The Stacks project

42.47 A baby case of localized Chern classes

In this section we discuss some properties of the bivariant classes constructed in the following lemma; most of these properties follow immediately from the characterization given in the lemma. We urge the reader to skip the rest of the section.

Lemma 42.47.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $i_ j : X_ j \to X$, $j = 1, 2$ be closed immersions such that $X = X_1 \cup X_2$ set theoretically. Let $E_2 \in D(\mathcal{O}_{X_2})$ be a perfect object. Assume

  1. Chern classes of $E_2$ are defined,

  2. the restriction $E_2|_{X_1 \cap X_2}$ is zero, resp. isomorphic to a finite locally free $\mathcal{O}_{X_1 \cap X_2}$-module of rank $< p$ sitting in cohomological degree $0$.

Then there is a canonical bivariant class

\[ P'_ p(E_2),\text{ resp. }c'_ p(E_2) \in A^ p(X_2 \to X) \]

characterized by the property

\[ P'_ p(E_2) \cap i_{2, *} \alpha _2 = P_ p(E_2) \cap \alpha _2 \quad \text{and}\quad P'_ p(E_2) \cap i_{1, *} \alpha _1 = 0, \]

respectively

\[ c'_ p(E_2) \cap i_{2, *} \alpha _2 = c_ p(E_2) \cap \alpha _2 \quad \text{and}\quad c'_ p(E_2) \cap i_{1, *} \alpha _1 = 0 \]

for $\alpha _ i \in \mathop{\mathrm{CH}}\nolimits _ k(X_ i)$ and similarly after any base change $X' \to X$ locally of finite type.

Proof. We are going to use the material of Section 42.46 without further mention.

Assume $E_2|_{X_1 \cap X_2}$ is zero. Consider a morphism of schemes $X' \to X$ which is locally of finite type and denote $i'_ j : X'_ j \to X'$ the base change of $i_ j$. By Lemma 42.19.4 we can write any element $\alpha ' \in \mathop{\mathrm{CH}}\nolimits _ k(X')$ as $i'_{1, *}\alpha '_1 + i'_{2, *}\alpha '_2$ where $\alpha '_2 \in \mathop{\mathrm{CH}}\nolimits _ k(X'_2)$ is well defined up to an element in the image of pushforward by $X'_1 \cap X'_2 \to X'_2$. Then we can set $P'_ p(E_2) \cap \alpha ' = P_ p(E_2) \cap \alpha '_2 \in \mathop{\mathrm{CH}}\nolimits _{k - p}(X'_2)$. This is well defined by our assumption that $E_2$ restricts to zero on $X_1 \cap X_2$.

If $E_2|_{X_1 \cap X_2}$ is isomorphic to a finite locally free $\mathcal{O}_{X_1 \cap X_2}$-module of rank $< p$ sitting in cohomological degree $0$, then $c_ p(E_2|_{X_1 \cap X_2}) = 0$ by rank considerations and we can argue in exactly the same manner. $\square$

Lemma 42.47.2. In Lemma 42.47.1 the bivariant class $P'_ p(E_2)$, resp. $c'_ p(E_2)$ in $A^ p(X_2 \to X)$ does not depend on the choice of $X_1$.

Proof. Suppose that $X_1' \subset X$ is another closed subscheme such that $X = X'_1 \cup X_2$ set theoretically and the restriction $E_2|_{X'_1 \cap X_2}$ is zero, resp. isomorphic to a finite locally free $\mathcal{O}_{X'_1 \cap X_2}$-module of rank $< p$ sitting in cohomological degree $0$. Then $X = (X_1 \cap X'_1) \cup X_2$. Hence we can write any element $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$ as $i_*\beta + i_{2, *}\alpha _2$ with $\alpha _2 \in \mathop{\mathrm{CH}}\nolimits _ k(X'_2)$ and $\beta \in \mathop{\mathrm{CH}}\nolimits _ k(X_1 \cap X'_1)$. Thus it is clear that $P'_ p(E_2) \cap \alpha = P_ p(E_2) \cap \alpha _2 \in \mathop{\mathrm{CH}}\nolimits _{k - p}(X_2)$, resp. $c'_ p(E_2) \cap \alpha = c_ p(E_2) \cap \alpha _2 \in \mathop{\mathrm{CH}}\nolimits _{k - p}(X_2)$, is independent of whether we use $X_1$ or $X'_1$. Similarly after any base change. $\square$

