I'm studiying the proof of Thm 1.51. in Laumon "Cohomology of Drinfeld Modular Varieties". In it, Laumon shows that the obstruction to the existence of a lift of a Drinfeld module $\phi: A \rightarrow k[\tau]$ to a Drinfeld module $\phi':A \rightarrow \mathcal{O}[\tau]$ is given by the second Hochschild cohomology $HH^2(A,\mathfrak{m}[\tau])$, where $\mathfrak{m}$ is the maximal ideal of $\mathcal{O}$.

I get that this is an obstruction to the existence of a lift of $\phi$ to a ring morphism $\phi':A \rightarrow \mathcal{O}[\tau]$. However, I don't see how the degree condition is controled by the Hochschild cohomology. I mean just because there exists a lift as a ring morphism $\phi'$, this does not need to be of the same rank as $\phi$.

Laumon does not explain this, and neither do Blum and Stuhler in "Drinfeld Modules and Elliptic Sheaves". Can some clarify this for me? Thanks