42.52 Properties of localized Chern classes

The main results in this section are additivity and multiplicativity for localized Chern classes.

Lemma 42.52.1. In Situation 42.50.1 assume $E|_{X \setminus Z}$ is zero. Then

\begin{align*} P_1(Z \to X, E) & = c_1(Z \to X, E), \\ P_2(Z \to X, E) & = c_1(Z \to X, E)^2 - 2c_2(Z \to X, E), \\ P_3(Z \to X, E) & = c_1(Z \to X, E)^3 - 3c_1(Z \to X, E)c_2(Z \to X, E) + 3c_3(Z \to X, E), \end{align*}

and so on where the products are taken in the algebra $A^{(1)}(Z \to X)$ of Remark 42.34.7.

Proof. The statement makes sense because the zero sheaf has rank $< 1$ and hence the classes $c_ p(Z \to X, E)$ are defined for all $p \geq 1$. The result itself follows immediately from the more general Lemma 42.49.6 as the localized Chern classes where defined using the procedure of Lemma 42.49.1 in Section 42.50. $\square$

Lemma 42.52.2. In Situation 42.50.1 let $Y \to X$ be locally of finite type and $c \in A^*(Y \to X)$. Then

$P_ p(Z \to X, E) \circ c = c \circ P_ p(Z \to X, E),$

respectively

$c_ p(Z \to X, E) \circ c = c \circ c_ p(Z \to X, E)$

in $A^*(Y \times _ X Z \to X)$.

Proof. This follows from Lemma 42.49.5. More precisely, let

$b : W \to \mathbf{P}^1_ X \quad \text{and}\quad Q \quad \text{and}\quad T' \subset T \subset W_\infty$

be as in the proof of Lemma 42.50.2. By definition $c_ p(Z \to X, E) = c'_ p(Q)$ as bivariant operations where the right hand side is the bivariant class constructed in Lemma 42.49.1 using $W, b, Q, T'$. By Lemma 42.49.5 we have $P'_ p(Q) \circ c = c \circ P'_ p(Q)$, resp. $c'_ p(Q) \circ c = c \circ c'_ p(Q)$ in $A^*(Y \times _ X Z \to X)$ and we conclude. $\square$

Remark 42.52.3. In Situation 42.50.1 it is convenient to define

$c^{(p)}(Z \to X, E) = 1 + c_1(E) + \ldots + c_{p - 1}(E) + c_ p(Z \to X, E) + c_{p + 1}(Z \to X, E) + \ldots$

as an element of the algebra $A^{(p)}(Z \to X)$ considered in Remark 42.34.7.

Lemma 42.52.4. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $Z \to X$ be a closed immersion. Let

$E_1 \to E_2 \to E_3 \to E_1[1]$

be a distinguished triangle of perfect objects in $D(\mathcal{O}_ X)$. Assume

1. the restrictions $E_1|_{X \setminus Z}$ and $E_3|_{X \setminus Z}$ are isomorphic to finite locally free $\mathcal{O}_{X \setminus Z}$-modules of rank $< p_1$ and $< p_3$ placed in degree $0$, and

2. at least one of the following is true: (a) $X$ is quasi-compact, (b) $X$ has quasi-compact irreducible components, (c) $E_3 \to E_1[1]$ can be represented by a map of locally bounded complexes of finite locally free $\mathcal{O}_ X$-modules, or (d) there exists an envelope $f : Y \to X$ such that $Lf^*E_3 \to Lf^*E_1[1]$ can be represented by a map of locally bounded complexes of finite locally free $\mathcal{O}_ Y$-modules.

With notation as in Remark 42.52.3 we have

$c^{(p_1 + p_3)}(Z \to X, E_2) = c^{(p_1)}(Z \to X, E_1)c^{(p_3)}(Z \to X, E_3)$

in $A^{(p_1 + p_3)}(Z \to X)$.

Proof. Observe that the assumptions imply that $E_2|_{X \setminus Z}$ is zero, resp. isomorphic to a finite locally free $\mathcal{O}_{X \setminus Z}$-module of rank $< p_1 + p_3$. Thus the statement makes sense.

Let $f : Y \to X$ be an envelope. Expaing the left and right hand sides of the formula in the statement of the lemma we see that we have to prove some equalities of classes in $A^*(X)$ and in $A^*(Z \to X)$. By the uniqueness in Lemma 42.35.6 it suffices to prove the corresponding relations in $A^*(Y)$ and $A^*(Z \to Y)$. Since moreover the construction of the classes involved is compatible with base change (Lemma 42.50.4) we may replace $X$ by $Y$ and the distinguished triangle by its pullback.

