# stacks/stacks-project

Better handling commutativity property loc chern

Wish I had realized this a week ago. Argh!
 @@ -7875,39 +7875,28 @@ \section{A baby case of localized chern classes} \begin{lemma} \label{lemma-silly-commutes} Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Suppose we have closed subschemes $X_1, X_2, X'_1, X'_2$ such that In Lemma \ref{lemma-silly} let $f : Y \to X$ be locally of finite type and say $c \in A^*(Y \to X)$. Then $$X = X_1 \cup X_2 = X'_1 \cup X'_2 c \circ P'_p(E_2) = P'_p(Lf_2^*E_2) \circ c \quad\text{resp.}\quad c \circ c'_p(E_2) = c'_p(Lf_2^*E_2) \circ c$$ set theoretically. Let $E_2 \in D(\mathcal{O}_{X_2})$ and $E'_2 \in D(\mathcal{O}_{X'_2})$ be perfect objects. Assume \begin{enumerate} \item chern classes of $E_2$ and $E'_2$ are defined, \item the restrictions $E_2|_{X_1 \cap X_2}$ and $E'_2|_{X'_1 \cap X'_2}$ are isomorphic to finite locally free modules of rank $< p$ and $< p'$ sitting in cohomological degree $0$. \end{enumerate} Then the classes $c'_p(E_2) \in A(X_2 \to X)$ and $c'_{p'}(E'_2) \in A(X'_2 \to X)$ of Lemma \ref{lemma-silly} commute in the sense of Remark \ref{remark-bivariant-commute}. in $A^*(Y_2 \to Y)$ where $f_2 : Y_2 \to X_2$ is the base change of $f$. \end{lemma} \begin{proof} Let $\alpha \in A_k(X)$. We may write $$\alpha = \alpha_{11} + \alpha_{12} + \alpha_{21} + \alpha_{22} \alpha = \alpha_1 + \alpha_2$$ with $\alpha_{ij} \in A_k(X_i \cap X'_j)$; we are omitting the pushforwards by the closed immersions $X_i \cap X'_j \to X$. Then $c'_p(E_2) \cap \alpha = c(E_2) \cap (\alpha_{21} + \alpha_{22})$ and hence $c'_{p'}(E'_2) \cap c'_p(E_2) \cap \alpha = c_{p'}(E'_2) \cap c_p(E_2) \cap \alpha_{22}$ where the cap products are taking place on $X_2 \cap X'_2$. Since chern classes commute with each other as bivariant classes (Lemma \ref{lemma-commutative-chern-perfect}) we conclude. with $\alpha_i \in A_k(X_i)$; we are omitting the pushforwards by the closed immersions $X_i \to X$. The reader then checks that $c'_p(E_2) \cap \alpha = c_p(E_2) \cap \alpha_2$, $c \cap c'_p(E_2) \cap \alpha = c \cap c_p(E_2) \cap \alpha_2$, $c \cap \alpha = c \cap \alpha_1 + c \cap \alpha_2$, and $c'_p(Lf_2^*E_2) \cap c \cap \alpha = c_p(Lf_2^*E_2) \cap c \cap \alpha_2$. We conclude by Lemma \ref{lemma-commutative-chern-perfect}. \end{proof} \begin{lemma} @@ -8127,7 +8116,41 @@ \section{Gysin at infinity} and we get the desired result. \end{proof} \begin{lemma} \label{lemma-gysin-at-infty-commutes} In Lemma \ref{lemma-gysin-at-infty} let $f : Y \to X$ be a morphism locally of finite type and $c \in A^*(Y \to X)$. Then $C \circ c = c \circ C$ in $A^*(W_\infty \times_X Y)$. \end{lemma} \begin{proof} Consider the commutative diagram $$\xymatrix{ W_\infty \times_X Y \ar@{=}[r] & W_{Y, \infty} \ar[r]_{i_{Y, \infty}} \ar[d] & W_Y \ar[r]_{b_Y} \ar[d] & \mathbf{P}^1_Y \ar[r]_{p_Y} \ar[d] & Y \ar[d]^f \\ & W_\infty \ar[r]^{i_\infty} & W \ar[r]^b & \mathbf{P}^1_X \ar[r]^p & X }$$ with cartesian squares. For an elemnent $\alpha \in A_k(X)$ choose $\beta \in A_{k + 1}(W)$ whose restriction to $b^{-1}(\mathbf{A}^1_X)$ is the flat pullback of $\alpha$. Then $c \cap \beta$ is a class in $A_*(W_Y)$ whose restriction to $b_Y^{-1}(\mathbf{A}^1_Y)$ is the flat pullback of $c \cap \alpha$. Next, we have $$i_{Y, \infty}^*(c \cap \beta) = c \cap i_\infty^*\beta$$ because $c$ is a bivariant class. This exactly