The Stacks project

Lemma 42.55.3. Let $(S, \delta )$ be as in Situation 42.7.1. Let $i : Z \to X$ be a regular closed immersion of codimension $r$ between schemes locally of finite type over $S$. Let $\mathcal{N} = \mathcal{C}_{Z/X}^\vee $ be the normal sheaf. If $X$ is quasi-compact (or has quasi-compact irreducible components), then $c_ t(Z \to X, i_*\mathcal{O}_ Z) = 0$ for $t = 1, \ldots , r - 1$ and

\[ c_ r(Z \to X, i_*\mathcal{O}_ Z) = (-1)^{r - 1} (r - 1)! c(Z \to X, \mathcal{N}) \quad \text{in}\quad A^ r(Z \to X) \]

where $c_ t(Z \to X, i_*\mathcal{O}_ Z)$ is the localized Chern class of Definition 42.50.3.

Proof. For any $x \in Z$ we can choose an affine open neighbourhood $\mathop{\mathrm{Spec}}(A) \subset X$ such that $Z \cap \mathop{\mathrm{Spec}}(A) = V(f_1, \ldots , f_ r)$ where $f_1, \ldots , f_ r \in A$ is a regular sequence. See Divisors, Definition 31.21.1 and Lemma 31.20.8. Then we see that the Koszul complex on $f_1, \ldots , f_ r$ is a resolution of $A/(f_1, \ldots , f_ r)$ for example by More on Algebra, Lemma 15.30.2. Hence $A/(f_1, \ldots , f_ r)$ is perfect as an $A$-module. It follows that $F = i_*\mathcal{O}_ Z$ is a perfect object of $D(\mathcal{O}_ X)$ whose restriction to $X \setminus Z$ is zero. The assumption that $X$ is quasi-compact (or has quasi-compact irreducible components) means that the localized Chern classes $c_ t(Z \to X, i_*\mathcal{O}_ Z)$ are defined, see Situation 42.50.1 and Definition 42.50.3. All in all we conclude that the statement makes sense.

Denote $b : W \to \mathbf{P}^1_ X$ the blowing up in $\infty (Z)$ as in Section 42.53. By (8) we have a closed immersion

\[ i' : \mathbf{P}^1_ Z \longrightarrow W \]

We claim that $Q = i'_*\mathcal{O}_{\mathbf{P}^1_ Z}$ is a perfect object of $D(\mathcal{O}_ W)$ and that $F$ and $Q$ satisfy the assumptions of Lemma 42.51.2.

Assume the claim. The output of Lemma 42.51.2 is that we have

\[ c_ p(Z \to X, F) = c'_ p(Q) = (E \to Z)_* \circ c'_ p(Q|_ E) \circ C \]

for all $p \geq 1$. Observe that $Q|_ E$ is equal to the pushforward of the structure sheaf of $Z$ via the morphism $Z \to E$ which is the base change of $i'$ by $\infty $. Thus the vanishing of $c_ t(Z \to X, F)$ for $1 \leq t \leq r - 1$ by Lemma 42.55.2 applied to $E \to Z$. Because $\mathcal{C}_{Z/X} = \mathcal{N}^\vee $ is locally free the bivariant class $c(Z \to X, \mathcal{N})$ is characterized by the relation

\[ j^* \circ C = p^* \circ c(Z \to X, \mathcal{N}) \]

where $j : C_ ZX \to W_\infty $ and $p : C_ ZX \to Z$ are the given maps. (Recall $C \in A^0(W_\infty \to X)$ is the class of Lemma 42.48.1.) Thus the displayed equation in the statement of the lemma follows from the corresponding equation in Lemma 42.55.2.

Proof of the claim. Let $A$ and $f_1, \ldots , f_ r$ be as above. Consider the affine open $\mathop{\mathrm{Spec}}(A[s]) \subset \mathbf{P}^1_ X$ as in Section 42.53. Recall that $s = 0$ defines $(\mathbf{P}^1_ X)_\infty $ over this open. Hence over $\mathop{\mathrm{Spec}}(A[s])$ we are blowing up in the ideal generated by the regular sequence $s, f_1, \ldots , f_ r$. By More on Algebra, Lemma 15.31.2 the $r + 1$ affine charts are global complete intersections over $A[s]$. The chart corresponding to the affine blowup algebra

\[ A[s][f_1/s, \ldots , f_ r/s] = A[s, y_1, \ldots , y_ r]/(sy_ i - f_ i) \]

contains $i'(Z \cap \mathop{\mathrm{Spec}}(A))$ as the closed subscheme cut out by $y_1, \ldots , y_ r$. Since $y_1, \ldots , y_ r, sy_1 - f_1, \ldots , sy_ r - f_ r$ is a regular sequence in the polynomial ring $A[s, y_1, \ldots , y_ r]$ we find that $i'$ is a regular immersion. Some details omitted. As above we conclude that $Q = i'_*\mathcal{O}_{\mathbf{P}^1_ Z}$ is a perfect object of $D(\mathcal{O}_ W)$. All the other assumptions on $F$ and $Q$ in Lemma 42.51.2 (and Lemma 42.49.1) are immediately verified. $\square$

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