Let $X$ be a set of points in $\mathbb{P}^n$. Then we can write a resolution $$0\rightarrow P_n \rightarrow \cdots \rightarrow P_0\rightarrow \mathcal{O}_X$$ where each $P_i=\bigoplus_j\mathcal{O}_{\mathbb{P}^n}(-a_{ij})$ (and the length of the resolution is $n$ as a zero dimensional ideal is Cohen-Macaulay).

Now suppose we start with a morphism $\varphi\colon X\rightarrow \mathbb{P}^n$ such that $\varphi_*\mathcal{O}_X$ is a locally free sheaf of rank $d$. Then we have $X\hookrightarrow \mathbb{P}(\varphi_*\mathcal{O}_X)\twoheadrightarrow \mathbb{P}^n$.

If $\varphi_*\mathcal{O}_X$ is not a sum of line bundles then the ring of sections of $X$ is not Cohen-Macaulay but as $\varphi_*\mathcal{O}_X$ is locally free it will be Cohen-Macaulay after localization (restriction to the fibres).

Question: Can I still write a resolution like the first one but now with $P_i=\bigoplus_j\mathcal{O}_{\mathbb{P}(\varphi_*\mathcal{O}_X)}(-a_{ij})$?