Let $π\colon Y = \mathrm{Proj}\_B \mathcal{A} \rightarrow B$ be a morphism constructed from a coherent graded sheaf of $\mathcal{O}\_B$-algebras $\mathcal{A} = \bigoplus\_k \mathcal{A}$.


I am looking for (minimal) hypotheses for the natural morphism
$$
\mathcal{A\_k} \rightarrow \mathcal π\_\*O\_Y(k)
$$ 
to be surjective for $k\gg 0$. Is this perhaps always the case? Please excuse me if this follows directly from the definitions. 


Stacks (Ex. 21.2) contains an example of a projectivisation of a coherent sheaf where injectivity at degree $k=1$ fails. That is all I could find.



Relative Proj and generation of sections
If the base is affine Noetherian, this is discussed in [this section of the Stacks project](http://stacks.math.columbia.edu/tag/01YR). See especially the last two lemmas with graded module $M = A$. Also, I want to mention that I do not know what you mean by a "coherent sheaf of $\mathcal{O}\_B$-algebras".


