If $A$ is a Noetherian ring, and $X = \mathbb{P}^r_A,$ then every coherent sheaf on $X$ admits a free resolution by direct sums of twisted structure sheaves, since there is always a surjection $\mathcal{O}_X(-n)^{\oplus m} \to \mathcal{F}$ for appropriate integers $n, m$. Additionally, as Hartshorne exercise III.6.9(a) implies, every coherent sheaf has a finite locally free resolution.
Can I choose a finite resolution by direct sums of twisted structure sheaves, though?
Best Answer
For $X = \mathbb{P}^r_A$ the answer is positive when $A$ is a field; this can be obtained by means of the Beilinson's resolution of the diagonal.
If $A$ is not a field the same approach gives a resolution with terms $M_n \otimes_A \mathcal{O}_X(-n)$, where $M_n$ are $A$-modules, not necessarily free.
If additionally $A$ has finite global dimension, resolving $M_n$ by projective $A$-modules, one can obtain a resolutions with terms $P_n \otimes_A \mathcal{O}_X(-n)$, where $P_n$ are projective $A$-modules, but still not necessarily free.