Adjust notation

chow.tex

@@ -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 $i_\infty^*[W]_{n + 1} = [W_\infty \cap W]_n = [W_\infty]_n$.