The Stacks project

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 $\omega _{Y/X} = H^{-r}(f^!\mathcal{O}_ Y)$ 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. $\square$

Lemma 48.29.3. Let $f : X \to Y$ be a morphism of schemes. Let $r \geq 0$. Assume

  1. $Y$ is Cohen-Macaulay (Properties, Definition 28.8.1),

  2. $f$ factors as $X \to P \to Y$ where the first morphism is an immersion and the second is smooth and proper,

  3. if $x \in X$ and $\dim (\mathcal{O}_{X, x}) \leq 1$, then $f$ is Koszul at $x$ (More on Morphisms, Definition 37.59.2), and

  4. 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 $\omega _{Y/X} = H^{-r}(f^!\mathcal{O}_ Y)$ there is a map

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

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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