Smooth closed immersion is regular with exact conormal sequence

Fix any point $x\in X$, including a nonclosed point. Put $R={\mathcal{O}}_{{\mathbb{P}}^N,x}$, $I={\mathcal{I}}_x$, and $S=R\mathrm{/}I={\mathcal{O}}_{X,x}$. The closed immersion identifies the residue fields of $R$ and $S$ with the same $\kappa (x)$; write $t={\mathrm{trdeg}}_k\kappa (x)$. By [F3], $R$ and $S$ are regular local rings. The local scheme dimensions of ${\mathbb{P}}_k^N$ and $X$ at $x$ are $N$ and $n$, so [F4] gives $\dim R=N-t$ and $\dim S=n-t$. Thus $\dim R-\dim S=N-n=c$ at every $x$, although the individual local-ring dimensions equal $N,n$ only when $t=0$.

Apply [F2] to the regular local ring $R$ and its regular quotient $S$. There is a regular system of parameters of $R$ whose first $c$ members $f_1,\dots ,f_c$ generate $I$; in particular this is an $R$-regular sequence. Each germ $f_i$ has a representative on an affine neighbourhood of $x$. By [F6] the ambient affine chart is Noetherian, so its ideal is finitely generated; clearing the finitely many denominators in the generation equalities at the stalk shrinks the neighbourhood until those representatives generate $\mathcal{I}$ there. Their germs at $x$ remain the stated regular sequence. This proves assertion (ii), with $S$ regular of its actual local dimension $n-t$.

The classes of $f_1,\dots ,f_c$ give a surjection $S^c\twoheadrightarrow I\mathrm{/}{I}^2$. If ${\sum }_i{\bar{a}}_i[f_i]=0$, lift ${\bar{a}}_i$ to $a_i\in R$ and write ${\sum }_i{a}_i{f}_i={\sum }_i{b}_i{f}_i$ with every $b_i\in I$, because the left side lies in $I^2$. Then ${\sum }_i(a_i-b_i)f_i=0$, and [F5] makes every $a_i-b_i$ lie in $I$; hence every ${\bar{a}}_i=0$ in $S$. Thus $S^c\stackrel{\sim }{\to }I\mathrm{/}{I}^2$. Since $x$ was arbitrary, $\mathcal{I}\mathrm{/}{\mathcal{I}}^2$ is locally free of rank $c$, proving (i).

By [F1] the conormal map $I\mathrm{/}{I}^2\to j^{\ast }{\mathrm{\Omega }}_{{\mathbb{P}}^N\mathrm{/}k,x}^1$ surjects onto $K=\ker (j^{\ast }{\mathrm{\Omega }}_{{\mathbb{P}}^N\mathrm{/}k,x}^1\to {\mathrm{\Omega }}_{X\mathrm{/}k,x}^1)$. By [F3] the middle and target modules are free over the local ring $S$ of ranks $N$ and $n$. The surjection onto the free target splits, so $K$ is a finite projective, hence free, $S$-module of rank $N-n=c$. Step 3.1 makes the conormal module free of the same rank. A surjection between free rank-$c$ modules over a local ring has determinant nonzero modulo the maximal ideal, hence unit determinant and an inverse by the adjugate formula. Therefore $I\mathrm{/}{I}^2\to K$ is an isomorphism, so the conormal map is injective at every $x$. This argument uses the actual differential-module ranks and does not treat an arbitrary regular parameter system as a basis of relative differentials.

The stalkwise isomorphism in step 3.1 gives the locally free conormal sheaf of rank $c$, step 2.1 gives the local regular-sequence presentation, and step 4.1 upgrades the right-exact conormal sequence [F1] to the short exact sequence in (iii). By [F3] the other terms have ranks $N$ and $n$; all maps are canonical sheaf morphisms, so their stalkwise exactness proves exactness globally. The Axiom of Choice is assumed as declared; this finite local calculation makes no further arbitrary family of choices.