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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

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.

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