# stacks/stacks-project

Another commutation relation

 @@ -7193,7 +7193,7 @@ \section{Chern classes and sections} Remark \ref{remark-top-chern-class}. By its very definition, in order to verify the formula, we may assume that $\mathcal{E}$ has constant rank. We may similarly assume $\mathcal{N}'$ and $\mathcal{N}$ have constant ranks, say $r$ and $r'$, so $\mathcal{N}$ have constant ranks, say $r'$ and $r$, so $\mathcal{E}$ has rank $r - r'$ and $c_{top}(\mathcal{E}) = c_{r - r'}(\mathcal{E})$. Observe that $p^*\mathcal{E}$ has a canonical section @@ -7208,9 +7208,9 @@ \section{Chern classes and sections} subscheme of $\delta$-dimension $n$. Then we have \begin{enumerate} \item $p^*[Y] = [p^{-1}(Y)]$ since $p^{-1}(Y)$ is integral of dimension $n + r$, $\delta$-dimension $n + r$, \item $(p')^*[Y] = [(p')^{-1}(Y)]$ since $(p')^{-1}(Y)$ is integral of dimension $n + r'$, $\delta$-dimension $n + r'$, \item the restriction of $s$ to $p^{-1}Y$ has vanishing scheme $(p')^{-1}Y$ and the closed immersion $(p')^{-1}Y \to p^{-1}Y$ is a regular immersion (locally cut out by a regular sequence). @@ -9598,7 +9598,7 @@ \section{Higher codimension gysin homomorphisms} exact sequence $0 \to \mathcal{N}' \to \mathcal{N} \to \mathcal{E} \to 0$ of finite locally free $\mathcal{O}_Z$-modules such that the given surjection $\sigma : \mathcal{N}^\vee \to \mathcal{C}_{Z/X}$ factors through a map $\sigma' : (\mathcal{N}')^\vee \to \mathcal{C}_{Z/x}$. $\sigma' : (\mathcal{N}')^\vee \to \mathcal{C}_{Z/X}$. Then $$c(Z \to X, \mathcal{N}) = c_{top}(\mathcal{E}) \circ c(Z \to X, \mathcal{N}') @@ -9882,12 +9882,12 @@ \section{Exterior product} Pullback along f is a bivariant class f^* \in A^p(\coprod X_i \to \Spec(k)) by Lemma \ref{lemma-flat-pullback-bivariant}. Denote c' \in A^0(\coprod X_i) the bivariant class Denote \nu \in A^0(\coprod X_i) the bivariant class which multiplies a cycle by n_i on the ith component. Thus c' \circ f^* \in A^p(\coprod X_i \to X). Thus \nu \circ f^* \in A^p(\coprod X_i \to X). Finally, we have a bivariant class$$ g_* \circ c' \circ f^* g_* \circ \nu \circ f^*  by Lemma \ref{lemma-push-proper-bivariant}. The reader easily verifies that $c_\alpha$ is equal to this class and hence @@ -9910,6 +9910,23 @@ \section{Exterior product} agree when capped against $[X']$ and the proof is complete. \end{proof} \begin{lemma} \label{lemma-chow-cohomology-towards-point-commutes} Let $k$ be a field. Let $X$ be a scheme locally of finite type over $k$. Let $c \in A^p(X \to \Spec(k))$. Let $Y \to Z$ be a morphism of schemes locally of finite type over $k$. Let $c' \in A^q(Y \to Z)$. Then $c \circ c' = c' \circ c$ in $A^{p + q}(X \times_k Y \to X \times_k Z)$. \end{lemma} \begin{proof} In the proof of Lemma \ref{lemma-chow-cohomology-towards-point} we have seen that $c$ is given by a combination of proper pushforward, multiplying by integers over connected components, and flat pullback. Since $c'$ commutes with each of these operations by definition of bivariant classe, we conclude. Some details omitted. \end{proof} @@ -10106,6 +10123,23 @@ \section{Exterior product over a $1$-dimensional base} commute with flat pullback and gysin maps) and the proof is complete. \end{proof} \begin{lemma} \label{lemma-chow-cohomology-towards-base-dim-1-commutes} Let $(S, \delta)$ be as above. Let $X$ be a scheme locally of finite type over $S$. Let $c \in A^p(X \to S)$. Let $Y \to Z$ be a morphism of schemes locally of finite type over $S$. Let $c' \in A^q(Y \to Z)$. Then $c \circ c' = c' \circ c$ in $A^{p + q}(X \times_S Y \to X \times_S Z)$. \end{lemma} \begin{proof} In the proof of Lemma \ref{lemma-chow-cohomology-towards-base-dim-1} we have seen that $c$ is given by a combination of proper pushforward, multiplying by integers over connected components, flat pullback, and gysin maps. Since $c'$ commutes with each of these operations by definition of bivariant classes, we conclude. Some details omitted. \end{proof}