Lemma 42.50.2 (0FB2)—The Stacks project

Lemma 42.50.2. In Situation 42.50.1 there exists a canonical bivariant class

\[ P_ p(Z \to X, E) \in A^ p(Z \to X), \quad \text{resp.}\quad c_ p(Z \to X, E) \in A^ p(Z \to X) \]

with the property that

42.50.2.1

\begin{equation} \label{chow-equation-defining-property-localized-classes} i_* \circ P_ p(Z \to X, E) = P_ p(E), \quad \text{resp.}\quad i_* \circ c_ p(Z \to X, E) = c_ p(E) \end{equation}

as bivariant classes on $X$ (with $\circ $ as in Lemma 42.33.4).

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

\[ b : W \longrightarrow \mathbf{P}^1_ X \quad \text{and}\quad T \longrightarrow W_\infty \quad \text{and}\quad Q \]

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

\[ T' \to T \to W_\infty \to X \]

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

\[ P_ p(Z \to X, E) = P'_ p(Q) \in A^ p(Z \to X) \]

and

\[ c_ p(Z \to X, E) = c'_ p(Q) \in A^ p(Z \to X) \]

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. $\square$

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