Computing Sheaf of differentials of a scheme

Question

Can some one please explain how to see that sheaf of differentials of the scheme $\mathbb A^n_Y =Spec(\mathbb A^n_\mathbb Z)\times _Z Y$ is $\mathcal O ^n_X$

Answer 1

If $A$ is a ring, then $\Omega^1_{A[T_1,\dotsc,T_n]/A}$ is free of rank $n$, generated by $d(T_1),\dotsc,d(T_n)$. It follows more generally that if $Y$ is a scheme (or just a locally ringed space), then $\Omega^1_{\mathbb{A}^n_Y/Y}$ is free of rank $n$, generated by the global sections $d(T_1),\dotsc,d(T_n)$.