Do we have $\alpha^{-1}\widetilde B=\mathcal O_{\operatorname{Spec}B}$?

Question

Let $\alpha:\operatorname{Spec}B\to\operatorname{Spec}A$ be a morphism of schemes associated to $f:A\to B$, then we have an $\mathcal O_{\operatorname{Spec}A}$-module $\widetilde B$.

Do we have $\alpha^{-1}\widetilde B=\mathcal O_{\operatorname{Spec}B}$?

If $f$ is surjective, do we have $\alpha^{-1}\widetilde B=\mathcal O_{\operatorname{Spec}B}$?