Lemma 48.29.2. Let $Y$ be a Noetherian scheme. Let $f : X \to Y$ be a local complete intersection morphism which factors as an immersion $X \to P$ followed by a proper smooth morphism $P \to Y$. Let $r$ be the locally constant function on $X$ such that $\omega _{Y/X} = H^{-r}(f^!\mathcal{O}_ Y)$ is the unique nonzero cohomology sheaf of $f^!\mathcal{O}_ Y$, see Lemma 48.17.11. Then there is a map

$\wedge ^ r\Omega _{X/Y} \longrightarrow \omega _{Y/X}$

which is an isomorphism on the stalk at a point $x$ if and only if $f$ is smooth at $x$.

Proof. The assumption implies that $X$ is compactifyable over $Y$ hence $f^!$ is defined, see Section 48.16. Let $j : W \to P$ be an open subscheme such that $X \to P$ factors through a closed immersion $i : X \to W$. Moreover, we have $f^! = i^! \circ j^! \circ g^!$ where $g : P \to Y$ is the given morphism. We have $g^!\mathcal{O}_ Y = \wedge ^ d\Omega _{P/Y}[d]$ by Lemma 48.15.7 where $d$ is the locally constant function giving the relative dimension of $P$ over $Y$. We have $j^! = j^*$. We have $i^!\mathcal{O}_ W = \wedge ^ c\mathcal{N}[-c]$ where $c$ is the codimension of $X$ in $W$ (a locally constant function on $X$) and where $\mathcal{N}$ is the normal sheaf of the Koszul-regular immersion $i$, see Lemma 48.15.6. Combining the above we find

$f^!\mathcal{O}_ Y = \left(\wedge ^ c\mathcal{N} \otimes _{\mathcal{O}_ X} \wedge ^ d\Omega _{P/Y}|_ X\right)[d - c]$

where we have also used Lemma 48.17.9. Thus $r = d|_ X - c$ as locally constant functions on $X$. The conormal sheaf of $X \to P$ is the module $\mathcal{I}/\mathcal{I}^2$ where $\mathcal{I} \subset \mathcal{O}_ W$ is the ideal sheaf of $i$, see Morphisms, Section 29.31. Consider the canonical exact sequence

$\mathcal{I}/\mathcal{I}^2 \to \Omega _{P/Y}|_ X \to \Omega _{X/Y} \to 0$

of Morphisms, Lemma 29.32.15. We obtain our map by an application of Lemma 48.29.1.

If $f$ is smooth at $x$, then the map is an isomorphism by an application of Lemma 48.29.1 and the fact that $\Omega _{X/Y}$ is locally free at $x$ of rank $r$. Conversely, assume that our map is an isomorphism on stalks at $x$. Then the lemma shows that $\Omega _{X/Y}$ is free of rank $r$ after replacing $X$ by an open neighbourhood of $x$. On the other hand, we may also assume that $X = \mathop{\mathrm{Spec}}(A)$ and $Y = \mathop{\mathrm{Spec}}(R)$ where $A = R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ m)$ and where $f_1, \ldots , f_ m$ is a Koszul regular sequence (this follows from the definition of local complete intersection morphisms). Clearly this implies $r = n - m$. We conclude that the rank of the matrix of partials $\partial f_ j/\partial x_ i$ in the residue field at $x$ is $m$. Thus after reordering the variables we may assume the determinant of $(\partial f_ j/\partial x_ i)_{1 \leq i, j \leq m}$ is invertible in an open neighbourhood of $x$. It follows that $R \to A$ is smooth at this point, see for example Algebra, Example 10.137.8. $\square$

Comment #4631 by Noah Olander on

Sorry for being pedantic but in the statement you should probably first put all the assumptions on $f$ before you start writing $f^{!}\mathcal{O}_Y$.

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