# stacks/stacks-project

Exterior product over dim 1 base

 @@ -9777,8 +9777,9 @@ \section{Exterior product} \label{section-exterior-product} \noindent Let $k$ be a field and let $(S, \delta)$ be as in Example \ref{example-field} which is a special case of our general Situation \ref{situation-setup}. Let $k$ be a field. Set $S = \Spec(k)$ and define $\delta : S \to \mathbf{Z}$ by sending the unique point to $0$. Then $(S, \delta)$ is a special case of our general Situation \ref{situation-setup}, see Example \ref{example-field}. Consider a cartesian square $$\xymatrix{ @@ -9849,8 +9850,7 @@ \section{Exterior product} \begin{lemma} \label{lemma-chow-cohomology-towards-point} Let k be a field and let (S, \delta) be as in Example \ref{example-field}. Let X \to \Spec(k) \to S be morphisms locally of finite type. Let k be a field. Let X be a scheme locally of finite type over k. Then we have a canonical identification$$ A^p(X \to \Spec(k)) = A_{-p}(X) @@ -9859,10 +9859,9 @@ \section{Exterior product} \end{lemma} \begin{proof} Observe that $A_*(\Spec(k))$ is nonzero only in degree $d$ Observe that $A_*(\Spec(k))$ is nonzero only in degree $0$ with generator $[\Spec(k)]$. Hence we get a map $A^p(X \to \Spec(k)) \to A_{-p}(X)$ by sending $c$ to $c \cap [\Spec(k)]$. $A^p(X \to \Spec(k)) \to A_{-p}(X)$ by sending $c$ to $c \cap [\Spec(k)]$. \medskip\noindent Conversely, suppose we have $\alpha \in A_{-p}(X)$. @@ -9873,8 +9872,8 @@ \section{Exterior product} c_\alpha \cap \alpha' = \alpha \times \alpha' $$in A_{n - p}(X \times_k X'). To show that this is a bivariant class we write \alpha = \sum n_i[X_i] as a locally finite sum of cycles on X. Consider the composition class we write \alpha = \sum n_i[X_i] as in Definition \ref{definition-cycles}. Consider the composition$$ \coprod X_i \xrightarrow{g} X \to \Spec(k) $$@@ -9915,6 +9914,197 @@ \section{Exterior product} \section{Exterior product over a 1-dimensional base} \label{section-exterior-product-dim-1} \noindent Let S be a locally Noetherian scheme which has an open covering by spectra of Dedekind domains. Set \delta(s) = 0 for s \in S closed and \delta(s) = 1 otherwise. Then (S, \delta) is a special case of our general Situation \ref{situation-setup}; see Example \ref{example-domain-dimension-1}. Observe that S is normal (Algebra, Lemma \ref{algebra-lemma-characterize-Dedekind}) and hence a disjoint union of normal integral schemes (Properties, Lemma \ref{properties-lemma-normal-locally-Noetherian}). Thus all of the arguments below reduce to the case where S is irreducible. On the other hand, we allow S to be nonseparated (so S could be the affine line with 0 doubled for example). \medskip\noindent Consider a cartesian square$$ \xymatrix{ X \times_S Y \ar[r] \ar[d] & Y \ar[d] \\ X \ar[r] & S } $$of schemes locally of finite type over S. We claim there is a canonical map$$ \times : A_n(X) \otimes_{\mathbf{Z}} A_m(Y) \longrightarrow A_{n + m - 1}(X \times_S Y) $$which is uniquely determined by the following rule: given integral closed subschemes X' \subset X and Y' \subset Y of \delta-dimensions n and m we set \begin{enumerate} \item [X'] \times [Y'] = [X' \times_S Y']_{n + m - 1} if X' or Y' dominates an irreducible component of S, \item [X'] \times [Y'] = 0 if neither X' nor Y' dominates an irreducible component of S. \end{enumerate} \begin{lemma} \label{lemma-exterior-product-well-defined-dim-1} The map \times : A_n(X) \otimes_{\mathbf{Z}} A_m(Y) \to A_{n + m - 1}(X \times_S Y) is well defined. \end{lemma} \begin{proof} Consider n and m cycles \alpha = \sum_{i \in I} n_i[X_i] and \beta = \sum_{j \in J} m_j[Y_j] with X_i \subset X and Y_j \subset Y locally finite families of integral closed subschemes of \delta-dimensions n and m. Let K \subset I \times J be the set of pairs (i, j) \in I \times J such that X_i or Y_j dominates an irreducible component of S. Then \{X_i \times_S Y_j\}_{(i, j) \in K} is a locally finite collection of closed subschemes of X \times_S Y of \delta-dimension n + m - 1. This means we can indeed consider$$ \alpha \times \beta = \sum\nolimits_{(i, j) \in K} n_i m_j [X_i \times_S Y_j]_{n + m - 1} $$as a (n + m - 1)-cycle on X \times_S Y. In this way we obtain an additive map \times : Z_n(X) \otimes_{\mathbf{Z}} Z_m(Y) \to Z_{n + m}(X \times_S Y). The problem is to show that this procedure is compatible with rational equivalence. \medskip\noindent Let i : X' \to X be the inclusion morphism of an integral closed subscheme of \delta-dimension n which dominates an irreducible component of