I am trying to find a reference for the following theorem:


Let $R$ be a complete DVR, and let $Y$ be a scheme projective and flat over $R$. Suppose that $X\_0 \longrightarrow Y\_0$ is a finite etale morphism, where $Y\_0$ is the fiber of $Y$ over the unique closed point of Spec $R$. Then there exists a scheme $X$, finite etale over $Y$, together with a closed immersion $X\_0 \longrightarrow X$ such that $X\_0 = X \times\_Y Y\_0$.


In a special case, this allows us to lift etale morphisms in characteristic $p$ to etale morphisms in characteristic zero, and can thus be important in studying the etale fundamental group of varieties in finite characteristic.


Note: While any reference would be appreciated, what I am really interested in is a reference that I can give for attribution purposes. My advisor thinks that this result was proved by Deligne about 40 years ago, but I would like something a little more solid.



Lifting Etale Morphisms
SGA 1, IX, 1.10: Let $Y$ be a scheme proper over a complete local noetherian ring $R$,
and let $Y\_0$ be the closed fibre of $Y/R$. Then the functor $X\mapsto X\_0$ from
finite etale coverings of $Y$ to finite etale coverings of $Y\_0$ is
an equivalence of categories.


