# stacks/stacks-project

 @@ -9023,15 +9023,15 @@ \section{Higher codimension gysin homomorphisms} Section \ref{section-blowup-Z-first}. By Lemma \ref{lemma-gysin-at-infty} we have a canonical bivariant class in $$c_1 \in A^0(W_\infty \to X) C \in A^0(W_\infty \to X)$$ Consider the open immersion $j : C_ZX \to W_\infty$ of (\ref{item-cone-is-open}) and the closed immersion $i : C_ZX \to N$ constructed above. By Lemma \ref{lemma-vectorbundle} for every $\alpha \in A_k(X)$ there exists a unique $\beta \in A_*(Z)$ such that $$i_*j^*(c_1 \cap \alpha) = p^*\beta i_*j^*(C \cap \alpha) = p^*\beta$$ If $\mathcal{N}$ has constant rank $r$, then $\beta \in A_{k - r}(Z)$. @@ -9046,11 +9046,11 @@ \section{Higher codimension gysin homomorphisms} \end{lemma} \begin{proof} Since both $i_* \circ j^* \circ c_1$ and $p^*$ are bivariant classes Since both $i_* \circ j^* \circ C$ and $p^*$ are bivariant classes (see Lemmas \ref{lemma-flat-pullback-bivariant} and \ref{lemma-push-proper-bivariant}) we can use the equation $$i_* \circ j^* \circ c_1 = p^* \circ c(Z \to X, \mathcal{N}) i_* \circ j^* \circ C = p^* \circ c(Z \to X, \mathcal{N})$$ (suitably interpreted) to define $c(Z \to X, \mathcal{N})$ as a bivariant class. This works because $p^*$ is always @@ -9084,16 +9084,16 @@ \section{Higher codimension gysin homomorphisms} W'_\infty \to W_\infty \times_X X',\quad C_{Z'}X' \to C_ZX \times_Z Z' $$To get c \cap \alpha' we use the class c_1 \cap \alpha' To get c \cap \alpha' we use the class C \cap \alpha' defined using the morphism W \times_{\mathbf{P}^1_X} \mathbf{P}^1_{X'} \to \mathbf{P}^1_{X'} in Lemma \ref{lemma-gysin-at-infty}. To get c' \cap \alpha' on the other hand, we use the class c'_1 \cap \alpha' defined using the morphism W' \to \mathbf{P}^1_{X'}. C' \cap \alpha' defined using the morphism W' \to \mathbf{P}^1_{X'}. By Lemma \ref{lemma-gysin-at-infty-independent} the pushforward of c'_1 \cap \alpha' by the closed immersion C' \cap \alpha' by the closed immersion W'_\infty \to (W \times_{\mathbf{P}^1_X} \mathbf{P}^1_{X'})_\infty, is equal to c_1 \cap \alpha'. Hence the same is true for the pullbacks is equal to C \cap \alpha'. Hence the same is true for the pullbacks to the opens$$ C_{Z'}X' \subset W'_\infty,\quad @@ -9153,11 +9153,11 @@ \section{Higher codimension gysin homomorphisms} Let us trace through the steps in the definition of $c(Z \to X, \mathcal{N}) \cap [X]_n$. Let $b : W \to \mathbf{P}^1_X$ be the blowing up of $\infty(Z)$. We first have to compute $c_1 \cap [X]_n$ where $c_1 \in A^0(W_\infty \to X)$ is $C \cap [X]_n$ where $C \in A^0(W_\infty \to X)$ is the class of Lemma \ref{lemma-gysin-at-infty}. To do this, note that $[W]_{n + 1}$ is a cycle on $W$ whose restriction to $\mathbf{A}^1_X$ is equal to the flat pullback of $[X]_n$. Hence $c_1 \cap [X]_n$ equal to the flat pullback of $[X]_n$. Hence $C \cap [X]_n$ is equal to $i_\infty^*[W]_{n + 1}$. Since none of the components of $W$ of $\delta$-dimension $n + 1$ is contained in $W_\infty$ we see that $i_\infty^*[W]_{n + 1} = [W_\infty \cap W]_n = [W_\infty]_n$.