# stacks/stacks-project

A bit more in weil and in chow

 @@ -11667,8 +11667,8 @@ \section{Gysin maps for local complete intersection morphisms} as the assumptions are preserved by base change by $X' \to X$ locally of finite type. After replacing $P$ by an open neighbourhood of $s(Z)$ we may assume $P \to X$ is smooth of fixed relative dimension $r$. Say $\dim_\delta(Z) = n$. Then $p^{-1}(Z)$ is equidimensional of dimension $r + n$ and $p^*[Z]$ is given by $[p^{-1}(Z)]_{n + r}$. Say $\dim_\delta(Z) = n$. Then every irreducible component of $p^{-1}(Z)$ has dimension $r + n$ and $p^*[Z]$ is given by $[p^{-1}(Z)]_{n + r}$. Observe that $s(X) \cap p^{-1}(Z) = s(Z)$ scheme theoretically. Hence by the same reference as used above $s(X) \cap p^{-1}(Z)$ is a closed subscheme regularly embedded in $\overline{p}^{-1}(Z)$ of codimension $r$. @@ -11785,8 +11785,8 @@ \section{Gysin maps for local complete intersection morphisms} $g$ is smooth of relative dimension $t$. Then $f^*[Y] = [X]_{n + s}$ and $g^*[Y] = [P]_{n + t}$. On the other hand $i$ is a regular immersion of codimension $t - s$. Thus $i^![P]_{n + t} = [X]_{n + 2}$ follows from Lemma \ref{lemma-gysin-fundamental} and the proof is complete. Thus $i^![P]_{n + t} = [X]_{n + s}$ (Lemma \ref{lemma-lci-gysin-easy}) and the proof is complete. \end{proof} \begin{lemma} @@ -11857,6 +11857,84 @@ \section{Gysin maps for local complete intersection morphisms} immediately from Lemma \ref{lemma-gysin-commutes}. \end{proof} \begin{lemma} \label{lemma-lci-gysin-easy} Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Consider a cartesian diagram $$\xymatrix{ X' \ar[d]_{f'} \ar[r] & X \ar[d]^f \\ Y' \ar[r] & Y }$$ of schemes locally of finite type over $S$. Assume \begin{enumerate} \item $f$ is a local complete intersection morphism and the gysin map exists for $f$, \item $X$, $X'$, $Y$, $Y'$ satisfy the equivalent conditions of Lemma \ref{lemma-locally-equidimensional}, \item for $x' \in X'$ with images $x$, $y'$, and $y$ in $X$, $Y'$, and $Y$ we have $n_{x'} - n_{y'} = n_x - n_y$ where $n_{x'}$, $n_x$, $n_{y'}$, and $n_y$ are as in the lemma, and \item for every generic point $\xi \in X'$ the local ring $\mathcal{O}_{Y', f'(\xi)}$ is Cohen-Macaulay. \end{enumerate} Then $f^![Y'] = [X']$ where $[Y']$ and $[X']$ are defined in the proof. \end{lemma} \begin{proof} Recall that $n_{x'}$ is the common value of $\delta(\xi)$ where $\xi$ is the generic point of an irreducible component passing through $x'$. Moreover, the functions $x' \mapsto n_{x'}$, $x \mapsto n_x$, $y' \mapsto n_{y'}$, and $y \mapsto n_y$ are locally constant. Let $X'_n$, $X_n$, $Y'_n$, and $Y_n$ be the open and closed subscheme of $X'$, $X$, $Y'$, and $Y$ where the function has value $n$. We set $[X'] = \sum [X'_n]_n$ and $[Y'] = \sum [Y'_n]_n$. Having said this, it is clear that to prove the lemma we may replace $X'$ by one of its connected components and $X$, $Y'$, $Y$ by the connected component that it maps into. Then we know that $X'$, $X$, $Y'$, and $Y$ are $\delta$-equidimensional in the sense that each irreducible component has the same $\delta$-dimension. Say $n'$, $n$, $m'$, and $m$ is this common value for $X'$, $X$, $Y'$, and $Y$. The last assumption means that $n' - m' = n - m$. \medskip\noindent Choose a factorization $f = g \circ i$ where $i : X \to P$ is an immersion and $g : P \to Y$ is smooth. As $X$ is connected, we see that the relative dimension of $P \to Y$ at points of $i(X)$ is constant. Hence after replacing $P$ by an open neighbourhood of $i(X)$, we may assume that $P \to Y$ has constant relative dimension and $i : X \to P$ is a closed immersion. Denote $g' : Y' \times_Y P \to Y'$ the base change of $g$ and denote $i' : X' \to Y' \times_Y P$ the base change of $i$. It is clear that $g^*[Y] = [P]$ and $(g')^*[Y'] = [Y' \times_Y P]$. Finally, if $\xi' \in X'$ is a generic point, then $\mathcal{O}_{Y' \times_Y P, i'(\xi)}$ is Cohen-Macaulay. Namely, the local ring map $\mathcal{O}_{Y', f'(\xi)} \to \mathcal{O}_{Y' \times_Y P, i'(\xi)}$ is flat with regular fibre (see Algebra, Section \ref{algebra-section-smooth-overview}), a regular local ring is Cohen-Macaulay (Algebra, Lemma \ref{algebra-lemma-regular-ring-CM}), $\mathcal{O}_{Y', f'(\xi)}$ is Cohen-Macaulay by assumption (4) and we get what we want from Algebra, Lemma \ref{algebra-lemma-CM-goes-up}. Thus we reduce to the case discussed in the next paragraph. \medskip\noindent Assume $f$ is a regular closed immersion and $X'$, $X$, $Y'$, and $Y$ are $\delta$-equidimensional of $\delta$-dimensions $n'$, $n$, $m'$, and $m$ and $m' - n' = m - n$. In this case we obtain the result immediately from Lemma \ref{lemma-gysin-easy}. \end{proof} @@ -12346,8 +12424,7 @@ \section{Intersection products} connected components, hence we may assume $X$ is smooth over $k$ and equidimensional of dimension $d$ and $Y$ is smooth over $k$ and equidimensional of dimension $e$. Observe that $f^![Y]_e = [X]_d$ (because this is true both if $f$ is smooth and if $f$ is a regular immersion; small detail omitted). $f^![Y]_e = [X]_d$ (see for example Lemma \ref{lemma-lci-gysin-easy}). Write $\alpha = c \cap [Y]_e$ and $\beta = c' \cap [Y]_e$ and hence $\alpha \cdot \beta = c \cap c' \cap [Y]_e$, see Lemma \ref{lemma-identify-chow-for-smooth}. @@ -12392,8 +12469,7 @@ \section{Intersection products} connected components, hence we may assume $X$ is smooth over $k$ and equidimensional of dimension $d$ and $Y$ is smooth over $k$ and equidimensional of dimension $e$. Observe that $f^![Y]_e = [X]_d$ (because this is true both if $f$ is smooth and if $f$ is a regular immersion; small detail omitted). $f^![Y]_e = [X]_d$ (see for example Lemma \ref{lemma-lci-gysin-easy}). Write $\alpha = c \cap [X]_d$ and $\beta = c' \cap [Y]_e$, see Lemma \ref{lemma-identify-chow-for-smooth}. We have \begin{align*} @@ -12459,6 +12535,30 @@ \section{Intersection products} Lemma \ref{lemma-gysin-easy} are satisfied and we conclude. \end{proof} \begin{lemma} \label{lemma-intersect-regularly-embedded} Let $k$ be a field. Let $X$ be a scheme smooth over $k$. Let $i : Y \to X$ be a regular closed immersion. Let $\alpha \in \CH_*(X)$. If $Y$ is equidimensional of dimension $e$, then $\alpha \cdot [Y]_e = i_*(i^!(\alpha))$ in $\CH_*(X)$. \end{lemma} \begin{proof} After decomposing $X$ into connected components we may and do assume $X$ is equidimensional of dimension $d$. Write $\alpha = c \cap [X]_n$ with $x \in A^*(X)$, see Lemma \ref{lemma-identify-chow-for-smooth}. Then $$i_*(i^!(\alpha)) = i_*(i^!(c \cap [X]_n)) = i_*(c \cap i^![X]_n) = i_*(c \cap [Y]_e) = c \cap i_*[Y]_e = \alpha \cdot [Y]_e$$ The first equality by choice of $c$. Then second equality by Lemma \ref{lemma-lci-gysin-commutes}. The third because $i^![X]_d = [Y]_e$ in $\CH_*(Y)$ (Lemma \ref{lemma-lci-gysin-easy}). The fourth because bivariant classes commute with proper pushforward. The last equality by Lemma \ref{lemma-identify-chow-for-smooth}. \end{proof}
 @@ -758,9 +758,9 @@ \section{Correspondences} Example \ref{example-graph-correspondence}. Then \begin{enumerate} \item pushforward of cycles by the correspondence $[\Gamma_f]$ corresponds to pullback of cycles by $f$, agrees with the gysin map $f^! : \CH^*(Y) \to \CH^*(X)$, \item pullback of cycles by the correspondence $[\Gamma_f]$ corresponds to pushforward of cycles by $f$, agrees with the pushforward map $f_* : \CH_*(Y) \to \CH_*(X)$, \item if $X$ and $Y$ are equidimensional of dimensions $d$ and $e$, then \begin{enumerate} @@ -769,13 +769,43 @@ \section{Correspondences} corresponds to pushforward of cycles by $f$, and \item pullback of cycles by the correspondence $[\Gamma_f^t]$ of Remark \ref{remark-transpose} corresponds to pullback of cycles by $f$. corresponds to the gysin map $f^!$. \end{enumerate} \end{enumerate} \end{lemma} \begin{proof} Omitted. Proof of (1). Recall that $[\Gamma_f]_*(\alpha) = \text{pr}_{2, *}([\Gamma_f] \cdot \text{pr}_1^*\alpha)$. We have $$[\Gamma_f] \cdot \text{pr}_1^*\alpha = (f, 1)_*((f, 1)^! \text{pr}_1^*\alpha) = (f, 1)_*((f, 1)^! \text{pr}_1^!