S. Then p' : X' \to S is flat of relative dimension n - 1, see More on Algebra, Lemma \ref{more-algebra-lemma-dedekind-torsion-free-flat}. Hence flat pullback along p' is an element (p')^* \in A^{-n + 1}(X' \to S) by Lemma \ref{lemma-flat-pullback-bivariant} and hence c' = i_* \circ (p')^* \in A^{-n + 1}(X \to S) by Lemma \ref{lemma-push-proper-bivariant}. This produces maps$$ c' \cap - : A_m(Y) \longrightarrow A_{m + n - 1}(X \times_S Y) $$which sends [Y'] to [X' \times_S Y']_{n + m - 1} for any integral closed subscheme Y' \subset Y of \delta-dimension m. \medskip\noindent Let i : X' \to X be the inclusion morphism of an integral closed subscheme of \delta-dimension n such that the composition X' \to X \to S factors through a closed point s \in S. Since s is a closed point of the spectrum of a Dedekind domain, we see that s is an effective Cartier divisor on S whose normal bundle is trivial. Denote c \in A^1(s \to S) the gysin homomorphism, see Lemma \ref{lemma-gysin-bivariant}. The morphism p' : X' \to s is flat of relative dimension n. Hence flat pullback along p' is an element (p')^* \in A^{-n}(X' \to S) by Lemma \ref{lemma-flat-pullback-bivariant}. Thus$$ c' = i_* \circ (p')^* \circ c \in A^{-n}(X \to S) $$by Lemma \ref{lemma-push-proper-bivariant}. This produces maps$$ c' \cap - : A_m(Y) \longrightarrow A_{m + n - 1}(X \times_S Y) $$which for any integral closed subscheme Y' \subset Y of \delta-dimension m sends [Y'] to either [X' \times_S Y']_{n + m - 1} if Y' dominates an irreducible component of S or to 0 if not. \medskip\noindent From the previous two paragraphs we conclude the construction ([X'], [Y']) \mapsto [X' \times_S Y']_{n + m - 1} factors through rational equivalence in the second variable, i.e., gives a well defined map Z_n(X) \otimes_{\mathbf{Z}} A_m(Y) \to A_{n + m - 1}(X \times_S Y). By symmetry the same is true for the other variable and we conclude. \end{proof} \begin{lemma} \label{lemma-chow-cohomology-towards-base-dim-1} Let (S, \delta) be as above. Let X be a scheme locally of finite type over S. Then we have a canonical identification$$ A^p(X \to S) = A_{1 - p}(X) $$for all p \in \mathbf{Z}. \end{lemma} \begin{proof} Consider the element [S]_1 \in A_1(S). We get a map A^p(X \to S) \to A_{1 - p}(X) by sending c to c \cap [S]_1. \medskip\noindent Conversely, suppose we have \alpha \in A_{1 - p}(X). Then we can define c_\alpha \in A^p(X \to S) as follows: given X' \to S and \alpha' \in A_n(X') we let$$ c_\alpha \cap \alpha' = \alpha \times \alpha' $$in A_{n - p}(X \times_S X'). To show that this is a bivariant class we write \alpha = \sum_{i \in I} n_i[X_i] as in Definition \ref{definition-cycles}. In particular the morphism$$ g : \coprod\nolimits_{i \in I} X_i \longrightarrow X $$is proper. Pick i \in I. If X_i dominates an irreducible component of S, then the structure morphism p_i : X_i \to S is flat and we have \xi_i = p_i^* \in A^p(X_i \to S). On the other hand, if p_i factors as p'_i : X_i \to s_i followed by the inclusion s_i \to S of a closed point, then we have \xi_i = (p'_i)^* \circ c_i \in A^p(X_i \to S) where c_i \in A^1(s_i \to S) is the gysin homomorphism and (p'_i)^* is flat pullback. Observe that$$ A^p(\coprod\nolimits_{i \in I} X_i \to S) = \prod\nolimits_{i \in I} A^p(X_i \to S) $$Thus we have$$ \xi = \sum n_i \xi_i \in A^p(\coprod\nolimits_{i \in I} X_i \to S) $$Finally, since g is proper we have a bivariant class$$ g_* \circ \xi \in A^p(X \to S) $$by Lemma \ref{lemma-push-proper-bivariant}. The reader easily verifies that c_\alpha is equal to this class (please compare with the proof of Lemma \ref{lemma-exterior-product-well-defined-dim-1}) and hence is itself a bivariant class. \medskip\noindent To finish the proof we have to show that the two constructions are mutually inverse. Since c_\alpha \cap [S]_1 = \alpha this is clear for one of the two directions. For the other, let c \in A^p(X \to S) and set \alpha = c \cap [S]_1. It suffices to prove that$$ c \cap [X'] = c_\alpha \cap [X']  when $X'$ is an integral scheme locally of finite type over $S$, see Lemma \ref{lemma-bivariant-zero}. However, either $p' : X' \to S$ is flat of relative dimension $\dim_\delta(X') - 1$ and hence $[X'] = (p')^*[S]_1$ or $X' \to S$ factors as $X' \to s \to S$ and hence $[X'] = (p')^*(s \to S)^*[S]_1$. Thus the fact that the bivariant classes $c$ and $c_\alpha$ agree on $[S]_1$ implies they agree when capped against $[X']$ (since bivariant classes commute with flat pullback and gysin maps) and the proof is complete. \end{proof}