# stacks/stacks-project

Improve statement + proof result of cb18842

 @@ -9806,54 +9806,49 @@ \section{Higher codimension gysin homomorphisms} \begin{lemma} \label{lemma-gysin-commutes} Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. Let $Z_i \subset X$ be closed subschemes endowed with virtual normal sheaves $\mathcal{N}_i$. The bivariant classes $c(Z_1 \to X, \mathcal{N}_1)$ and $c(Z_2 \to X, \mathcal{N}_2)$ commute in the sense of Remark \ref{remark-bivariant-commute}. locally of finite type over $S$. Let $Z \subset X$ be a closed subscheme with virtual normal sheaf $\mathcal{N}$. Let $Y \to X$ be locally of finite type and $c \in A^*(Y \to X)$. Then $$c \circ c(Z \to X, \mathcal{N}) = c(Z \to X, \mathcal{N}) \circ c$$ in $A^*(Z \times_X Y \to X)$. \end{lemma} \begin{proof} Write $c_1 = c(Z_1 \to X, \mathcal{N}_1)$ and $c_2 = c(Z_2 \to X, \mathcal{N}_2)$. To check $res(c_1) \circ c_2 = res(c_2) \circ c_1$ in $A^*(Z_1 \cap Z_2 \to X)$ we use Lemma \ref{lemma-bivariant-zero}. Thus we may assume $X$ is an integral scheme and we have to show $c_1 \cap c_2 \cap [X] = c_2 \circ c_1 \cap [X]$ in $A^*(Z_1 \cap Z_2)$. To check this we may use Lemma \ref{lemma-bivariant-zero}. Thus we may assume $X$ is an integral scheme and we have to show $c \cap c(Z \to X, \mathcal{N}) \cap [X] = c(Z \to X, \mathcal{N}) \circ c \cap [X]$ in $A_*(Z \times_X Y)$. \medskip\noindent If $Z_1 = X$, then $c_1 = c_{top}(\mathcal{N}_1)$ by If $Z = X$, then $c(Z \to X, \mathcal{N}) = c_{top}(\mathcal{N})$ by Lemma \ref{lemma-gysin-fundamental} which commutes with the bivariant class $c_2$, see Lemma \ref{lemma-cap-commutative-chern}. The same argument works if $Z_2 = X$. with the bivariant class $c$, see Lemma \ref{lemma-cap-commutative-chern}. \medskip\noindent Assume that both $Z_1$ and $Z_2$ are not equal to $X$. By Lemma \ref{lemma-bivariant-zero} it even suffices to prove the result after blowing up $X$ (in a nonzero ideal). Let us blowup $X$ in the product of the ideal sheaves of $Z_1$ and $Z_2$. This transforms both $Z_1$ and $Z_2$ into effective Cartier divisors, see Divisors, Lemmas Assume that $Z$ is not equal to $X$. By Lemma \ref{lemma-bivariant-zero} it even suffices to prove the result after blowing up $X$ (in a nonzero ideal). Let us blowup $X$ in the ideal sheaf of $Z$. This reduces us to the case where $Z$ is an effective Cartier divisor, see Divisors, Lemma \ref{divisors-lemma-blowing-up-gives-effective-Cartier-divisor}, \ref{divisors-lemma-blow-up-pullback-effective-Cartier}, and \ref{divisors-lemma-blowing-up-two-ideals}. Thus we may assume $Z_1$ and $Z_2$ are effective Cartier divisors. \medskip\noindent If $Z_1$ and $Z_2$ are effective Cartier divisors, then we have If $Z$ is an effective Cartier divisor, then we have $$c(Z_i \to X, \mathcal{N}_i) = c_{top}(\mathcal{E}_i) \circ i_i^* c(Z \to X, \mathcal{N}) = c_{top}(\mathcal{E}) \circ i^*$$ where $i_i^*$ is the gysing homomorphism associated to $i_i : Z_i \to X$ and $\mathcal{E}_i$ is the dual of the kernel of $\mathcal{N}_i^\vee \to \mathcal{C}_{Z_i/X}$, see where $i^*$ is the gysing homomorphism associated to $i : Z \to X$ and $\mathcal{E}$ is the dual of the kernel of $\mathcal{N}^\vee \to \mathcal{C}_{Z/X}$, see Lemmas \ref{lemma-gysin-decompose} and \ref{lemma-gysin-agrees}. Then we conclude because chern classes are in the center of the bivariant ring (in the strong sense formulated in Lemma \ref{lemma-cap-commutative-chern}) and we have the commutation relation for gysin homomorphisms $i_i^*$ by Remark \ref{example-gysin-commute}. Lemma \ref{lemma-cap-commutative-chern}) and commute with the gysin homomorphism $i^*$ by definition of bivariant classes. \end{proof}