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
$f$ 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 $Q_ t = Q|_{W_ t}$,
$q_ t : W_ t \to X$ is an isomorphism and $Q_ t = q_ t^*E$ for $t \in \mathbf{A}^1_ F$,
$q_\infty : W_\infty \to X$ is an isomorphism over $X \setminus Z$,
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
$E$ is a perfect object of $D(\mathcal{O}_ X)$,
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
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
\begin{equation} \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) \end{equation}
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.
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
\[ ch(Z \to X, E) = \sum \nolimits _{p = 0, 1, 2, \ldots } \frac{P_ p(Z \to X, E)}{p!} \quad \text{in}\quad \prod \nolimits _{p \geq 0} A^ p(Z \to X) \otimes \mathbf{Q} \]
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.
$\square$
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 15.75.6. 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.
$\square$
Comments (0)