## 42.50 Localized Chern classes

Outline of the construction. Let $F$ be a field, let $X$ be a variety over $F$, let $E$ be a perfect object of $D(\mathcal{O}_ X)$, and let $Z \subset X$ be a closed subscheme such that $E|_{X \setminus Z} = 0$. Then we want to construct elements

$c_ p(Z \to X, E) \in A^ p(Z \to X)$

We will do this by constructing a diagram

$\xymatrix{ W \ar[d]_ f \ar[r]_ q & X \\ \mathbf{P}^1_ F }$

and a perfect object $Q$ of $D(\mathcal{O}_ W)$ such that

1. $f$ is flat, and $f$, $q$ are proper; for $t \in \mathbf{P}^1_ F$ denote $W_ t$ the fibre of $f$, $q_ t : W_ t \to X$ the restriction of $q$, and $Q_ t = Q|_{W_ t}$,

2. $q_ t : W_ t \to X$ is an isomorphism and $Q_ t = q_ t^*E$ for $t \in \mathbf{A}^1_ F$,

3. $q_\infty : W_\infty \to X$ is an isomorphism over $X \setminus Z$,

4. if $T \subset W_\infty$ is the closure of $q_\infty ^{-1}(X \setminus Z)$ then $Q_\infty |_ T$ is zero.

The idea is to think of this as a family $\{ (W_ t, Q_ t)\}$ parametrized by $t \in \mathbf{P}^1$. For $t \not= \infty$ we see that $c_ p(Q_ t)$ is just $c_ p(E)$ on the chow groups of $Q_ t = X$. But for $t = \infty$ we see that $c_ p(Q_\infty )$ sends classes on $Q_\infty$ to classes supported on $E = q_\infty ^{-1}(Z)$ since $Q_\infty |_ T = 0$. We think of $E$ as the exceptional locus of $q_\infty : W_\infty \to X$. Since any $\alpha \in \mathop{\mathrm{CH}}\nolimits _*(X)$ gives rise to a “family” of cycles $\alpha _ t \in \mathop{\mathrm{CH}}\nolimits _*(W_ t)$ it makes sense to define $c_ p(Z \to X, E) \cap \alpha$ as the pushforward $(E \to Z)_*(c_ p(Q_\infty ) \cap \alpha _\infty )$.

To make this work there are two main ingredients: (1) the construction of $W$ and $Q$ is a sort of algebraic Macpherson's graph construction; it is done in More on Flatness, Section 38.44. (2) the construction of the actual class given $W$ and $Q$ is done in Section 42.49 relying on Sections 42.48 and 42.47.

Situation 42.50.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a scheme locally of finite type over $S$. Let $i : Z \to X$ be a closed immersion. Let $E \in D(\mathcal{O}_ X)$ be an object. Let $p \geq 0$. Assume

1. $E$ is a perfect object of $D(\mathcal{O}_ X)$,

2. the restriction $E|_{X \setminus Z}$ is zero, resp. isomorphic to a finite locally free $\mathcal{O}_{X \setminus Z}$-module of rank $< p$ sitting in cohomological degree $0$, and

3. at least one1 of the following is true: (a) $X$ is quasi-compact, (b) $X$ has quasi-compact irreducible components, (c) there exists a locally bounded complex of finite locally free $\mathcal{O}_ X$-modules representing $E$, or (d) there exists a morphism $X \to X'$ of schemes locally of finite type over $S$ such that $E$ is the pullback of a perfect object on $X'$ and the irreducible components of $X'$ are quasi-compact.

Lemma 42.50.2. In Situation 42.50.1 there exists a canonical bivariant class

$P_ p(Z \to X, E) \in A^ p(Z \to X), \quad \text{resp.}\quad c_ p(Z \to X, E) \in A^ p(Z \to X)$

with the property that

42.50.2.1
$$\label{chow-equation-defining-property-localized-classes} i_* \circ P_ p(Z \to X, E) = P_ p(E), \quad \text{resp.}\quad i_* \circ c_ p(Z \to X, E) = c_ p(E)$$

as bivariant classes on $X$ (with $\circ$ as in Lemma 42.33.4).

