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:**


1. An effective Cartier divisor $W\_1 \subset X\_1$ is flat over ${\rm Spec~}k[t]/(t^2)$.
2. As in 1, $W\_0$ is an effective Cartier divisor in $X\_0$.
3. 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]](http://stacks.math.columbia.edu/tag/00MD)
4. $\bar {Z\_0}$ 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 $Z\_1={\rm Spec~} k[x,y,t]/(t^2,x+ty^2)$.
5. [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.



The closure of an effective Cartier divisor in a special situation
Sorry, I do not understand what you say in 5. What if A = k[x, y] and X\_0 = Spec(A) and U\_0 = D(x) and the effective Cartier divisor is given by the zero locus of y + t/x ? Then (y + t/x)(y - t/x) = y^2 and the scheme theoretic closure of Z\_1 is given by an ideal containing both y^2 and xy + t which cannot be principal.


There are results in the literature concerning this question when one takes out something of sufficiently high codimension and X\_0 is sufficiently nice. Look in Grothendieck's Cohomologie locale des faisceaux coh´erents et th´eoremes de Lefschetz locaux et globaux (SGA 2) for example.


