# stacks/stacks-project

 @@ -9842,13 +9842,19 @@ \section{Gysin maps for diagonals} \begin{proof} After decomposing $X$ into connected components we may and do assume that $X \to Y$ is smooth of constant relative dimension $d$. Let $X' \to X$ be locally of finite type with $X'$ integral of $\delta$-dimension $n$. Then $\text{pr}_i^*[X'] = [X \times_Y X']_{n + d}$. Observe that $X \times_{\Delta, X \times_Y X} (X \times_Y X')$ is the graph of $X' \to X \times_Y X'$ as a locally closed subscheme. Since the dimension is correct we conclude that $\Delta^! \cap [X \times_Y X']_{n + d} = [X']$ by Let $X' \to X$ be locally of finite type with $\dim_\delta(X') = n$. Then $\text{pr}_i^*[X'] = [X \times_Y X']_{n + d}$. We have a cartesian diagram $$\xymatrix{ X' \ar[d] \ar[r] & X \ar[d]^\Delta \\ X \times_Y X' \ar[r] & X \times_Y X }$$ The left vertical arrow is a regular immersion of codimension $d$ since it is a section of the smooth morphism $X \times_Y X' \to X'$, see Divisors, Lemma \ref{divisors-lemma-section-smooth-regular-immersion}. It follows that $\Delta^! \cap [X \times_Y X']_{n + d} = [X']$ by Lemma \ref{lemma-gysin-fundamental}. \end{proof} @@ -10178,6 +10184,14 @@ \section{Intersection products for smooth varieties} \alpha \cdot \beta = \Delta^!(\alpha \times \beta) \in A_*(Y \times_X Z) $$In the special case where X = Y = Z we obtain a multiplication$$ A_*(X) \times A_*(X) \to A_*(X),\quad (\alpha, \beta) \mapsto \alpha \cdot \beta $$which is called the {\it intersection product}. We observe that this product is clearly symmetric. Associativity follows from the next lemma (as well as the one following). \begin{lemma} \label{lemma-associative} @@ -10240,11 +10254,6 @@ \section{Intersection products for smooth varieties} equation. \end{proof} \noindent In the special case where X = Y = Z we obtain a multiplication A_*(X) \times A_*(X) \to A_*(X). There is an alternative description of this product. \begin{lemma} \label{lemma-identify-chow-for-smooth} Let k be a field. Let X be a smooth scheme over k, equidimensional @@ -10258,10 +10267,25 @@ \section{Intersection products for smooth varieties} \end{lemma} \begin{proof} The map is an isomorphism by combining Lemmas \ref{lemma-chow-cohomology-towards-point} and Proposition \ref{proposition-compute-bivariant}. We omit the verification about composition and products. Denote g : X \to \Spec(k) the structure morphism. The map is the composition of the isomorphisms$$ A^p(X) \to A^{p - d}(X \to \Spec(k)) \to A_{d - p}(X)  The first is the isomorphism $c \mapsto c \circ g^*$ of Proposition \ref{proposition-compute-bivariant} and the second is the isomorphism $c \mapsto c \cap [\Spec(k)]$ of Lemma \ref{lemma-chow-cohomology-towards-point}. From the proof of Lemma \ref{lemma-chow-cohomology-towards-point} we see that the inverse to the second arrow sends $\alpha \in A_{d - p}(X)$ to the bivariant class $c_\alpha$ which sends $\beta \in A_*(Y)$ for $Y$ locally of finite type over $k$ to $\alpha \times \beta$ in $A_*(X \times_k Y)$. From the proof of Proposition \ref{proposition-compute-bivariant} we see the inverse to the first arrow in turn sends $c_\alpha$ to the bivariant class which sends $\beta \in A_*(Y)$ for $Y \to X$ locally of finite type to $\Delta^!(\alpha \times \beta) = \alpha \cdot \beta$. From this the final result of the lemma follows. \end{proof}