Pullback of a Cartier divisor

Let $f:X\to Y$ be a morphism of schemes. Write ${\mathcal{K}}_X$ and ${\mathcal{K}}_Y$ for the sheaves of meromorphic functions (Sheaf total quotient rings), so that over an open $U$ the value ${\mathcal{K}}_X(U)$ is the sheafification at $U$ of $U^{\mathrm{\prime }}\mapsto S_X(U^{\mathrm{\prime }}{)}^{-1}{\mathcal{O}}_X(U^{\mathrm{\prime }})$, where $S_X(U^{\mathrm{\prime }})$ consists of the sections of ${\mathcal{O}}_X$ that are nonzerodivisors at every stalk of $U^{\mathrm{\prime }}$. The structure maps ${\mathcal{O}}_X\to {\mathcal{K}}_X$ and ${\mathcal{O}}_Y\to {\mathcal{K}}_Y$ are injective. Call a section of ${\mathcal{O}}_Y$ over an open regular when its germ at every point of that open is a nonzerodivisor; these are exactly the elements of $S_Y$. A Cartier divisor on $Y$ is a global section of ${\mathcal{K}}_Y^{\times }\mathrm{/}{\mathcal{O}}_Y^{\times }$ (Cartier divisor).

Pullback of meromorphic functions. We say that pullbacks of meromorphic functions are defined for $f$ if for all opens $U\subseteq X$ and $V\subseteq Y$ with $f(U)\subseteq V$ the ring homomorphism $f^{\mathrm{\#}}:{\mathcal{O}}_Y(V)\to {\mathcal{O}}_X(U)$ carries regular sections of ${\mathcal{O}}_Y(V)$ to regular sections of ${\mathcal{O}}_X(U)$, that is, $f^{\mathrm{\#}}(S_Y(V))\subseteq S_X(U)$. In that case the universal property of localisation turns $f^{\mathrm{\#}}$ into ring homomorphisms $S_Y(V{)}^{-1}{\mathcal{O}}_Y(V)\to S_X(U{)}^{-1}{\mathcal{O}}_X(U)$, compatible with restriction; these assemble into a morphism of presheaves $f^{-1}{\mathcal{P}}_Y\to {\mathcal{P}}_X$, where ${\mathcal{P}}_X(U)=S_X(U{)}^{-1}{\mathcal{O}}_X(U)$, and sheafifying gives a morphism of sheaves of rings $f^{-1}{\mathcal{K}}_Y⟶{\mathcal{K}}_X,s⟼f^{\ast }(s),$ the pullback map on meromorphic functions. Since ring homomorphisms carry units to units, it restricts to a morphism of sheaves of abelian groups $f^{-1}{\mathcal{K}}_Y^{\times }\to {\mathcal{K}}_X^{\times }$, and it carries $f^{-1}{\mathcal{O}}_Y^{\times }$ into ${\mathcal{O}}_X^{\times }$.

Pullback of a Cartier divisor. Let $D\in \mathrm{CaDiv}(Y)$ be represented by a local-equation datum $\{(U_i,g_i){\}}_{i\in I}$, so the $U_i$ cover $Y$, each $g_i\in {\mathcal{K}}_Y(U_i{)}^{\times }$ is a meromorphic unit, and $g_i\mathrm{/}{g}_j\in {\mathcal{O}}_Y^{\times }(U_i\cap U_j)$ for all $i,j$ (Cartier divisor). Because ${\mathcal{K}}_Y$ is the sheafification of ${\mathcal{P}}_Y$, after refining the cover we may assume that each $g_i$ is the image of an element $a_i\mathrm{/}{s}_i\in S_Y(U_i{)}^{-1}{\mathcal{O}}_Y(U_i)$ with $a_i\in {\mathcal{O}}_Y(U_i)$ and $s_i\in S_Y(U_i)$; the refinement changes neither the datum nor the divisor. Say that the datum is $f$-admissible when $f^{\mathrm{\#}}(a_i)\in S_X(f^{-1}{U}_i)\text{and}{f}^{\mathrm{\#}}(s_i)\in S_X(f^{-1}{U}_i)\text{for every }i\mathrm{.}$ Since $s_i$ is regular, its image in ${\mathcal{K}}_Y(U_i)$ is a unit, so $g_i=a_i\mathrm{/}{s}_i$ is a unit of ${\mathcal{K}}_Y(U_i)$ automatically once the representation exists. We say that the pullback $f^{\ast }D$ is defined if $D$ admits an $f$-admissible local-equation datum on some open cover of $Y$.

