Lemma 42.55.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a scheme locally of finite type over $S$. Let $\mathcal{E}$ be a locally free $\mathcal{O}_ X$-module of rank $r$. Then

## 42.55 Calculating some classes

To get further we need to compute the values of some of the classes we've constructed above.

**Proof.**
By the splitting principle we can turn this into a calculation in the polynomial ring on the Chern roots $x_1, \ldots , x_ r$ of $\mathcal{E}$. See Section 42.43. Observe that

Thus the logarithm of the left hand side of the equation in the lemma is

Please notice the minus sign in front. However, we have

Hence we see that the first nonzero term in our Chern class is in degree $r$ and equal to the predicted value. $\square$

Lemma 42.55.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a scheme locally of finite type over $S$. Let $\mathcal{C}$ be a locally free $\mathcal{O}_ X$-module of rank $r$. Consider the morphisms

Then $c_ t(i_*\mathcal{O}_ X) = 0$ for $t = 1, \ldots , r - 1$ and in $A^0(C \to E)$ we have

where $j : C \to E$ and $p : C \to X$ are the inclusion and structure morphism of the vector bundle $C = \underline{\mathop{\mathrm{Spec}}}(\text{Sym}^*(\mathcal{C}))$.

**Proof.**
The canonical map $\pi ^*\mathcal{C} \to \mathcal{O}_ E(1)$ vanishes exactly along $i(X)$. Hence the Koszul complex on the map

is a resolution of $i_*\mathcal{O}_ X$. In particular we see that $i_*\mathcal{O}_ X$ is a perfect object of $D(\mathcal{O}_ E)$ whose Chern classes are defined. The vanishing of $c_ t(i_*\mathcal{O}_ X)$ for $t = 1, \ldots , t - 1$ follows from Lemma 42.55.1. This lemma also gives

On the other hand, by Lemma 42.43.3 we have

and $\pi ^*\mathcal{C}^\vee \otimes \mathcal{O}_ E(1)$ has a section $s$ vanishing exactly along $i(X)$.

After replacing $X$ by a scheme locally of finite type over $X$, it suffices to prove that both sides of the equality have the same effect on an element $\alpha \in \mathop{\mathrm{CH}}\nolimits _*(E)$. Since $C \to X$ is a vector bundle, every cycle class on $C$ is of the form $p^*\beta $ for some $\beta \in \mathop{\mathrm{CH}}\nolimits _*(X)$ (Lemma 42.36.3). Hence by Lemma 42.19.3 we can write $\alpha = \pi ^*\beta + \gamma $ where $\gamma $ is supported on $E \setminus C$. Using the equalities above it suffices to show that

when $W \subset E$ is an integral closed subscheme which is either (a) disjoint from $C$ or (b) is of the form $W = \pi ^{-1}Y$ for some integral closed subscheme $Y \subset X$. Using the section $s$ and Lemma 42.44.1 we find in case (a) $c_ r(\pi ^*\mathcal{C}^\vee \otimes \mathcal{O}_ E(1)) \cap [W] = 0$ and in case (b) $c_ r(\pi ^*\mathcal{C}^\vee \otimes \mathcal{O}_ E(1)) \cap [W] = [i(Y)]$. The result follows easily from this; details omitted. $\square$

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

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

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

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

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

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. $\square$

Lemma 42.55.4. In the situation of Lemma 42.55.3 say $\dim _\delta (X) = n$. Then we have

$c_ t(Z \to X, i_*\mathcal{O}_ Z) \cap [X]_ n = 0$ for $t = 1, \ldots , r - 1$,

$c_ r(Z \to X, i_*\mathcal{O}_ Z) \cap [X]_ n = (-1)^{r - 1}(r - 1)![Z]_{n - r}$,

$ch_ t(Z \to X, i_*\mathcal{O}_ Z) \cap [X]_ n = 0$ for $t = 0, \ldots , r - 1$, and

$ch_ r(Z \to X, i_*\mathcal{O}_ Z) \cap [X]_ n = [Z]_{n - r}$.

**Proof.**
Parts (1) and (2) follow immediately from Lemma 42.55.3 combined with Lemma 42.54.5. Then we deduce parts (3) and (4) using the relationship between $ch_ p = (1/p!)P_ p$ and $c_ p$ given in Lemma 42.52.1. (Namely, $(-1)^{r - 1}(r - 1)!ch_ r = c_ r$ provided $c_1 = c_2 = \ldots = c_{r - 1} = 0$.)
$\square$

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