says that $C \cap c \cap \alpha = c \cap C \cap \alpha$. The same argument works after any base change by $X' \to X$ locally of finite type. This proves the lemma. \end{proof} @@ -8322,43 +8345,30 @@ \section{Preparation for localized chern classes} \begin{lemma} \label{lemma-homomorphism-commute} In Lemma \ref{lemma-localized-chern-pre} assume $Q|_T$ is isomorphic to a finite locally free $\mathcal{O}_T$-module of rank $< p$. Assume we have another perfect object $Q' \in D(\mathcal{O}_W)$ whose chern classes are defined with $Q'|_T$ is isomorphic to a finite locally free $\mathcal{O}_T$-module of rank $< p'$ placed in cohomological degree $0$. Then $c'_p(Q) \in A^p(Z \to X)$ and $c'_{p'}(Q') \in A^{p'}(Z \to X)$ commute in the sense of Remark \ref{remark-bivariant-commute}. In Lemma \ref{lemma-localized-chern-pre} let $Y \to X$ be a morphism locally of finite type and let $c \in A^*(Y \to X)$ be a bivariant class. Then $$P'_p(Q) \circ c = c \circ P'_p(Q) \quad\text{resp.}\quad c'_p(Q) \circ c = c \circ c'_p(Q)$$ in $A^*(Y \times_X Z \to X)$. \end{lemma} \begin{proof} Let $E \subset W_\infty$ be the inverse image of $Z$. Denote $c'_p(Q|_E)$ and $c'_{p'}(Q'|_E)$ the classes constructed in Lemma \ref{lemma-silly}. We have \begin{align*} res(c'_p(Q)) \circ c'_{p'}(Q') & = c'_p(Q) \circ (Z \to X)_* \circ c'_{p'}(Q') \\ & = (E \to Z)_* \circ c'_p(Q|_E) \circ C \circ (Z \to X)_* \circ c'_{p'}(Q') \\ & = (E \to Z)_* \circ c'_p(Q|_E) \circ c_{p'}(Q'|_{W_\infty}) \circ C \\ & = (E \to Z)_* \circ c'_p(Q|_E) \circ (E \to W_\infty)_* \circ c'_{p'}(Q'|_E) \circ C \\ & = (E \to Z)_* \circ res(c'_p(Q|_E)) \circ c'_{p'}(Q'|_E) \circ C \end{align*} The first equality holds by Remark \ref{remark-res-push}. The second equality is the definition of $c'_p(Q)$. The third equality is Lemma \ref{lemma-homomorphism}. The fourth equality is Lemma \ref{lemma-silly-silly}. The fifth equality is Remark \ref{remark-res-push} once again but this time using $E \subset W_\infty$. By Lemma \ref{lemma-silly-commutes} the order in the product $res(c'_p(Q|_E)) \circ c'_{p'}(Q'|_E)$ may be switched and we conclude. Recall that $P'_p(Q) = (E \to Z)_* \circ P'_p(Q|_E) \circ C$, resp.\ $c'_p(Q) = (E \to Z)_* \circ c'_p(Q|_E) \circ C$ where $C$ is as in Lemma \ref{lemma-gysin-at-infty} and $P'_p(Q|_E)$, resp.\ $c'_p(Q|_E)$ are as in Lemma \ref{lemma-silly}. By Lemma \ref{lemma-gysin-at-infty-commutes} we see that $C$ commutes with $c$ and by Lemma \ref{lemma-silly-commutes} we see that $P'_p(Q|_E)$, resp.\ $c'_p(Q|_E)$ commutes with $c$. Since $c$ is a bivariant class it commutes with proper pushforward by $E \to Z$ by definition. This finishes the proof. \end{proof} \begin{lemma} @@ -8907,146 +8917,50 @@ \section{Properties of localized chern classes} \begin{lemma} \label{lemma-loc-chern-classes-commute} Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. Let $Z_i \subset X$, $i = 1, 2$ be closed subschemes. Let $E_i \in D(\mathcal{O}_X)$, $i = 1, 2$ be a perfect objects whose chern classes are defined. Assume the restriction $E_i|_{X \setminus Z_i}$ is isomorphic to a finite locally free $\mathcal{O}_{X \setminus Z_i}$-module of rank $< p_i$ sitting in cohomological degree $0$. Then $c_{p_1}(Z_1 \to X, E_1)$ and $c_{p_2}(Z_2 \to X, E_2)$ commute in the sense of Remark \ref{remark-bivariant-commute}. In the situation of Definition \ref{definition-localized-chern} assume $P_p(Z \to X, E)$, resp.