48.29 The fundamental class of an lci morphism
In this section we will use the computations made in Section 48.15. Thus our result will suffer from the same kind of non-uniqueness as we have in that section.
Lemma 48.29.1. Let $X$ be a locally ringed space. Let
\[ \mathcal{E}_1 \xrightarrow {\alpha } \mathcal{E}_0 \to \mathcal{F} \to 0 \]
be a short exact sequence of $\mathcal{O}_ X$-modules. Assume $\mathcal{E}_1$ and $\mathcal{E}_0$ are locally free of ranks $r_1, r_0$. Then there is a canonical map
\[ \wedge ^{r_0 - r_1}\mathcal{F} \longrightarrow \wedge ^{r_1}(\mathcal{E}_1^\vee ) \otimes \wedge ^{r_0}\mathcal{E}_0 \]
which is an isomorphism on the stalk at $x \in X$ if and only if $\mathcal{F}$ is locally free of rank $r_0 - r_1$ in an open neighbourhood of $x$.
Proof.
If $r_1 > r_0$ then $\wedge ^{r_0 - r_1}\mathcal{F} = 0$ by convention and the unique map cannot be an isomorphism. Thus we may assume $r = r_0 - r_1 \geq 0$. Define the map by the formula
\[ s_1 \wedge \ldots \wedge s_ r \mapsto t_1^\vee \wedge \ldots \wedge t_{r_1}^\vee \otimes \alpha (t_1) \wedge \ldots \wedge \alpha (t_{r_1}) \wedge \tilde s_1 \wedge \ldots \wedge \tilde s_ r \]
where $t_1, \ldots , t_{r_1}$ is a local basis for $\mathcal{E}_1$, correspondingly $t_1^\vee , \ldots , t_{r_1}^\vee $ is the dual basis for $\mathcal{E}_1^\vee $, and $s'_ i$ is a local lift of $s_ i$ to a section of $\mathcal{E}_0$. We omit the proof that this is well defined.
If $\mathcal{F}$ is locally free of rank $r$, then it is straightforward to verify that the map is an isomorphism. Conversely, assume the map is an isomorphism on stalks at $x$. Then $\wedge ^ r\mathcal{F}_ x$ is invertible. This implies that $\mathcal{F}_ x$ is generated by at most $r$ elements. This can only happen if $\alpha $ has rank $r$ modulo $\mathfrak m_ x$, i.e., $\alpha $ has maximal rank modulo $\mathfrak m_ x$. This implies that $\alpha $ has maximal rank in a neighbourhood of $x$ and hence $\mathcal{F}$ is locally free of rank $r$ in a neighbourhood as desired.
$\square$
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 $\omega _{X/Y} = H^{-r}(f^!\mathcal{O}_ Y)$ is the unique nonzero cohomology sheaf of $f^!\mathcal{O}_ Y$, see Lemma 48.17.11. Then there is a map
\[ \wedge ^ r\Omega _{X/Y} \longrightarrow \omega _{X/Y} \]
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 compactifiable 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.
$\square$
Lemma 48.29.3. Let $f : X \to Y$ be a morphism of schemes. Let $r \geq 0$. Assume
$Y$ is Cohen-Macaulay (Properties, Definition 28.8.1),
$f$ factors as $X \to P \to Y$ where the first morphism is an immersion and the second is smooth and proper,
if $x \in X$ and $\dim (\mathcal{O}_{X, x}) \leq 1$, then $f$ is Koszul at $x$ (More on Morphisms, Definition 37.62.2), and
if $\xi $ is a generic point of an irreducible component of $X$, then we have $\text{trdeg}_{\kappa (f(\xi ))} \kappa (\xi ) = r$.
Then with $\omega _{X/Y} = H^{-r}(f^!\mathcal{O}_ Y)$ there is a map
\[ \wedge ^ r\Omega _{X/Y} \longrightarrow \omega _{X/Y} \]
which is an isomorphism on the locus where $f$ is smooth.
Proof.
Let $U \subset X$ be the open subscheme over which $f$ is a local complete intersection morphism. Since $f$ has relative dimension $r$ at all generic points by assumption (4) we see that the locally constant function of Lemma 48.29.2 is constant with value $r$ and we obtain a map
\[ \wedge ^ r\Omega _{X/Y}|_ U = \wedge ^ r \Omega _{U/Y} \longrightarrow \omega _{U/Y} = \omega _{X/Y}|_ U \]
which is an isomorphism in the smooth points of $f$ (this locus is contained in $U$ because a smooth morphism is a local complete intersection morphism). By Lemma 48.21.5 and the assumption that $Y$ is Cohen-Macaulay the module $\omega _{X/Y}$ is $(S_2)$. Since $U$ contains all the points of codimension $1$ by condition (3) and using Divisors, Lemma 31.5.11 we see that $j_*\omega _{U/Y} = \omega _{X/Y}$. Hence the map over $U$ extends to $X$ and the proof is complete.
$\square$
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