In the proof of Lemma 42.46.7 we have seen that conditions (2)(a), (2)(b), and (2)(c) imply condition (2)(d). Combined with the discussion in the previous paragraph we reduce to the case discussed in the next paragraph.

Let $\varphi ^\bullet : \mathcal{E}_3^\bullet [-1] \to \mathcal{E}_1^\bullet$ be a map of locally bounded complexes of finite locally free $\mathcal{O}_ X$-modules representing the map $E_3[-1] \to E_1$ in the derived category. Consider the scheme $X' = \mathbf{A}^1 \times X$ with projection $g : X' \to X$. Let $Z' = g^{-1}(Z) = \mathbf{A}^1 \times Z$. Denote $t$ the coordinate on $\mathbf{A}^1$. Consider the cone $\mathcal{C}^\bullet$ of the map of complexes

$t g^*\varphi ^\bullet : g^*\mathcal{E}_3^\bullet [-1] \longrightarrow g^*\mathcal{E}_1^\bullet$

over $X'$. We obtain a distinguished triangle

$g^*\mathcal{E}_1^\bullet \to \mathcal{C}^\bullet \to g^*\mathcal{E}_3^\bullet \to g^*\mathcal{E}_1^\bullet [1]$

where the first three terms form a termwise split short exact sequence of complexes. Clearly $\mathcal{C}^\bullet$ is a bounded complex of finite locally free $\mathcal{O}_{X'}$-modules whose restriction to $X' \setminus Z'$ is isomorphic to a finite locally free $\mathcal{O}_{X' \setminus Z'}$-module of rank $< p_1 + p_3$ placed in degree $0$. Thus we have the localized Chern classes

$c_ p(Z' \to X', \mathcal{C}^\bullet ) \in A^ p(Z' \to X')$

for $p \geq p_1 + p_3$. For any $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$ consider

$c_ p(Z' \to X', \mathcal{C}^\bullet ) \cap g^*\alpha \in \mathop{\mathrm{CH}}\nolimits _{k + 1 - p}(\mathbf{A}^1 \times X)$

If we restrict to $t = 0$, then the map $t g^*\varphi ^\bullet$ restricts to zero and $\mathcal{C}^\bullet |_{t = 0}$ is the direct sum of $\mathcal{E}_1^\bullet$ and $\mathcal{E}_3^\bullet$. By compatibility of localized Chern classes with base change (Lemma 42.50.4) we conclude that

$i_0^* \circ c^{(p_1 + p_3)}(Z' \to X', \mathcal{C}^\bullet ) \circ g^* = c^{(p_1 + p_2)}(Z \to X, E_1 \oplus E_3)$

in $A^{(p_1 + p_3)}(Z \to X)$. On the other hand, if we restrict to $t = 1$, then the map $t g^*\varphi ^\bullet$ restricts to $\varphi$ and $\mathcal{C}^\bullet |_{t = 1}$ is a bounded complex of finite locally free modules representing $E_2$. We conclude that

$i_1^* \circ c^{(p_1 + p_3)}(Z' \to X', \mathcal{C}^\bullet ) \circ g^* = c^{(p_1 + p_2)}(Z \to X, E_2)$

in $A^{(p_1 + p_3)}(Z \to X)$. Since $i_0^* = i_1^*$ by definition of rational equivalence (more precisely this follows from the formulae in Lemma 42.32.4) we conclude that

$c^{(p_1 + p_2)}(Z \to X, E_2) = c^{(p_1 + p_2)}(Z \to X, E_1 \oplus E_3)$

This reduces us to the case discussed in the next paragraph.

Assume $E_2 = E_1 \oplus E_3$ and the triples $(X, Z, E_ i)$ are as in Situation 42.50.1. For $i = 1, 3$ let

$b_ i : W_ i \to \mathbf{P}^1_ X \quad \text{and}\quad Q_ i \quad \text{and}\quad T'_ i \subset T_ i \subset W_{i, \infty }$

be as in the proof of Lemma 42.50.2. By definition

$c_ p(Z \to X, E_ i) = c'_ p(Q_ i)$

where the right hand side is the bivariant class constructed in Lemma 42.49.1 using $W_ i, b_ i, Q_ i, T'_ i$. Set $W = W_1 \times _{b_1, \mathbf{P}^1_ X, b_2} W_2$ and consider the cartesian diagram

$\xymatrix{ W \ar[d]_{g_1} \ar[rd]^ b \ar[r]_{g_3} & W_3 \ar[d]^{b_3} \\ W_1 \ar[r]^{b_1} & \mathbf{P}^1_ X }$

