Weil pullback not automatic

An arbitrary morphism of schemes carries no pullback of Weil divisors. Even when source and target are Noetherian normal, a morphism can send a local equation of a prime divisor to the zero section of the source and can have that prime divisor's inverse image of codimension zero. The recipe that pulls back a local equation and records its orders along the prime divisors of the source then has no nonzero rational function to evaluate and no height-one cycle of the source to receive a coefficient.

Set $Y={\mathbb{A}}_k^1=\mathrm{Spec}k[t]$ and let $Z=V(t)=\{0\}$, a prime divisor with local equation $t$ (Weil divisor normal noetherian scheme); the element $t\in K(Y{)}^{\times }$ is a unit of the rational-function field $k(t)$. Consider two morphisms out of Noetherian normal sources.

Constant morphism. Let $f:{\mathbb{A}}_k^1\to {\mathbb{A}}_k^1$ be the morphism with $f^{\mathrm{\#}}(t)=0$, so every point of the source maps to $0$. The inverse image $f^{-1}(Z)$ is then the whole source ${\mathbb{A}}_k^1$, a closed subscheme of codimension zero rather than a formal sum of prime divisors of the source. On the meromorphic side, the regular section $t$ is a nonzerodivisor of ${\mathcal{O}}_Y(Y)$ while its image $f^{\mathrm{\#}}(t)=0$ is not a nonzerodivisor of ${\mathcal{O}}_X(X)$; hence pullbacks of meromorphic functions are not defined for this $f$, and no meromorphic function $f^{\ast }(t)$ on the source exists whose orders along prime divisors could be recorded (Pullback of a Cartier divisor).

Inclusion of the origin. Let $i:\mathrm{Spec}k\to {\mathbb{A}}_k^1$ be the inclusion of the origin, the morphism with $i^{\mathrm{\#}}(t)=0$ in ${\mathcal{O}}_{\mathrm{Spec}k}=k$. Again the local equation pulls back to zero. The source has no prime divisors at all, so $\mathrm{Div}(\mathrm{Spec}k)=0$ and no nonzero Weil divisor of the source is available to receive the pullback (Weil divisor normal noetherian scheme).

In both examples the inverse image is the whole source with ideal sheaf zero, so the pullback of the effective Cartier divisor $Z$ is itself undefined: the local-equation criterion requires the pulled-back regular equation to be regular again, which fails because $t$ is sent to $0$ (Pullback of a Cartier divisor). The failure is thus not an artefact of the Weil formalism, but of the absence of a hypothesis such as flatness: for flat morphisms pullbacks of meromorphic functions are defined and every Cartier divisor has a defined pullback (Pullback of a Cartier divisor), and divisor pullback is built from that Cartier description under suitable hypotheses. No formula $Z\mapsto f^{-1}(Z)$ on height-one cycles is contravariant for arbitrary morphisms.