$ onto $Y$.

Suppose that $f : X \to Y$ is a morphism between smooth curves. Is it true that the pullback divisor $f^*(D)$ agrees with the preimage scheme $f^{-1}(D)$?

It clearly suffice to consider the case of a point $Q \in Y$ of multiplicity $n$. Then for each preimage point $P$, the pullback divisor gives that point multiplicity $n*e_p$, where $e_p$ is the degree of vanishing of the pullback of a local equation for $\phi_Q$ for $Q$. To compute that order at $P$, we find a local equation $\phi_P$ for $P$ on $X$ and then apply the valuation in $O_{X,P}$ given by $(\phi_P) \subseteq O_{X,P}$ to the function $\phi_Q \circ f$.

On the other hand, the length of the preimage scheme is given by $\dim_k (k \otimes_{k[y]} k[x])$, where $k[x]$ is a $k[y]$ module via the map $f^*$, and where $k$ is $k[y]$ module via the map $y \mapsto 0$.

I'm not sure how to relate these notions now. There must be some subtlty because of course (?) the preimage scheme cannot have negative length.

Thank you for your help!

Elle Najt
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1 Answers1

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Over a smooth curve $Y$ there is a bijective correspondence between effective divisors $D \in \operatorname {\mathcal Div_+(Y)}$ and closed subschemes $S\subset Y$.
$\bullet $ To $D$ associate the subscheme $S$ given by the coherent ideal subsheaf $\mathcal I_S=\mathcal O_Y(-D)\subset \mathcal O_Y$.
$\bullet \bullet$ To the subscheme $S\subset Y$ associate the divisor $D=\sum_{s\in S} (\operatorname {length} \mathcal O_{S,s})\cdot [s]$

It is then true that for a finite map $f:X\to Y$ of smooth curves the subscheme $f^{-1}(S)\subset X$ corresponds in the above correspondence to the divisor $f^{*}(D)\in \operatorname {\mathcal Div_+(X)}$.
The proof is the one you sketched. A wide generalization to flat morphisms between varieties of arbitrary dimensions can be be found in Fulton's Intersection Theory, page 18, Lemma 1.7.1 .

Edit
A very general version of this correspondence, valid for a completely arbitrary scheme, can be found on page 302 of Görsten-Wedhorn's book.
The result on the correspondence between pull-backs of subschemes and effective divisors under flat morphisms is Corollary 11.49, page 313.

Georges Elencwajg
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  • What is meant by the notation $O_Y(-D)$? – Elle Najt Sep 07 '15 at 23:51
  • It's the sheaf of rational functions $f$ with $div(f)\geq D$ – Georges Elencwajg Sep 07 '15 at 23:54
  • I see. So in the case $D = \Sigma n_i P_i$, the sections of $O_Y(-D)$ are those rational functions that vanish with order $n_i$ at each $P_i$. I think this agrees with my construction, at least in the algebraically closed case: I can assume that $Y$ is affine, by removing a hypersurface that avoids these points. Then $O_Y(-D)$ is given by $\Pi m_i^{n)i}$, where $m_i$ is the maximal ideal corresponding to the point $P_i$ ($f$ vanishing with order $n_i$ at $P_i$ means $f \in m_i^{n_i}$?) which is exactly the kernel of the embedding I constructed. Does that seem right? – Elle Najt Sep 08 '15 at 00:13
  • Yes, it is correct. – Georges Elencwajg Sep 08 '15 at 00:14