Proof. The construction of these bivariant classes is as follows. Let

$b : W \longrightarrow \mathbf{P}^1_ X \quad \text{and}\quad T \longrightarrow W_\infty \quad \text{and}\quad Q$

be the blowing up, the perfect object $Q$ in $D(\mathcal{O}_ W)$, and the closed immersion constructed in More on Flatness, Section 38.44 and Lemma 38.44.1. Let $T' \subset T$ be the open and closed subscheme such that $Q|_{T'}$ is zero, resp. isomorphic to a finite locally free $\mathcal{O}_{T'}$-module of rank $< p$ sitting in cohomological degree $0$. By condition (2) of Situation 42.50.1 the morphisms

$T' \to T \to W_\infty \to X$

are all isomorphisms of schemes over the open subscheme $X \setminus Z$ of $X$. Below we check the chern classes of $Q$ are defined. Recalling that $Q|_{X \times \{ 0\} } \cong E$ by construction, we conclude that the bivariant class constructed in Lemma 42.49.1 using $W, b, Q, T'$ gives us classes

$P_ p(Z \to X, E) = P'_ p(Q) \in A^ p(Z \to X)$

and

$c_ p(Z \to X, E) = c'_ p(Q) \in A^ p(Z \to X)$

satisfying (42.50.2.1).

In this paragraph we prove that the chern classes of $Q$ are defined (Definition 42.46.3); we suggest the reader skip this. If assumption (3)(a) or (3)(b) of Situation 42.50.1 holds, i.e., if $X$ has quasi-compact irreducible components, then the same is true for $W$ (because $W \to X$ is proper). Hence we conclude that the chern classes of any perfect object of $D(\mathcal{O}_ W)$ are defined by Lemma 42.46.4. If (3)(c) hold, i.e., if $E$ can be represented by a locally bounded complex of finite locally free modules, then the object $Q$ can be represented by a locally bounded complex of finite locally free $\mathcal{O}_ W$-modules by part (5) of More on Flatness, Lemma 38.44.1. Hence the chern classes of $Q$ are defined. Finally, assume (3)(d) holds, i.e., assume we have a morphism $X \to X'$ of schemes locally of finite type over $S$ such that $E$ is the pullback of a perfect object $E'$ on $X'$ and the irreducible components of $X'$ are quasi-compact. Let $b' : W' \to \mathbf{P}^1_{X'}$ and $Q' \in D(\mathcal{O}_{W'})$ be the morphism and perfect object constructed as in More on Flatness, Section 38.44 starting with the triple $(\mathbf{P}^1_{X'}, (\mathbf{P}^1_{X'})_\infty , L(p')^*E')$. By the discussion above we see that the chern classes of $Q'$ are defined. Since $b$ and $b'$ were constructed via an application of More on Flatness, Lemma 38.43.6 it follows from More on Flatness, Lemma 38.43.8 that there exists a morphism $W \to W'$ such that $Q = L(W \to W')^*Q'$. Then it follows from Lemma 42.46.4 that the chern classes of $Q$ are defined. $\square$

Definition 42.50.3. With $(S, \delta )$, $X$, $E \in D(\mathcal{O}_ X)$, and $i : Z \to X$ as in Situation 42.50.1.

1. If the restriction $E|_{X \setminus Z}$ is zero, then for all $p \geq 0$ we define

$P_ p(Z \to X, E) \in A^ p(Z \to X)$

by the construction in Lemma 42.50.2 and we define the localized Chern character by the formula

$ch(Z \to X, E) = \sum \nolimits _{p = 0, 1, 2, \ldots } \frac{P_ p(Z \to X, E)}{p!} \quad \text{in}\quad \prod \nolimits _{p \geq 0} A^ p(Z \to X) \otimes \mathbf{Q}$
2. If the restriction $E|_{X \setminus Z}$ is isomorphic to a finite locally free $\mathcal{O}_{X \setminus Z}$-module of rank $< p$ sitting in cohomological degree $0$, then we define the localized $p$th Chern class $c_ p(Z \to X, E)$ by the construction in Lemma 42.50.2.

In the situation of the definition assume $E|_{X \setminus Z}$ is zero. Then, to be sure, we have the equality

$i_* \circ ch(Z \to X, E) = ch(E)$

in $A^*(X) \otimes \mathbf{Q}$ because we have shown the equality (42.50.2.1) above.

Here is an important sanity check.

Lemma 42.50.4. In Situation 42.50.1 let $f : X' \to X$ be a morphism of schemes which is locally of finite type. Denote $E' = f^*E$ and $Z' = f^{-1}(Z)$. Then the bivariant class of Definition 42.50.3

$P_ p(Z' \to X', E') \in A^ p(Z' \to X'), \quad \text{resp.}\quad c_ p(Z' \to X', E') \in A^ p(Z' \to X')$

constructed as in Lemma 42.50.2 using $X', Z', E'$ is the restriction (Remark 42.33.5) of the bivariant class $P_ p(Z \to X, E) \in A^ p(Z \to X)$, resp. $c_ p(Z \to X, E) \in A^ p(Z \to X)$.

