I am studying first order deformations and a natural question arises.

**Situation:** Let $X_1$ be a scheme. $\pi: X_1 \to {\rm Spec}~ k[t]/(t^2)$ is a flat morphism of finite type, where $k$ is an algebraically closed field. $X_0$ is the central fiber, i.e., $X_1 \times_{{\rm Spec}~ k} {\rm Spec~} k[t]/(t^2)$. The subscript $0$ will always denote the central fiber of the corresponding object. Assume $X_0$ is regular. $U_1\subset X_1$ is an open subset and $Z_1\subset U_1$ is an effective Cartier divisor.

**Question:** Is the scheme-theoretic closure of $Z_1$ in $X_1$ an effective Cartier divisor?

**My Observations:**

- An effective Cartier divisor $W_1 \subset X_1$ is flat over ${\rm Spec~}k[t]/(t^2)$.
- As in 1, $W_0$ is an effective Cartier divisor in $X_0$.
- If $W_1$ is a subscheme of $X_1$, flat over ${\rm Spec~}k[t]/(t^2)$, assume $W_0$ is an effective Cartier divisor, then $W_1$ is an effective Cartier divisor. [Lemma 10.93.1, Stacks Project, Tag 00MD]
- $\bar {Z_0}$ need not equal $\bar {Z_1}\cap X_0$. Example: $X_1=\mathbb P^2_{{\rm Spec~}k[t]/(t^2)}$, $U_1={\rm Spec~} k[x,y,t]/(t^2)$ and $Z_1={\rm Spec~} k[x,y,t]/(t^2,x+ty^2)$.
- [This is wrong. See the answer.]If for every $x\in X_0$, there is a neighborhood $U$ such that the class group $Cl(U)=0$, then the answer to the above question is positive. This is because locally the deformation is trivial, so we may assume that $X_1={\rm Spec~}\left( A\oplus tA\right)$ where $A$ is some $k-$alegebra which is an UFD. Then we can compute the scheme theoretic closure easily.

[This is wrong. See the answer.]By observation 5, if there would be a counterexample, $X_0$ cannot be as simple as affine planes.