42.55 Calculating some classes

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

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

$\prod \nolimits _{n = 0, \ldots , r} c(\wedge ^ n \mathcal{E})^{(-1)^ n} = 1 - (r - 1)! c_ r(\mathcal{E}) + \ldots$

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

$c(\wedge ^ n \mathcal{E}) = \prod \nolimits _{1 \leq i_1 < \ldots < i_ n \leq r} (1 + x_{i_1} + \ldots + x_{i_ n})$

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

$- \sum \nolimits _{p \geq 1} \sum \nolimits _{n = 0}^ r \sum \nolimits _{1 \leq i_1 < \ldots < i_ n \leq r} \frac{(-1)^{p + n}}{p}(x_{i_1} + \ldots + x_{i_ n})^ p$

$\sum \nolimits _{p \geq 0} \sum \nolimits _{n = 0}^ r \sum \nolimits _{1 \leq i_1 < \ldots < i_ n \leq r} \frac{(-1)^{p + n}}{p!}(x_{i_1} + \ldots + x_{i_ n})^ p = \prod (1 - e^{-x_ i})$

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

$X = \underline{\text{Proj}}_ X(\mathcal{O}_ X[T]) \xrightarrow {i} E = \underline{\text{Proj}}_ X(\text{Sym}^*(\mathcal{C})[T]) \xrightarrow {\pi } X$

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

$p^* \circ \pi _* \circ c_ r(i_*\mathcal{O}_ X) = (-1)^{r - 1}(r - 1)! j^*$

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

$\pi ^*\mathcal{C} \otimes \mathcal{O}_ E(-1) \to \mathcal{O}_ E$

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

$c_ r(i_*\mathcal{O}_ X) = - (r - 1)! c_ r(\pi ^*\mathcal{C} \otimes \mathcal{O}_ E(-1))$

On the other hand, by Lemma 42.43.3 we have

$c_ r(\pi ^*\mathcal{C} \otimes \mathcal{O}_ E(-1)) = (-1)^ r c_ r(\pi ^*\mathcal{C}^\vee \otimes \mathcal{O}_ E(1))$

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

$p^*(\pi _*(c_ r(\pi ^*\mathcal{C}^\vee \otimes \mathcal{O}_ E(1)) \cap [W])) = j^*[W]$

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

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

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. $\square$

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

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

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

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

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

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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