In that case, on $f^{-1}{U}_i$ the two sections $f^{\mathrm{\#}}(a_i)$ and $f^{\mathrm{\#}}(s_i)$ are regular, so $f^{\mathrm{\#}}(a_i)\mathrm{/}{f}^{\mathrm{\#}}(s_i)$ is a unit of $S_X(f^{-1}{U}_i{)}^{-1}{\mathcal{O}}_X(f^{-1}{U}_i)$, hence a unit of ${\mathcal{K}}_X(f^{-1}{U}_i)$; we denote it by $g_i^{\mathrm{\prime }}$. On an overlap $W=U_i\cap U_j$ the ratio $u=g_i\mathrm{/}{g}_j$ is a unit of ${\mathcal{O}}_Y(W)$, and multiplying the identity $g_i=u{g}_j$ by $s_i{s}_j$ and using that $g_i,g_j$ are the images of $a_i\mathrm{/}{s}_i$ and $a_j\mathrm{/}{s}_j$ gives that the function $a_i{s}_j-u{a}_j{s}_i\in {\mathcal{O}}_Y(W)$ has image $0$ in ${\mathcal{K}}_Y(W)$; since ${\mathcal{O}}_Y\to {\mathcal{K}}_Y$ is injective, this function is $0$, so $a_i{s}_j=u{a}_j{s}_i$ in ${\mathcal{O}}_Y(W)$. Applying $f^{\mathrm{\#}}$ and dividing by the regular sections $f^{\mathrm{\#}}(s_i),f^{\mathrm{\#}}(s_j)$ gives $g_i^{\mathrm{\prime }}=f^{\mathrm{\#}}(u)g_j^{\mathrm{\prime }}$ in ${\mathcal{K}}_X(f^{-1}W)$, and $f^{\mathrm{\#}}(u)$ is a unit of ${\mathcal{O}}_X(f^{-1}W)$; hence $g_i^{\mathrm{\prime }}\mathrm{/}{g}_j^{\mathrm{\prime }}$ is a unit of ${\mathcal{O}}_X(f^{-1}W)$. Therefore the family $\{(f^{-1}{U}_i,g_i^{\mathrm{\prime }}){\}}_i$ is a local-equation datum on $X$ and determines a Cartier divisor (Cartier divisor); we define $f^{\ast }D$ to be that divisor.

This is well defined. Indeed, if $g_i=a_i\mathrm{/}{s}_i=b_i\mathrm{/}{t}_i$ are two representations with all four pullbacks regular, then multiplying $a_i\mathrm{/}{s}_i=b_i\mathrm{/}{t}_i$ by $s_i{t}_i$ shows that the function $a_i{t}_i-b_i{s}_i\in {\mathcal{O}}_Y(U_i)$ has image $0$ in ${\mathcal{K}}_Y(U_i)$, hence is $0$; applying $f^{\mathrm{\#}}$ and dividing by the regular sections $f^{\mathrm{\#}}(s_i),f^{\mathrm{\#}}(t_i)$ gives $f^{\mathrm{\#}}(a_i)\mathrm{/}{f}^{\mathrm{\#}}(s_i)=f^{\mathrm{\#}}(b_i)\mathrm{/}{f}^{\mathrm{\#}}(t_i)$ in ${\mathcal{K}}_X(f^{-1}{U}_i)$. Similarly, if two $f$-admissible data represent the same $D$ and on a common refinement $W$ their equations satisfy $g=uh$ with $u\in {\mathcal{O}}_Y^{\times }(W)$, the same clearing-denominators argument gives $g^{\mathrm{\prime }}=f^{\mathrm{\#}}(u)h^{\mathrm{\prime }}$ with $f^{\mathrm{\#}}(u)$ a unit of ${\mathcal{O}}_X$, so the two resulting local-equation data determine the same Cartier divisor after refinement. In particular $f^{\ast }D$ is independent of the chosen $f$-admissible datum, and restriction of an admissible datum to a refinement is again admissible with the same pullback.