Of course $b^{-1}(\mathbf{A}^1)$ maps isomorphically to $\mathbf{A}^1_ X$. Observe that $T' = g_1^{-1}(T'_1) \cap g_2^{-1}(T'_2)$ still contains all the points of $W_\infty$ lying over $X \setminus Z$. By Lemma 42.49.3 we may use $W$, $b$, $g_ i^*\mathcal{Q}_ i$, and $T'$ to construct $c_ p(Z \to X, E_ i)$ for $i = 1, 3$. Also, by the stronger independence given in Lemma 42.51.2 we may use $W$, $b$, $g_1^*Q_1 \oplus g_3^*Q_3$, and $T'$ to compute the classes $c_ p(Z \to X, E_2)$. Thus the desired equality follows from Lemma 42.49.7. $\square$

Lemma 42.52.5. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $Z \to X$ be a closed immersion. Let

$E_1 \to E_2 \to E_3 \to E_1[1]$

be a distinguished triangle of perfect objects in $D(\mathcal{O}_ X)$. Assume

1. the restrictions $E_1|_{X \setminus Z}$ and $E_3|_{X \setminus Z}$ are zero, and

2. at least one of the following is true: (a) $X$ is quasi-compact, (b) $X$ has quasi-compact irreducible components, (c) $E_3 \to E_1[1]$ can be represented by a map of locally bounded complexes of finite locally free $\mathcal{O}_ X$-modules, or (d) there exists an envelope $f : Y \to X$ such that $Lf^*E_3 \to Lf^*E_1[1]$ can be represented by a map of locally bounded complexes of finite locally free $\mathcal{O}_ Y$-modules.

Then we have

$P_ p(Z \to X, E_2) = P_ p(Z \to X, E_1) + P_ p(Z \to X, E_3)$

for all $p \in \mathbf{Z}$ and consequently $ch(Z \to X, E_2) = ch(Z \to X, E_1) + ch(Z \to X, E_3)$.

Proof. The proof is exactly the same as the proof of Lemma 42.52.4 except it uses Lemma 42.49.8 at the very end. For $p > 0$ we can deduce this lemma from Lemma 42.52.4 with $p_1 = p_3 = 1$ and the relationship between $P_ p(Z \to X, E)$ and $c_ p(Z \to X, E)$ given in Lemma 42.52.1. The case $p = 0$ can be shown directly (it is only interesting if $X$ has a connected component entirely contained in $Z$). $\square$

Lemma 42.52.6. In Situation 42.7.1 let $X$ be locally of finite type over $S$. Let $Z_ i \subset X$, $i = 1, 2$ be closed subschemes. Let $F_ i$, $i = 1, 2$ be perfect objects of $D(\mathcal{O}_ X)$. Assume for $i = 1, 2$ that $F_ i|_{X \setminus Z_ i}$ is zero1 and that $F_ i$ on $X$ satisfies assumption (3) of Situation 42.50.1. Denote $r_ i = P_0(Z_ i \to X, F_ i) \in A^0(Z_ i \to X)$. Then we have

$c_1(Z_1 \cap Z_2 \to X, F_1 \otimes _{\mathcal{O}_ X}^\mathbf {L} F_2) = r_1 c_1(Z_2 \to X, F_2) + r_2 c_1(Z_1 \to X, F_1)$

in $A^1(Z_1 \cap Z_2 \to X)$ and

\begin{align*} c_2(Z_1 \cap Z_2 \to X, F_1 \otimes _{\mathcal{O}_ X}^\mathbf {L} F_2) & = r_1 c_2(Z_2 \to X, F_2) + r_2 c_2(Z_1 \to X, F_1) + \\ & {r_1 \choose 2} c_1(Z_2 \to X, F_2)^2 + \\ & (r_1r_2 - 1) c_1(Z_2 \to X, F_2)c_1(Z_1 \to X, F_1) + \\ & {r_2 \choose 2} c_1(Z_1 \to X, F_1)^2 \end{align*}

in $A^2(Z_1 \cap Z_2 \to X)$ and so on for higher Chern classes. Similarly, we have

$ch(Z_1 \cap Z_2 \to X, F_1 \otimes _{\mathcal{O}_ X}^\mathbf {L} F_2) = ch(Z_1 \to X, F_1) ch(Z_2 \to X, F_2)$

in $A^*(Z_1 \cap Z_2 \to X) \otimes \mathbf{Q}$. More precisely, we have

$P_ p(Z_1 \cap Z_2 \to X, F_1 \otimes _{\mathcal{O}_ X}^\mathbf {L} F_2) = \sum \nolimits _{p_1 + p_2 = p} {p \choose p_1} P_{p_1}(Z_1 \to X, F_1) P_{p_2}(Z_2 \to X, F_2)$

in $A^ p(Z_1 \cap Z_2 \to X)$.

