I am reading Illusie's lecture notes "topics in algebraic geometry", and I have difficulty in following his proof of unobstructedness of deformation of curves. Here is the statement of the proposition:


**Prop** Let $f\_0: X\_0 \to Y\_0$ be a smooth proper morphism with relative dimension 1, and $i : Y\_0 \to Y$ a first-order thickening with ideal $I$. If moreover $Y$ is affine, then there always exists a lifting of $X\_0$ over $Y$ .


**Proof**. First, since $Y\_0$ is affine, we note that
$H^q(X\_0, T\_{X\_0/Y\_0}
 \otimes f\_{0}^{\*}I)= \Gamma (Y\_0,R^qf\_{0\*}(T\_{X\_0/Y\_0}\otimes f\_{0}^{\*}I))$  

By Zariski’s main theorem, for any $q > 1$,  

$R^qf\_{0\*}(T\_{X\_0/Y\_0}\otimes f\_{0}^{\*}I))=0$ (\*)  

Hence the obstruction $o(f\_0, i)\in H^2(X\_0, T\_{X\_0/Y\_0}\otimes.
 f\_0^\*I)$ vanishes. # 


And I wonder how to derive (\*) by Zariski's main theorem. 


Thank you!



The proof of unobstructedness of deformations for curves
Probably Illusie wrote "Zariski's Main Theorem", but he intended the Theorem of Formal Functions (which is the key result needed in the modern proof of Zariski's Theorem).


In fact, the Theorem of Formal Functions implies the following result, see [Hartshorne, Algebraic Geometry, Corollary 11.2 page 279].



> 
> **Proposition.** Let $f \colon X \to Y$ be a projective morphism of noetherian schemes, and let $r= \textrm{max} \{\dim X\_y | y \in Y \}$. Then $R^qf\_\* (\mathscr{F})=0$ for all $q >r$ and for all coherent sheaves $\mathscr{F}$ on $X$.
> 
> 
> 


Now by projection formula we have $$R^qf\_{0\*}(T\_{X\_0/Y\_0}\otimes f\_{0}^{\*}I))=R^qf\_{0\*}(T\_{X\_0/Y\_0}) \otimes I,$$ 
so you can apply the previous proposition with $r=1$ and $\mathscr{F}=T\_{X\_0/Y\_0}$ in order to get the desired vanishing. 


