The Stacks project

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 $\prod _{p \geq 0} A^ p(Z_1 \cap Z_2 \to X) \otimes \mathbf{Q}$. 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

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


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

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

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


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

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 $< p_ i$.

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0FBF. Beware of the difference between the letter 'O' and the digit '0'.