Lemma 42.47.3. In Lemma 42.47.1 let $X' \to X$ be a morphism which is locally of finite type. Denote $X' = X'_1 \cup X'_2$ and $E'_2 \in D(\mathcal{O}_{X'_2})$ the pullbacks to $X'$. Then the class $P'_ p(E_2')$, resp. $c'_ p(E_2')$ in $A^ p(X_2' \to X')$ constructed in Lemma 42.47.1 using $X' = X'_1 \cup X'_2$ and $E_2'$ is the restriction (Remark 42.33.5) of the class $P'_ p(E_2)$, resp. $c'_ p(E_2)$ in $A^ p(X_2 \to X)$.

Proof. Immediate from the characterization of these classes in Lemma 42.47.1. $\square$

Lemma 42.47.4. In Lemma 42.47.1 say $E_2$ is the restriction of a perfect $E \in D(\mathcal{O}_ X)$ such that $E|_{X_1}$ is zero, resp. isomorphic to a finite locally free $\mathcal{O}_{X_1}$-module of rank $< p$ sitting in cohomological degree $0$. If Chern classes of $E$ are defined, then $i_{2, *} \circ P'_ p(E_2) = P_ p(E)$, resp. $i_{2, *} \circ c'_ p(E_2) = c_ p(E)$ (with $\circ $ as in Lemma 42.33.4).

Proof. First, assume $E|_{X_1}$ is zero. With notations as in the proof of Lemma 42.47.1 the lemma in this case follows from

\begin{align*} P_ p(E) \cap \alpha ' & = i'_{1, *}(P_ p(E) \cap \alpha '_1) + i'_{2, *}(P_ p(E) \cap \alpha '_2) \\ & = i'_{1, *}(P_ p(E|_{X_1}) \cap \alpha '_1) + i'_{2, *}(P'_ p(E_2) \cap \alpha ') \\ & = i'_{2, *}(P'_ p(E_2) \cap \alpha ') \end{align*}

The case where $E|_{X_1}$ is isomorphic to a finite locally free $\mathcal{O}_{X_1}$-module of rank $< p$ sitting in cohomological degree $0$ is similar. $\square$

Lemma 42.47.5. In Lemma 42.47.1 suppose we have closed subschemes $X'_2 \subset X_2$ and $X_1 \subset X'_1 \subset X$ such that $X = X'_1 \cup X'_2$ set theoretically. Assume $E_2|_{X'_1 \cap X_2}$ is zero, resp. isomorphic to a finite locally free module of rank $< p$ placed in degree $0$. Then we have $(X'_2 \to X_2)_* \circ P'_ p(E_2|_{X'_2}) = P'_ p(E_2)$, resp. $(X'_2 \to X_2)_* \circ c'_ p(E_2|_{X'_2}) = c_ p(E_2)$ (with $\circ $ as in Lemma 42.33.4).

Proof. This follows immediately from the characterization of these classes in Lemma 42.47.1. $\square$

Lemma 42.47.6. In Lemma 42.47.1 let $f : Y \to X$ be locally of finite type and say $c \in A^*(Y \to X)$. Then

\[ c \circ P'_ p(E_2) = P'_ p(Lf_2^*E_2) \circ c \quad \text{resp.}\quad c \circ c'_ p(E_2) = c'_ p(Lf_2^*E_2) \circ c \]

in $A^*(Y_2 \to Y)$ where $f_2 : Y_2 \to X_2$ is the base change of $f$.