\ $c_p(Z \to X, E)$ is defined. Let $Y \to X$ be locally of finite type and $c \in A^*(Y \to X)$. Then $$P_p(Z \to X, E) \circ c = c \circ P_p(Z \to X, E),$$ respectively $$c_p(Z \to X, E) \circ c = c \circ c_p(Z \to X, E)$$ in $A^*(Y \times_X Z \to X)$. \end{lemma} \begin{proof} The difficulty in this proof is that we don't assume that $Z_1 = Z_2$ so we can't reduce this lemma to Lemma \ref{lemma-homomorphism-commute}. We'll actually have to use something about the precise complexes we use in the construction of localized chern classes. \medskip\noindent Our assumptions say $E_i$ is represented to a bounded complex $\mathcal{E}_i^\bullet$ of finite locally free $\mathcal{O}_X$-modules. For $i = 1, 2$ let This follows from Lemma \ref{lemma-homomorphism-commute}. Namely, our assumptions say $E$ is represented to a bounded complex $\mathcal{E}^\bullet$ of finite locally free $\mathcal{O}_X$-modules. Let $$b_i : W_i \to \mathbf{P}^1_X b : W \to \mathbf{P}^1_X \quad\text{and}\quad \mathcal{Q}_i^\bullet \mathcal{Q}^\bullet$$ be the blowing up and complex of $\mathcal{O}_{W_i}$-modules constructed in be the blowing up and complex of $\mathcal{O}_W$-modules constructed in More on Flatness, Section \ref{flat-section-blowup-complexes-III}. Let $T_i \subset W_{i, \infty}$ be the closed subscheme whose existence is Let $T \subset W_\infty$ be the closed subscheme whose existence is averted in More on Flatness, Lemma \ref{flat-lemma-graph-construction}. Let $T'_i \subset T_i$ be the open and closed subscheme such that $\mathcal{Q}_i^\bullet|_{T'_i}$ is isomorphic to a finite locally free sheaf of rank $< p_i$ in degree $0$. By definition Let $T' \subset T$ be the open and closed subscheme such that $\mathcal{Q}_i^\bullet|_{T'_i}$ is zero, resp.\ isomorphic to a finite locally free sheaf of rank $< p$ placed in degree $0$. By definition $$c_p(Z \to X, E_i) = c'_p(\mathcal{Q}_i^\bullet) c_p(Z \to X, E) = c'_p(\mathcal{Q}^\bullet)$$ as bivariant operations (and not just on cycles over $X$) where the right hand side is the bivariant class constructed in Lemma \ref{lemma-localized-chern-pre} using $W_i, b_i, \mathcal{Q}_i^\bullet, T'_i$. \medskip\noindent By Divisors, Lemma \ref{divisors-lemma-blowing-up-two-ideals} we can choose a commutative diagram $$\xymatrix{ W \ar[d]^{g_1} \ar[rd]^b \ar[r]_{g_2} & W_2 \ar[d]^{b_2} \\ W_1 \ar[r]^{b_1} & \mathbf{P}^1_X }$$ where all morphisms are blowing ups which are isomorphisms over $\mathbf{A}^1_X$. By Lemma \ref{lemma-localized-chern-pre-independent} we may use $W$, $b = b_i \circ g_i$, $Q_i = g_i^*\mathcal{Q}_i^\bullet$, and $g_i^{-1}(T'_i)$ to construct $c_{p_i}(Z \to X, E_i) = c'_{p_i}(Q_i)$. By More on Flatness, Lemma \ref{flat-lemma-complex-and-divisor-eta-pull} applied to the morphisms $g_i : W \to W_i$ we find that Lemma \ref{lemma-localized-chern-pre} using $W, b, \mathcal{Q}^\bullet, T'$. By Lemma \ref{lemma-homomorphism-commute} we have $$Q_i = g_i^*\mathcal{Q}_i^\bullet = \eta_\mathcal{I}b^*p^*\mathcal{E}_i^\bullet$$ where $\mathcal{I}$ is the invertible ideal sheaf of the effective Cartier divisor $W_\infty$ and $p :\mathbf{P}^1_X \to X$ is the projection morphism. \medskip\noindent Set $W^i = b^{-1}(p^{-1}(Z_i)) \subset W$. Picture $$\xymatrix{ W^1 \ar[d]_{b^1} \ar[r] & W \ar[d]^b & W^2 \ar[l] \ar[d]^{b^2} \\ \mathbf{P}^1_{Z_1} \ar[r] & \mathbf{P}^1_X & \mathbf{P}^1_{Z_2} \ar[l] }$$ Since $\mathcal{E}_i^\bullet|_{X \setminus Z_i}$ is isomorphic to a finite locally free $\mathcal{O}_{X \setminus Z_i}$-module of rank $< p_i$ sitting in cohomological degree $0$, we see that $b^*p^*\mathcal{E}_i^\bullet|_{W \setminus W^i}$ is isomorphic to a finite locally free $\mathcal{O}_{W \setminus W^i}$-module of rank $< p_i$ sitting in cohomological degree $0$. Hence, for example by More on Flatness, Lemma \ref{flat-lemma-eta-first-property}, we see $Q_i|_{W \setminus W^i}$ is isomorphic a finite locally free $\mathcal{O}_{W \setminus W^i}$-module of rank $< p_i$ sitting in cohomological degree $0$. \medskip\noindent Let $\alpha \in A_k(X)$. We are going to compute $c_{p_2}(Z_2 \to X, E_2) \cap c_{p_1}(Z_1 \to X, E_1) \cap \alpha$ in $A_{k - p_1 - p_2}(X)$. Choose $\beta \in A_{k + 1}(W)$ whose restriction to $b^{-1}(\mathbf{A}^1_X)$ is the flat pullback of $\alpha$. Since $W^1_\infty$ is the inverse image of $Z_1$ under the morphism $W_\infty \to X$ we have the first equality of \begin{align*} c_{p_1}(Z_1 \to X, E_1) \cap \alpha & = (W^1_\infty \to Z_1)_*(c'_{p_1}(Q_1|_{W^1_\infty}) \cap i_\infty^*\beta) \\ & = (W^1_\infty \to Z_1)_*( c_{p_1}(W^1_\infty \to W_\infty, Q_1|_{W_\infty}) \cap i_\infty^*\beta) \\ & = (W^1_\infty \to Z_1)_* (i^1_\infty)^* (c_{p_1}(W^1 \to W, Q_1) \cap \beta) \end{align*} The second equality is Lemma \ref{lemma-loc-chern-agree}. The third equality is the fact that $c_{p_1}(W^1 \to W, Q_1)$ is a bivariant class whose formation commutes with base change and $i^1_\infty : W^1_\infty \to W^1$ is the base change of $i_\infty$. Note that the proper morphism $$b^1 : W^1 \longrightarrow \mathbf{P}^1_{Z_1}$$ and the restriction $Q_2|_{W^1}$ is the pair which defines the restriction of $c_{p_2}(Z_2 \to X, E_2) = c'_{p_2}(Q_2)$ to $Z_1$. Denote $C_1 \in A^0(W^1_\infty \to Z_1)$ the class of Lemma \ref{lemma-gysin-at-infty}. Then we have to compute $$((W^1 \cap W^2)_\infty \to Z_1 \cap Z_2)_*\Big( c'_{p_2}(Q_2|_{(W^1 \cap W^2)_\infty}) \cap C_1 \cap (W^1_\infty \to Z_1)_* (i^1_\infty)^* (c_{p_1}(W^1 \to W, Q_1) \cap \beta)\Big)$$ with apologies for the horrible notation. By Lemma \ref{lemma-homomorphism-pre} we have $C_1 \circ (W^1_\infty \to Z_1)_* \circ (i^1_\infty)^* = (i^1_\infty)^*$ hence this simplifies to $$((W^1 \cap W^2)_\infty \to Z_1 \cap Z_2)_*\Big( c'_{p_2}(Q_2|_{(W^1 \cap W^2)_\infty}) \cap (i^1_\infty)^* (c_{p_1}(W^1 \to W, Q_1) \cap \beta)\Big)$$ which in turn is equal to $$((W^1 \cap W^2)_\infty \to Z_1 \cap Z_2)_*( c'_{p_2}(Q_2|_{(W^1 \cap W^2)_\infty}) \cap c'_{p_1}(Q_1|_{W^1_\infty}) \cap i_\infty^*\beta) P'_p(\mathcal{Q}^\bullet) \circ c = c \circ P'_p(\mathcal{Q}^\bullet) \quad\text{resp.}\quad c'_p(\mathcal{Q}^\bullet) \circ c = c \circ c'_p(\mathcal{Q}^\bullet)$$ by reusing some of the equalities we used above (in the opposite direction). Since $c'_{p_2}(Q_2|_{(W^1 \cap W^2)_\infty})$ is the restriction of $c'_{p_2}(Q_2|_{W^2_\infty})$ we find that this is symmetric in the indices by Lemma \ref{lemma-silly-commutes} which concludes the proof. in $A^*(Y \times_X Z \to X)$ and we conclude. \end{proof} \begin{remark} @@ -9841,7 +9755,7 @@ \section{Higher codimension gysin homomorphisms} c(Z \to X, \mathcal{N}) = c_{top}(\mathcal{E}) \circ i^*  where $i^*$ is the gysing homomorphism associated to $i : Z \to X$ where $i^*$ is the gysin 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}.