Let X be a nonsingular variety of dim n over an algebraically closed field k. Let Y be an irreducible closed subscheme defined by a sheaf of ideals $\mathscr I$.
Then I want to prove that
Y is a nonsingular variety over k iff
(1) $ \Omega_{Y/k} $ is locally free, and
(2) the sequence $ 0 \rightarrow \mathscr I/\mathscr I^2 \rightarrow \Omega_{X/k} \otimes \mathcal O_Y \rightarrow \Omega_{Y/k} \rightarrow 0 $ is exact.
This is a theorem from Hartshorne's Algebraic Geometry (Chapter II, theorem 8.17).Assuming (1) and (2) and proving Y is nonsingular is okay. But the other way is not clear.
So I will assume Y is nonsingular. The subdivision (1) follows directly as Y is nonsingular (In fact $ \Omega_{Y/k} $ will be locally free of rank q = dim Y) . But am not able to do the 2nd subdivision.
We have the following exact sequence $ \mathscr I/\mathscr I^2 \rightarrow \Omega_{X/k} \otimes \mathcal O_Y \rightarrow \Omega_{Y/k} \rightarrow 0 $ (which is called the second exact sequence in literature). Let us name the map $ \mathscr I/\mathscr I^2 \rightarrow \Omega_{X/k} \otimes \mathcal O_Y $ as $ \delta $ and the map $\Omega_{X/k} \otimes \mathcal O_Y \rightarrow \Omega_{Y/k} $ as $ \phi $.
My aim is to prove that $\delta$ is injective. I will mimic the proof of Hartshorne and try to tell what I have not understood there-
Let y $\in$ Y be a closed point. Then ker $\phi$ is locally free of rank r= (n-q) at y, so it is possible to choose sections $ x_1, x_2,...x_r \in \mathscr I $ in a suitable neighbourhood of y, such that $dx_1,dx_2,...,dx_r$ generate ker $\phi$ (This is because X is nonsingular, so $\Omega_{X/k} \otimes \mathcal O_Y $ is locally free $ \mathcal O_Y $- module of rank n, and Y is nonsingular, so $\Omega_{Y/k}$ is locally free $\mathcal O_Y$-module of rank q). Let $\mathscr I'$ be the ideal sheaf generated by $ x_1, x_2,...x_r $, and Y' be the corresponding closed subscheme.
So far everything is fine. Now he says- By construction. the $dx_1,dx_2,...,dx_r$ generate a free subsheaf of rank r of $\Omega_{X/k} \otimes \mathcal O_{Y'} $ in a neighbourhood of y. I don't understand why $dx_1,dx_2,...,dx_r$ generate a free subsheaf of rank r of $\Omega_{X/k} \otimes \mathcal O_{Y'} $ in a neighbourhood of y? This is my first question.
Now suppose we assume this. It follows that in the exact sequence for Y' $ \mathscr I'/\mathscr I'^2 \rightarrow \Omega_{X/k} \otimes \mathcal O_{Y'} \rightarrow \Omega_{Y'/k} \rightarrow 0 $, we have $\delta $ injective ( Here also $\delta$ is the map $ \mathscr I'/\mathscr I'^2 \rightarrow \Omega_{X/k} \otimes \mathcal O_{Y'}$ and it is injective because its image is free), and $\delta $ is injective says that this is a short exact sequence and $ \mathscr I'/\mathscr I'^2 $ becomes locally free. So $\Omega_{Y'/k}$ also becomes locally free of rank (n-r).
Now he says that the previous part of the proof shows that Y' is irreducible and nonsingular. But according to me the previous part assumes irreducibility and proves nonsingularity.. I am not able to conclude that Y' is irreducible from the above data. This is my second question. Can any one please help?