\alpha) = (f, 1)_*(f^!\alpha)$$ The first equality by Chow Homology, Lemma \ref{chow-lemma-intersect-regularly-embedded}. The second by Chow Homology, Lemma \ref{chow-lemma-lci-gysin-flat}. The third because $\text{pr}_1 \circ (f, 1) = f$ and Chow Homology, Lemma \ref{chow-lemma-lci-gysin-composition}. Then we coclude because $\text{pr}_{2, *} \circ (f, 1)_* = 1_*$ by Chow Homology, Lemma \ref{chow-lemma-compose-pushforward}. \medskip\noindent Proof of (2). Recall that $[\Gamma_f]_*(\beta) = \text{pr}_{1, *}([\Gamma_f] \cdot \text{pr}_2^*\beta)$. Arguing exactly as above we have $$[\Gamma_f] \cdot \text{pr}_2^*\beta = (f, 1)_*\beta$$ Thus the result follows as before. \medskip\noindent Proof of (3). Proved in exactly the same manner as above. \end{proof} \begin{example} @@ -803,15 +833,8 @@ \section{Correspondences} \noindent The category of correspondences is a symmetric monoidal category. Given smooth projective schemes $X$ and $Y$ over $k$, we define $X \otimes Y = X \times Y$. As associativity constraint $$X \otimes (Y \otimes Z) = (X \otimes Y) \otimes Z$$ we use the usual associativity constraint on products of schemes. The unit object will be $\Spec(k)$. The commutativity will be given by the isomorphism $X \times Y \to Y \times X$ switching the factors. Given four smooth projective schemes $X, X', Y, Y'$ over $k$ we define a tensor product $X \otimes Y = X \times Y$. Given four smooth projective schemes $X, X', Y, Y'$ over $k$ we define a tensor product $$\otimes : \text{Corr}^r(X, Y) \times \text{Corr}^{r'}(X', Y') @@ -825,7 +848,13 @@ \section{Correspondences}$$ where $\text{pr}_{13} : X \times X' \times Y \times Y' \to X \times Y$ and $\text{pr}_{24} : X \times X' \times Y \times Y' \to X' \times Y'$ are the projections. are the projections. As associativity constraint $$X \otimes (Y \otimes Z) = (X \otimes Y) \otimes Z$$ we use the usual associativity constraint on products of schemes. The commutativity constraint will be given by the isomorphism $X \times Y \to Y \times X$ switching the factors. \begin{lemma} \label{lemma-tensor-product} @@ -846,11 +875,11 @@ \section{Correspondences} Set $\eta_X = [\Gamma_{X \to X \times X}] \in \text{Corr}^0(X \times X, X)$, $\eta_Y = [\Gamma_{Y \to Y \times Y}] \in \text{Corr}^0(Y \times Y, Y)$, $[X] \in \text{Corr}^d(X, \Spec(k))$, and $[Y] \in \text{Corr}^e(Y, \Spec(k))$. The diagram $[X] \in \text{Corr}^{-d}(X, \Spec(k))$, and $[Y] \in \text{Corr}^{-e}(Y, \Spec(k))$. The diagram $$\xymatrix{ X \otimes Y \ar[r]_{a^t \otimes \text{id}} \ar[d]_{\text{id} \otimes a} & X \otimes Y \ar[r]_{a \otimes \text{id}} \ar[d]_{\text{id} \otimes a^t} & Y \otimes Y \ar[r]_{\eta_Y} & Y \ar[d]^{[Y]} \\ X \otimes X \ar[r]^{\eta_X} & @@ -862,7 +891,30 @@ \section{Correspondences} \end{lemma} \begin{proof} Going either way around the diagram a computation shows that we Recall that \text{Corr}^r(W, \Spec(k)) = \CH_{-r}(W) for any smooth projective scheme W over k and given c \in \text{Corr}^s(W', W) the composition with c agrees with pullback by c as a map \CH_{-r}(W) \to \CH_{-r - s}(W') (Lemma \ref{lemma-composition-correspondences}). Finally, we have Lemma \ref{lemma-functor-and-cycles} which tells us how to convert this into usual pushforward and pullback of cycles. We have$$ (a \otimes \text{id})^* \eta_Y^* [Y] = (a \otimes \text{id})^* [\Delta_Y] = (f \times \text{id})_*\Delta_Y = [\Gamma_f] $$and the other way around we get$$ (\text{id} \otimes a^t)^* \eta_X^* [X] = (\text{id} \otimes a^t)^* [\Delta_X] = (\text{id} \times f)^![\Delta_X] = [\Gamma_f]  The last equality follows from Chow Homology, Lemma \ref{chow-lemma-lci-gysin-easy}. In other words, going either way around the diagram we obtain the element of $\text{Corr}^d(X \times Y, \Spec(k))$ corresponding to the cycle $\Gamma_f \subset X \times Y$. \end{proof}