I'm studying the proof of Thm 1.5.1. in Laumon's "Cohomology of Drinfeld Modular Varieties". Notation: $\mathfrak{m}$ is a square zero ideal of $\mathcal{O}$ and $k=\mathcal{O}/\mathfrak{m}$. Laumon shows that the obstruction to the existence of a lift of a Drinfeld module $\phi: A \rightarrow k[\tau]$ to a Drinfeld module $\phi':A \rightarrow \mathcal{O}[\tau]$ lies in the second Hochschild cohomology $HH^2(A,\mathfrak{m}[\tau])$.


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 controlled 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.



Smoothness of the moduli space of Drinfeld modules
At this point of the proof, Laumon assumes that $\phi$ is standard—I'll do so as well. Let $a\in A$ be nonzero. Then 


$$ \phi\_a = \sum\_i \phi\_{a,i} \tau^i$$


with $\phi\_{a,-d\deg(\infty)\infty(a)} \in k^{\times}$ and $\phi\_{a,i} = 0$ in higher degrees. Note that an element $x$ of $\mathcal{O}$ is a unit if and only if is a unit mod $\mathfrak{m}$, because $\mathfrak{m}$ consists of nilpotents. Hence 


$$ \phi'\_a = \sum\_i \phi'\_{a,i} \tau^i$$


with $\phi'\_{a,-d\deg(\infty)\infty(a)} \in \mathcal{O}^{\times}$ and $\phi'\_{a,i} \in \mathfrak{m}$ in higher degrees. Compare this with Definition 1.2.1 to see that $\phi'$ is a Drinfeld module of rank $d$ over $\mathcal{O}$.


