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# stacks/stacks-project

Improve exposition of a lemma in weil

It was just a trivial consequence of the blow up formula...
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aisejohan committed Nov 27, 2019
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1. +14 −0 chow.tex
2. +62 −0 obsolete.tex
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4. +40 −109 weil.tex
 @@ -11761,6 +11761,20 @@ \section{An Adams operator} complicated and another approach has to be used. \end{remark} \begin{lemma} \label{lemma-minus-adams-operator} Let $X$ be a scheme. There is a ring map $\psi^{-1} : K_0(\textit{Vect}(X)) \to K_0(\textit{Vect}(X))$ which sends $[\mathcal{E}]$ to $[\mathcal{E}^\vee]$ when $\mathcal{E}$ is finite locally free and is compatible with pullbacks. \end{lemma} \begin{proof} The only thing to check is that taking duals is compatible with short exact sequences and with pullbacks. This is clear. \end{proof} \begin{remark} \label{remark-chern-classes-K} Let $(S, \delta)$ be as in Situation \ref{situation-setup}.
 @@ -2531,6 +2531,68 @@ \section{Modifications} \section{Intersection theory} \label{section-intersection-theory} \begin{lemma} \label{lemma-good-blowing-up} Let $b : X' \to X$ be the blowing up of a smooth projective scheme over a field $k$ in a smooth closed subscheme $Z \subset X$. Picture $$\xymatrix{ E \ar[r]_j \ar[d]_\pi & X' \ar[d]^b \\ Z \ar[r]^i & X }$$ Assume there exists an element of $K_0(X)$ whose restriction to $Z$ is equal to the class of $\mathcal{C}_{Z/X}$ in $K_0(Z)$. Then $[Lb^*\mathcal{O}_Z] = [\mathcal{O}_E] \cdot \alpha''$ in $K_0(X')$ for some $\alpha'' \in K_0(X')$. \end{lemma} \begin{proof} The schemes $X$, $X'$, $E$, $Z$ are smooth and projective over $k$ and hence we have $K'_0(X) = K_0(X) = K_0(\textit{Vect}(X)) = K_0(D^b_{\textit{Coh}}(X)))$ and similarly for the other $3$. See Derived Categories of Schemes, Lemmas \ref{perfect-lemma-Noetherian-Kprime}, \ref{perfect-lemma-Kprime-K}, and \ref{perfect-lemma-K-is-old-K}. We will switch between these versions at will in this proof. Consider the short exact sequence $$0 \to \mathcal{F} \to \pi^*\mathcal{C}_{Z/X} \to \mathcal{C}_{E/X'} \to 0$$ of finite locally free $\mathcal{O}_E$-modules defining $\mathcal{F}$. Observe that $\mathcal{C}_{E/X'} = \mathcal{O}_{X'}(-E)|_E$ is the restriction of the invertible $\mathcal{O}_X$-module $\mathcal{O}_{X'}(-E)$. Let $\alpha \in K_0(X)$ be an element such that $i^*\alpha = [\mathcal{C}_{Z/X}]$ in $K_0(Z)$. Let $\alpha' = b^*\alpha - [\mathcal{O}_{X'}(-E)]$. Then $j^*\alpha' = [\mathcal{F}]$. We deduce that $j^*\lambda^i(\alpha') = [\wedge^i(\mathcal{F})]$ by Weil Cohomology Theories, Lemma \ref{weil-lemma-lambda-operations}. This means that $[\mathcal{O}_E] \cdot \alpha' = [\wedge^i\mathcal{F}]$ in $K_0(X)$, see Derived Categories of Schemes, Lemma \ref{perfect-lemma-projection-formula}. Let $r$ be the maximum codimension of an irreducible component of $Z$ in $X$. A computation which we omit shows that $H^{-i}(Lb^*\mathcal{O}_Z) = \wedge^i\mathcal{F}$ for $i \geq 0, 1, \ldots, r - 1$ and zero in other degrees. It follows that in $K_0(X)$ we have \begin{align*} [Lb^*\mathcal{O}_Z] & = \sum\nolimits_{i = 0, \ldots, r - 1} (-1)^i[\wedge^i\mathcal{F}] \\ & = \sum\nolimits_{i = 0, \ldots, r - 1} (-1)^i[\mathcal{O}_E] \lambda^i(\alpha') \\ & = [\mathcal{O}_E] \left(\sum\nolimits_{i = 0, \ldots, r - 1} (-1)^i \lambda^i(\alpha')\right) \end{align*} This proves the lemma with $\alpha'' = \sum_{i = 0, \ldots, r - 1} (-1)^i \lambda^i(\alpha')$. \end{proof} \begin{lemma} \label{lemma-gysin-factors-principal} Let $(S, \delta)$ be as in Chow Homology, Situation \ref{chow-situation-setup}.