Proof. Denote $p : \mathbf{P}^1_ X \to X$ and $p' : \mathbf{P}^1_{X'} \to X'$ the structure morphisms. Recall that $b : W \to \mathbf{P}^1_ X$ and $b' : W' \to \mathbf{P}^1_{X'}$ are the morphism constructed from the triples $(\mathbf{P}^1_ X, (\mathbf{P}^1_ X)\infty , p^*E)$ and $(\mathbf{P}^1_{X'}, (\mathbf{P}^1_{X'})\infty , (p')^*E')$ in More on Flatness, Lemma 38.43.6. Furthermore $Q = L\eta _{\mathcal{I}_\infty }p^*E$ and $Q = L\eta _{\mathcal{I}'_\infty }(p')^*E'$ where $\mathcal{I}_\infty \subset \mathcal{O}_ W$ is the ideal sheaf of $W_\infty$ and $\mathcal{I}'_\infty \subset \mathcal{O}_{W'}$ is the ideal sheaf of $W'_\infty$. Next, $h : \mathbf{P}^1_{X'} \to \mathbf{P}^1_ X$ is a morphism of schemes such that the pullback of the effective Cartier divisor $(\mathbf{P}^1_ X)_\infty$ is the effective Cartier divisor $(\mathbf{P}^1_{X'})_\infty$ and such that $h^*p^*E = (p')^*E'$. By More on Flatness, Lemma 38.43.8 we obtain a commutative diagram

$\xymatrix{ W' \ar[rd]_{b'} \ar[r]_-g & \mathbf{P}^1_{X'} \times _{\mathbf{P}^1_ X} W \ar[d]_ r \ar[r]_-q & W \ar[d]^ b \\ & \mathbf{P}^1_{X'} \ar[r] & \mathbf{P}^1_ X }$

such that $W'$ is the “strict transform” of $\mathbf{P}^1_{X'}$ with respect to $b$ and such that $Q' = (q \circ g)^*Q$. Now recall that $P_ p(Z \to X, E) = P'_ p(Q)$, resp. $c_ p(Z \to X, E) = c'_ p(Q)$ where $P'_ p(Q)$, resp. $c'_ p(Q)$ are constructed in Lemma 42.49.1 using $b, Q, T'$ where $T'$ is a closed subscheme $T' \subset W_\infty$ with the following two properties: (a) $T'$ contains all points of $W_\infty$ lying over $X \setminus Z$, and (b) $Q|_{T'}$ is zero, resp. isomorphic to a finite locally free module of rank $< p$ placed in degree $0$. In the construction of Lemma 42.49.1 we chose a particular closed subscheme $T'$ with properties (a) and (b) but the precise choice of $T'$ is immaterial, see Lemma 42.49.3.

Next, by Lemma 42.49.2 the restriction of the bivariant class $P_ p(Z \to X, E) = P'_ p(Q)$, resp. $c_ p(Z \to X, E) = c_ p(Q')$ to $X'$ corresponds to the class $P'_ p(q^*Q)$, resp. $c'_ p(q^*Q)$ constructed as in Lemma 42.49.1 using $r : \mathbf{P}^1_{X'} \times _{\mathbf{P}^1_ X} W \to \mathbf{P}^1_{X'}$, the complex $q^*Q$, and the inverse image $q^{-1}(T')$.

Now by the second statement of Lemma 42.49.3 we have $P'_ p(Q') = P'_ p(q^*Q)$, resp. $c'_ p(q^*Q) = c'_ p(Q')$. Since $P_ p(Z' \to X', E') = P'_ p(Q')$, resp. $c_ p(Z' \to X', E') = c'_ p(Q')$ we conclude that the lemma is true. $\square$

Remark 42.50.5. In Situation 42.50.1 it would have been more natural to replace assumption (3) with the assumption: “the chern classes of $E$ are defined”. In fact, combining Lemmas 42.50.2 and 42.50.4 with Lemma 42.35.6 it is easy to extend the definition to this (slightly) more general case. If we ever need this we will do so here.

Lemma 42.50.6. In Situation 42.50.1 we have

$P_ p(Z \to X, E) \cap i_*\alpha = P_ p(E|_ Z) \cap \alpha , \quad \text{resp.}\quad c_ p(Z \to X, E) \cap i_*\alpha = c_ p(E|_ Z) \cap \alpha$

in $\mathop{\mathrm{CH}}\nolimits _*(Z)$ for any $\alpha \in \mathop{\mathrm{CH}}\nolimits _*(Z)$.