Proof. Let $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$. We may write

\[ \alpha = \alpha _1 + \alpha _2 \]

with $\alpha _ i \in \mathop{\mathrm{CH}}\nolimits _ k(X_ i)$; we are omitting the pushforwards by the closed immersions $X_ i \to X$. The reader then checks that $c'_ p(E_2) \cap \alpha = c_ p(E_2) \cap \alpha _2$, $c \cap c'_ p(E_2) \cap \alpha = c \cap c_ p(E_2) \cap \alpha _2$, $c \cap \alpha = c \cap \alpha _1 + c \cap \alpha _2$, and $c'_ p(Lf_2^*E_2) \cap c \cap \alpha = c_ p(Lf_2^*E_2) \cap c \cap \alpha _2$. We conclude by Lemma 42.46.6. $\square$

Lemma 42.47.7. In Lemma 42.47.1 assume $E_2|_{X_1 \cap X_2}$ is zero. Then

\begin{align*} P'_1(E_2) & = c'_1(E_2), \\ P'_2(E_2) & = c'_1(E_2)^2 - 2c'_2(E_2), \\ P'_3(E_2) & = c'_1(E_2)^3 - 3c'_1(E_2)c'_2(E_2) + 3c'_3(E_2), \\ P'_4(E_2) & = c'_1(E_2)^4 - 4c'_1(E_2)^2c'_2(E_2) + 4c'_1(E_2)c'_3(E_2) + 2c'_2(E_2)^2 - 4c'_4(E_2), \end{align*}

and so on with multiplication as in Remark 42.34.7.

Proof. The statement makes sense because the zero sheaf has rank $< 1$ and hence the classes $c'_ p(E_2)$ are defined for all $p \geq 1$. The equalities follow immediately from the characterization of the classes produced by Lemma 42.47.1 and the corresponding result for capping with the Chern classes of $E_2$ given in Remark 42.46.8. $\square$

Lemma 42.47.8. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $i_ j : X_ j \to X$, $j = 1, 2$ be closed immersions such that $X = X_1 \cup X_2$ set theoretically. Let $E, F \in D(\mathcal{O}_ X)$ be perfect objects. Assume

  1. Chern classes of $E$ and $F$ are defined,

  2. the restrictions $E|_{X_1 \cap X_2}$ and $F|_{X_1 \cap X_2}$ are isomorphic to a finite locally free $\mathcal{O}_{X_1}$-modules of rank $< p$ and $< q$ sitting in cohomological degree $0$.

With notation as in Remark 42.34.7 set

\[ c^{(p)}(E) = 1 + c_1(E) + \ldots + c_{p - 1}(E) + c'_ p(E|_{X_2}) + c'_{p + 1}(E|_{X_2}) + \ldots \in A^{(p)}(X_2 \to X) \]

with $c'_ p(E|_{X_2})$ as in Lemma 42.47.1. Similarly for $c^{(q)}(F)$ and $c^{(p + q)}(E \oplus F)$. Then $c^{(p + q)}(E \oplus F) = c^{(p)}(E)c^{(q)}(F)$ in $A^{(p + q)}(X_2 \to X)$.

Proof. Immediate from the characterization of the classes in Lemma 42.47.1 and the additivity in Lemma 42.46.7. $\square$

Lemma 42.47.9. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $i_ j : X_ j \to X$, $j = 1, 2$ be closed immersions such that $X = X_1 \cup X_2$ set theoretically. Let $E, F \in D(\mathcal{O}_{X_2})$ be perfect objects. Assume

  1. Chern classes of $E$ and $F$ are defined,

  2. the restrictions $E|_{X_1 \cap X_2}$ and $F|_{X_1 \cap X_2}$ are zero,

Denote $P'_ p(E), P'_ p(F), P'_ p(E \oplus F) \in A^ p(X_2 \to X)$ for $p \geq 0$ the classes constructed in Lemma 42.47.1. Then $P'_ p(E \oplus F) = P'_ p(E) + P'_ p(F)$.