Effective divisors. Suppose $D$ is effective, with local regular equations $f_i\in S_Y(U_i)$ (Effective cartier divisor); choose the representation $f_i\mathrm{/}1$ with numerator $f_i$ and denominator $1$. Then the datum is $f$-admissible exactly when each pulled-back regular equation $f^{\mathrm{\#}}(f_i)$ is again regular on $f^{-1}{U}_i$, and in that case $f^{\ast }D$ is the effective Cartier divisor cut out locally by the equations $f^{\mathrm{\#}}(f_i)$. If the pullback of an effective $D$ is defined through some other representation $a_i\mathrm{/}{s}_i$ of $f_i$ with regular pullbacks, then $f_i{s}_i=a_i$ in ${\mathcal{O}}_Y(U_i)$ by the clearing-denominators argument, so $f^{\mathrm{\#}}(f_i)f^{\mathrm{\#}}(s_i)=f^{\mathrm{\#}}(a_i)$ with both factors on the right regular, and hence $f^{\mathrm{\#}}(f_i)$ is regular and $f^{\ast }D$ is effective. Thus for effective $D$ the assertion "$f^{\ast }D$ is defined" is equivalent to the regularity of the pulled-back regular equations.

Flat morphisms. If $f$ is flat (Flat morphism of schemes), then pullbacks of meromorphic functions are defined for $f$ and every Cartier divisor on $Y$ has a defined pullback. Indeed, let $x\in X$, $y=f(x)$, and let $s\in S_Y(V)$ for an open $V\ni y$. Flatness at $x$ says that ${\mathcal{O}}_{X,x}$ is a flat ${\mathcal{O}}_{Y,y}$-module; tensoring the injective multiplication map $s_y:{\mathcal{O}}_{Y,y}\to {\mathcal{O}}_{Y,y}$ with ${\mathcal{O}}_{X,x}$ over ${\mathcal{O}}_{Y,y}$ therefore gives the injective map $f^{\mathrm{\#}}(s_y):{\mathcal{O}}_{X,x}\to {\mathcal{O}}_{X,x}$, so the germ $f^{\mathrm{\#}}(s{)}_x$ is a nonzerodivisor. As $x$ was arbitrary, $f^{\mathrm{\#}}(s)$ is regular. Hence $f^{\mathrm{\#}}(S_Y(V))\subseteq S_X(f^{-1}V)$ for all $V$, every local-equation datum is $f$-admissible (regularity of the numerator is automatic as recalled above), and $f^{\ast }$ is defined on all of $\mathrm{CaDiv}(Y)$; this is the flat case of the source's list of sufficient conditions. When pullbacks of meromorphic functions are defined for $f$ in the sense above, the map $f^{-1}{\mathcal{K}}_Y\to {\mathcal{K}}_X$ descends to $f^{-1}({\mathcal{K}}_Y^{\times }\mathrm{/}{\mathcal{O}}_Y^{\times })\to {\mathcal{K}}_X^{\times }\mathrm{/}{\mathcal{O}}_X^{\times }$ and recovers the same pullback of every Cartier divisor.

Boundary cases. The zero Cartier divisor is represented by the equation $1=1\mathrm{/}1$; its pullback is represented by $f^{\mathrm{\#}}(1)=1$ and is the zero divisor, so it is defined for every morphism $f$. If $Y=\mathrm{\varnothing }$ then $\mathrm{CaDiv}(Y)=0$ and the only pullback is $0$; if $X=\mathrm{\varnothing }$ then every local-equation datum is $f$-admissible vacuously and $f^{\ast }D$ is the unique Cartier divisor of the empty scheme. The sign convention is that of Cartier divisor: zeros of the pulled-back equations are recorded with positive coefficients, poles with negative ones.