*Note: I [originally asked][1] this on MSE without any success.*

Let $f\colon X\to Y$ be a morphism  of schemes for which $f_*(\mathcal{O}_X)=\mathcal{O_Y}$. When this condition arises has been discussed in particular [in this MO question][5], and a related question is [this one][2] (there are, I think, some others too). My broad question is **what are some interesting things we can deduce from this when it does occur?**

My more specific question is about when $f_*$ sends locally free sheaves to locally free sheaves (which has been discussed [here][3]): **If we know $f_*(\mathcal{O}_X)=\mathcal{O}_Y$, is there an easy condition which implies that $f_*$ takes vector bundles to vector bundles?**

For example, it is clear if $f$ is open and surjective, or more generally if $Y$ has an open cover $\{V_i\}$ for which $f^{-1}(V_i)\subset X$ is contained in an open subscheme which admits only trivial bundles (e.g. $f^{-1}V_i\subset\mathbf{A}^n_k$).

I am happy to assume that $f$ is quasicompact and quasiseparated, or even that both $X$ and $Y$ are. The example I have in mind is the punctured plane $j\colon\mathbf{A}^2_k\setminus 0\hookrightarrow\mathbf{A}^2_k$. Since $\Gamma(\mathbf{A}^2_k,\mathcal{O_{\mathbf{A}^2_k}})=\Gamma(\mathbf{A}^2_k\setminus 0,\mathcal{O_{\mathbf{A}^2_k}})$, the qcqs lemma shows $j_*\mathcal{O_{\mathbf{A}^2_k\setminus0}}=\mathcal{O_{\mathbf{A}^2_k}}$. What might I show to conclude that $j_*\mathcal{E}$ is a vector bundle for any vector bundle $\mathcal{E}$, so in particular [all vector bundles over $\mathbf{A}^2_k\setminus0$ are trivial][4].

Any help is appreciated!


  [5]: https://mathoverflow.net/questions/63301/when-will-the-pushforward-of-a-structure-sheaf-still-be-a-structure-sheaf "When will the pushforward of a structure sheaf still be a structure sheaf?"
  [2]: https://mathoverflow.net/questions/23891/when-is-the-push-forward-of-the-structure-sheaf-locally-free "When is the push-forward of the structure sheaf locally free"
  [3]: https://mathoverflow.net/questions/67387/when-is-the-pushforward-of-a-vector-bundle-still-a-vector-bundle "When is the pushforward of a vector bundle still a vector bundle?"
  [4]: https://mathoverflow.net/questions/120776/algebraic-vector-bundles-on-affine-punctured-plane "Algebraic vector bundles on affine punctured plane"


  [1]: https://math.stackexchange.com/q/5010814/982236 "Deductions from the pushfoward of the structure sheaf being the structure sheaf."