Hartshorne Deformation theory Exercise7.1 This is a exercise in "Deformation Theory [Hartshorne]".
Let C be a local Artin ring with residue field k.
Let X be a scheme flat over C , and let$X_0=X\times _Ck$.
If F is a coherent sheaf on X which is flat over C, 
and $F_0=F⊗_{O_X}O_{X_0}$ is locally free on $X_0$,show that F is locally free on X.
what 'local Artin ring' implies in this question?
 A: It suffices to show that if $X$ is affine and $F_0$ is free, then $F$ is free. Since $F_0$ is free over $X_0$, we can find an isomorphism $\mathscr{O}_{X_0}^r\rightarrow F_0$. 
Arbitrarily lift this to a map $\mathscr{O}_X^r\rightarrow F$. We claim that this lift is an isomorphism. Let $K$ be the kernel and $M$ be the cokernel.
First, we see that, since $\mathscr{O}_X^r\rightarrow F$ is a surjection after applying $\cdot\otimes_{\mathscr{O}_X}\mathscr{O}_{X_0}$, $M\otimes_{\mathscr{O}_X}\mathscr{O}_{X_0}=0\Leftrightarrow M=mM$, where $m$ is the maximal ideal of $C$. Since $C$ is artinian, $m^N=0$ for $N>>0$, so $M=0$.
Now, we take $0\rightarrow K\rightarrow \mathscr{O}_X^r\rightarrow F\rightarrow0$ and tensor to get $0\rightarrow K\otimes_{\mathscr{O}_X}\mathscr{O}_{X_0}\rightarrow \mathscr{O}_{X_0}^r\rightarrow F_0\rightarrow 0$, which is left exact since $F$ is flat over $C$.
Therefore, we again see that $K\otimes_C k=0$, which means $K=mK$, so $K=0$ from the same argument as above.
(So we used $C$ is artinian to apply a version of Nakayama's lemma in the case where our coherent sheaves over $X$ aren't finitely generated over $C$.)
