Can you give two closed immersions $(f,f^\sharp),(g,g^\sharp):(X,\mathcal O_X)\to (Y,\mathcal O_Y)$ such that $f=g$ but $f^\sharp\neq g^\sharp$?

Question

Can you give two closed immersions of schemes $(f,f^\sharp),(g,g^\sharp):(X,\mathcal O_X)\to (Y,\mathcal O_Y)$ such that $f=g$ but $f^\sharp\neq g^\sharp$?

Answer 1

Let $k$ be a field, $Y=\mathbb{A}^2_k= \mathrm{Spec}k[x,y]$ and $X= \mathrm{Spec}k[z]/(z^2)$. Consider the morphisms defined by

$$k[x,y] \rightarrow k[z]/(z^2)$$

with

$$f^{\sharp}\colon x \mapsto z, y \mapsto 0$$

and

$$g^{\sharp} \colon x \mapsto 0, y \mapsto z.$$