 0FID,weil-section-c1 0FIE,weil-lemma-chern-classes 0FIF,weil-lemma-cycle-classes 0FIG,weil-lemma-good-blowing-up 0FIG,obsolete-lemma-good-blowing-up 0FIH,weil-lemma-divide-pullback-good-blowing-up 0FII,weil-lemma-A5-A6-imply 0FIJ,weil-lemma-poincare-duality
149 weil.tex
 @@ -4649,11 +4649,8 @@ \section{Weil cohomology theories, III} by Lemma \ref{lemma-adams-and-chern} as desired. \end{proof} \noindent The following couple of lemmas can be significantly generalized. \begin{lemma} \label{lemma-good-blowing-up} \label{lemma-divide-pullback-good-blowing-up} Let $b : X' \to X$ be the blowing up of a smooth projective scheme over $k$ in a smooth closed subscheme $Z \subset X$. Picture @@ -4665,119 +4662,53 @@ \section{Weil cohomology theories, III} $$Assume there exists an element of K_0(X) whose restriction to Z is equal to the class of \mathcal{C}_{Z/X} in K_0(Z). Then [Lb^*\mathcal{O}_Z] = [\mathcal{O}_E] \cdot \alpha'' in K_0(X') for some \alpha'' \in K_0(X'). \end{lemma} \begin{proof} The schemes X, X', E, Z are smooth and projective over k and hence we have K'_0(X) = K_0(X) = K_0(\textit{Vect}(X)) = K_0(D^b_{\textit{Coh}}(X))) and similarly for the other 3. See Derived Categories of Schemes, Lemmas \ref{perfect-lemma-Noetherian-Kprime}, \ref{perfect-lemma-Kprime-K}, and \ref{perfect-lemma-K-is-old-K}. We will switch between these versions at will in this proof. Consider the short exact sequence$$ 0 \to \mathcal{F} \to \pi^*\mathcal{C}_{Z/X} \to \mathcal{C}_{E/X'} \to 0 of finite locally free \mathcal{O}_E-modules defining \mathcal{F}. Observe that \mathcal{C}_{E/X'} = \mathcal{O}_{X'}(-E)|_E is the restriction of the invertible \mathcal{O}_X-module \mathcal{O}_{X'}(-E). Let \alpha \in K_0(X) be an element such that i^*\alpha = [\mathcal{C}_{Z/X}] in K_0(Z). Let \alpha' = b^*\alpha - [\mathcal{O}_{X'}(-E)]. Then j^*\alpha' = [\mathcal{F}]. We deduce that j^*\lambda^i(\alpha') = [\wedge^i(\mathcal{F})] by Lemma \ref{lemma-lambda-operations}. This means that [\mathcal{O}_E] \cdot \alpha' = [\wedge^i\mathcal{F}] in K_0(X), see Derived Categories of Schemes, Lemma \ref{perfect-lemma-projection-formula}. Let r be the maximum codimension of an irreducible component of Z in X. A computation which we omit shows that H^{-i}(Lb^*\mathcal{O}_Z) = \wedge^i\mathcal{F} for i \geq 0, 1, \ldots, r - 1 and zero in other degrees. It follows that in K_0(X) we have \begin{align*} [Lb^*\mathcal{O}_Z] & = \sum\nolimits_{i = 0, \ldots, r - 1} (-1)^i[\wedge^i\mathcal{F}] \\ & = \sum\nolimits_{i = 0, \ldots, r - 1} (-1)^i[\mathcal{O}_E] \lambda^i(\alpha') \\ & = [\mathcal{O}_E] \left(\sum\nolimits_{i = 0, \ldots, r - 1} (-1)^i \lambda^i(\alpha')\right) \end{align*} This proves the lemma with \alpha'' = \sum_{i = 0, \ldots, r - 1} (-1)^i \lambda^i(\alpha'). \end{proof} \begin{lemma} \label{lemma-divide-pullback-good-blowing-up} In Lemma \ref{lemma-good-blowing-up} assume every irreducible component of Z has codimension r in X. Then there exists a cycle \theta \in \CH^{r - 1}(X') \otimes \mathbf{Q} such that b^![Z] = [E] \cdot \theta in \CH^r(X') \otimes \mathbf{Q} and \pi_*j^!