The Stacks project

Proof. The construction of these bivariant classes is as follows. Let

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

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

and

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. $\square$

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

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. $\square$

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 $X = Z$. $\square$

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. $\square$

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. $\square$

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 15.75.7. 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. $\square$