Proof. We only prove the second equality and we omit the proof of the first. Since $c_ p(Z \to X, E)$ is a bivariant class and since the base change of $Z \to X$ by $Z \to X$ is $\text{id} : Z \to Z$ we have $c_ p(Z \to X, E) \cap i_*\alpha = c_ p(Z \to X, E) \cap \alpha$. By Lemma 42.50.4 the restriction of $c_ p(Z \to X, E)$ to $Z$ (!) is the localized Chern class for $\text{id} : Z \to Z$ and $E|_ Z$. Thus the result follows from (42.50.2.1) with $X = Z$. $\square$

Lemma 42.50.7. In Situation 42.50.1 if $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$ has support disjoint from $Z$, then $P_ p(Z \to X, E) \cap \alpha = 0$, resp. $c_ p(Z \to X, E) \cap \alpha = 0$.

Proof. This is immediate from the construction of the localized Chern classes. It also follows from the fact that we can compute $c_ p(Z \to X, E) \cap \alpha$ by first restricting $c_ p(Z \to X, E)$ to the support of $\alpha$, and then using Lemma 42.50.4 to see that this restriction is zero. $\square$

Lemma 42.50.8. In Situation 42.50.1 assume $Z \subset Z' \subset X$ where $Z'$ is a closed subscheme of $X$. Then $P_ p(Z' \to X, E) = (Z \to Z')_* \circ P_ p(Z \to X, E)$, resp. $c_ p(Z' \to X, E) = (Z \to Z')_* \circ c_ p(Z \to X, E)$ (with $\circ$ as in Lemma 42.33.4).

Proof. The construction of $P_ p(Z' \to X, E)$, resp. $c_ p(Z' \to X, E)$ in Lemma 42.50.2 uses the exact same morphism $b : W \to \mathbf{P}^1_ X$ and perfect object $Q$ of $D(\mathcal{O}_ W)$. Then we can use Lemma 42.47.5 to conclude. Some details omitted. $\square$

Lemma 42.50.9. In Lemma 42.47.1 say $E_2$ is the restriction of a perfect $E \in D(\mathcal{O}_ X)$ whose restriction to $X_1$ is zero, resp. isomorphic to a finite locally free $\mathcal{O}_{X_1}$-module of rank $< p$ sitting in cohomological degree $0$. Then the class $P'_ p(E_2)$, resp. $c'_ p(E_2)$ of Lemma 42.47.1 agrees with $P_ p(X_2 \to X, E)$, resp. $c_ p(X_2 \to X, E)$ of Definition 42.50.3 provided $E$ satisfies assumption (3) of Situation 42.50.1.

Proof. The assumptions on $E$ imply that there is an open $U \subset X$ containing $X_1$ such that $E|_ U$ is zero, resp. isomorphic to a finite locally free $\mathcal{O}_ U$-module of rank $< p$. See More on Algebra, Lemma 15.75.7. Let $Z \subset X$ be the complement of $U$ in $X$ endowed with the reduced induced closed subscheme structure. Then $P_ p(X_2 \to X, E) = (Z \to X_2)_* \circ P_ p(Z \to X, E)$, resp. $c_ p(X_2 \to X, E) = (Z \to X_2)_* \circ c_ p(Z \to X, E)$ by Lemma 42.50.8. Now we can prove that $P_ p(X_2 \to X, E)$, resp. $c_ p(X_2 \to X, E)$ satisfies the characterization of $P'_ p(E_2)$, resp. $c'_ p(E_2)$ given in Lemma 42.47.1. Namely, by the relation $P_ p(X_2 \to X, E) = (Z \to X_2)_* \circ P_ p(Z \to X, E)$, resp. $c_ p(X_2 \to X, E) = (Z \to X_2)_* \circ c_ p(Z \to X, E)$ just proven and the fact that $X_1 \cap Z = \emptyset$, the composition $P_ p(X_2 \to X, E) \circ i_{1, *}$, resp. $c_ p(X_2 \to X, E) \circ i_{1, *}$ is zero by Lemma 42.50.7. On the other hand, $P_ p(X_2 \to X, E) \circ i_{2, *} = P_ p(E_2)$, resp. $c_ p(X_2 \to X, E) \circ i_{2, *} = c_ p(E_2)$ by Lemma 42.50.6. $\square$

[1] Please ignore this technical condition on a first reading; see discussion in Remark 42.50.5.

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