Proof. Immediate from the characterization of the classes in Lemma 42.47.1 and the additivity in Lemma 42.46.7. $\square$

Lemma 42.47.10. In Lemma 42.47.1 assume $E_2$ has constant rank $0$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Then

\[ c'_ i(E_2 \otimes \mathcal{L}) = \sum \nolimits _{j = 0}^ i \binom {- i + j}{j} c'_{i - j}(E_2) c_1(\mathcal{L})^ j \]

Proof. The assumption on rank implies that $E_2|_{X_1 \cap X_2}$ is zero. Hence $c'_ i(E_2)$ is defined for all $i \geq 1$ and the statement makes sense. The actual equality follows immediately from Lemma 42.46.10 and the characterization of $c'_ i$ in Lemma 42.47.1. $\square$

Lemma 42.47.11. In Situation 42.7.1 let $X$ be locally of finite type over $S$. Let

\[ X = X_1 \cup X_2 = X'_1 \cup X'_2 \]

be two ways of writing $X$ as a set theoretic union of closed subschemes. Let $E$, $E'$ be perfect objects of $D(\mathcal{O}_ X)$ whose Chern classes are defined. Assume that $E|_{X_1}$ and $E'|_{X'_1}$ are zero1 for $i = 1, 2$. Denote

  1. $r = P'_0(E) \in A^0(X_2 \to X)$ and $r' = P'_0(E') \in A^0(X'_2 \to X)$,

  2. $\gamma _ p = c'_ p(E|_{X_2}) \in A^ p(X_2 \to X)$ and $\gamma '_ p = c'_ p(E'|_{X'_2}) \in A^ p(X'_2 \to X)$,

  3. $\chi _ p = P'_ p(E|_{X_2}) \in A^ p(X_2 \to X)$ and $\chi '_ p = P'_ p(E'|_{X'_2}) \in A^ p(X'_2 \to X)$

the classes constructed in Lemma 42.47.1. Then we have

\[ c'_1((E \otimes _{\mathcal{O}_ X}^\mathbf {L} E')|_{X_2 \cap X'_2}) = r \gamma '_1 + r' \gamma _1 \]

in $A^1(X_2 \cap X'_2 \to X)$ and

\[ c'_2((E \otimes _{\mathcal{O}_ X}^\mathbf {L} E')|_{X_2 \cap X'_2}) = r \gamma '_2 + r' \gamma _2 + {r \choose 2} (\gamma '_1)^2 + (rr' - 1) \gamma '_1\gamma _1 + {r' \choose 2} \gamma _1^2 \]

in $A^2(X_2 \cap X'_2 \to X)$ and so on for higher Chern classes. Similarly, we have

\[ P'_ p((E \otimes _{\mathcal{O}_ X}^\mathbf {L} E')|_{X_2 \cap X'_2}) = \sum \nolimits _{p_1 + p_2 = p} {p \choose p_1} \chi _{p_1} \chi '_{p_2} \]

in $A^ p(X_2 \cap X'_2 \to X)$.

Proof. First we observe that the statement makes sense. Namely, we have $X = (X_2 \cap X'_2) \cup Y$ where $Y = (X_1 \cap X'_1) \cup (X_1 \cap X'_2) \cup (X_2 \cap X'_1)$ and the object $E \otimes _{\mathcal{O}_ X}^\mathbf {L} E'$ restricts to zero on $Y$. The actual equalities follow from the characterization of our classes in Lemma 42.47.1 and the equalities of Lemma 42.46.11. We omit the details. $\square$

[1] Presumably there is a variant of this lemma where we only assume these restrictions are isomorphic to a finite locally free modules of rank $< p$ and $< p'$.

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