Proof. Choose proper morphisms $b_ i : W_ i \to \mathbf{P}^1_ X$ and $Q_ i \in D(\mathcal{O}_{W_ i})$ as well as closed subschemes $T_ i \subset W_{i, \infty }$ as in the construction of the localized Chern classes for $F_ i$ or more generally as in Lemma 42.51.2. Choose a commutative diagram

$\xymatrix{ W \ar[d]^{g_1} \ar[rd]^ b \ar[r]_{g_2} & W_2 \ar[d]^{b_2} \\ W_1 \ar[r]^{b_1} & \mathbf{P}^1_ X }$

where all morphisms are proper and isomorphisms over $\mathbf{A}^1_ X$. For example, we can take $W$ to be the closure of the graph of the isomorphism between $b_1^{-1}(\mathbf{A}^1_ X)$ and $b_2^{-1}(\mathbf{A}^1_ X)$. By Lemma 42.51.2 we may work with $W$, $b = b_ i \circ g_ i$, $Lg_ i^*Q_ i$, and $g_ i^{-1}(T_ i)$ to construct the localized Chern classes $c_ p(Z_ i \to X, F_ i)$. Thus we reduce to the situation described in the next paragraph.

Assume we have

1. a proper morphism $b : W \to \mathbf{P}^1_ X$ which is an isomorphism over $\mathbf{A}^1_ X$,

2. $E_ i \subset W_\infty$ is the inverse image of $Z_ i$,

3. perfect objects $Q_ i \in D(\mathcal{O}_ W)$ whose Chern classes are defined, such that

1. the restriction of $Q_ i$ to $b^{-1}(\mathbf{A}^1_ X)$ is the pullback of $F_ i$, and

2. there exists a closed subscheme $T_ i \subset W_\infty$ containing all points of $W_\infty$ lying over $X \setminus Z_ i$ such that $Q_ i|_{T_ i}$ is zero.

By Lemma 42.51.2 we have

$c_ p(Z_ i \to X, F_ i) = c'_ p(Q_ i) = (E_ i \to Z_ i)_* \circ c'_ p(Q_ i|_{E_ i}) \circ C$

and

$P_ p(Z_ i \to X, F_ i) = P'_ p(Q_ i) = (E_ i \to Z_ i)_* \circ P'_ p(Q_ i|_{E_ i}) \circ C$

for $i = 1, 2$. Next, we observe that $Q = Q_1 \otimes _{\mathcal{O}_ W}^\mathbf {L} Q_2$ satisfies (3)(a) and (3)(b) for $F_1 \otimes _{\mathcal{O}_ X}^\mathbf {L} F_2$ and $T_1 \cup T_2$. Hence we see that

$c_ p(Z_1 \cap Z_2 \to X, F_1 \otimes _{\mathcal{O}_ X}^\mathbf {L} F_2) = (E_1 \cap E_2 \to Z_1 \cap Z_2)_* \circ c'_ p(Q|_{E_1 \cap E_2}) \circ C$

and

$P_ p(Z_1 \cap Z_2 \to X, F_1 \otimes _{\mathcal{O}_ X}^\mathbf {L} F_2) = (E_1 \cap E_2 \to Z_1 \cap Z_2)_* \circ P'_ p(Q|_{E_1 \cap E_2}) \circ C$

by the same lemma. By Lemma 42.47.11 the classes $c'_ p(Q|_{E_1 \cap E_2})$ and $P'_ p(Q|_{E_1 \cap E_2})$ can be expanded in the correct manner in terms of the classes $c'_ p(Q_ i|_{E_ i})$ and $P'_ p(Q_ i|_{E_ i})$. Then finally Lemma 42.51.1 tells us that polynomials in $c'_ p(Q_ i|_{E_ i})$ and $P'_ p(Q_ i|_{E_ i})$ agree with the corresponding polynomials in $c'_ p(Q_ i)$ and $P'_ p(Q_ i)$ as desired. $\square$

[1] Presumably there is a variant of this lemma where we only assume $F_ i|_{X \setminus Z_ i}$ is isomorphic to a finite locally free $\mathcal{O}_{X \setminus Z_ i}$-module of rank $< p_ i$.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).