(\theta) = [Z] in \CH^r(Z) \otimes \mathbf{Q}. Assume every irreducible component of Z has codimension r in X. Then there exists a cycle \theta \in \CH^{r - 1}(X') such that b^![Z] = [E] \cdot \theta in \CH^r(X') and \pi_*j^!(\theta) = [Z] in \CH^r(Z). \end{lemma} \begin{proof} We resume the notation of the proof of Lemma \ref{lemma-good-blowing-up}. Recall that [Z] = ch_r(\mathcal{O}_Z) \cap [X], see Chow Homology, Lemma \ref{chow-lemma-actual-computation}. Since b^![X] = [X'] and since b^! commutes with chern classes (Chow Homology, Remark \ref{chow-remark-gysin-chern-classes}) we obtain \begin{align*} b^![Z] & = b^! (ch_r(\mathcal{O}_Z) \cap [X]) \\ & = ch_r(\mathcal{O}_Z) \cap b^![X] \\ & = ch_r(\mathcal{O}_Z) \cap [X'] \\ & = ch_r(Lb^*\mathcal{O}_Z) \cap [X'] \end{align*} The final equality because ch commutes with pullback and because pullback on b^* : K_0(X) \to K_0(X') is given by derived pullback if we make the identification K_0(X) = K_0(D_{perf}(\mathcal{O}_X)). Using the expression in the proof of Lemma \ref{lemma-good-blowing-up} we see ch(Lb^*\mathcal{O}_Z) = ch(\mathcal{O}_E) \circ ch\left( \sum\nolimits_{i = 0, \ldots, r - 1} (-1)^i\lambda^i(\alpha') \right) The scheme $X$ is smooth and projective over $k$ and hence we have $K_0(X) = K_0(\textit{Vect}(X))$. See Derived Categories of Schemes, Lemmas \ref{perfect-lemma-resolution-property-ample} and \ref{perfect-lemma-K-is-old-K}. Let $\alpha \in K_0(\text{Vect}(X))$ be an element whose restriction to $Z$ is $[\mathcal{F}]$. By Chow Homology, Lemma \ref{chow-lemma-minus-adams-operator} there exists an element $\alpha^\vee$ which restricts to $\mathcal{C}_{Z/X}^\vee$. By the blow up formula (Chow Homology, Lemma \ref{chow-lemma-blow-up-formula}) we have $$We have b^![Z] = b^!i_*[Z] = j_* res(b^!)([Z]) = j_*(c_{r - 1}(\mathcal{F}^\vee) \cap \pi^*[Z]) = j_*(c_{r - 1}(\mathcal{F}^\vee) \cap [E])$$ ch(\mathcal{O}_E) \cap [X'] = ch(\mathcal{O}_{X'}) \cap [X'] - ch(\mathcal{O}_{X'}(-E)) \cap [X'] = [E] - (1/2)[E]^2 + (1/6)[E]^3 - \ldots where $\mathcal{F}$ is the kernel of the surjection $\pi^*\mathcal{C}_{Z/X} \to \mathcal{C}_{E/X'}$. Observe that $b^*\alpha^\vee - [\mathcal{O}_{X'}(E)]$ is an element of $K_0(\text{Vect}(X'))$ which restricts to $[\pi^*\mathcal{C}_{Z/X}^\vee] - [\mathcal{C}_{E/X'}^\vee] = [\mathcal{F}^\vee]$ on $E$. Since capping with chern classes commutes with $j_*$ we conclude that the above is equal to $$We can indeed formally divide'' the expression above by [E]. Thus it makes sense to take c_{r - 1}(b^*\alpha^\vee - [\mathcal{O}_{X'}(E)]) \cap [E]$$ \theta = \text{degree }r - 1\text{ part of } ch(\sum (-1)^i\lambda^i(\alpha')) \cap ([X'] - (1/2)[E] + (1/6)[E]^2 - \ldots) in the chow group of $X'$. Hence we see that setting Then we have b^![Z] = [E] \cdot \theta by construction. Recall that the restriction of \alpha' to E is \mathcal{F}. Thus j^!\theta is equal to the degree r - 1 part of \begin{align*} & ch(\sum (-1)^i\wedge^i(\mathcal{F})) \cap j^!([X'] - (1/2)[E] + (1/6)[E]^2 - \ldots) \\ & = ((-1)^{r - 1}c_{r - 1}(\mathcal{F}) + \ldots) \cap ([E] - (1/2)j^![E] + (1/6)j^![E]^2 - \ldots) \\ & = (-1)^{r - 1}c_{r - 1}(\mathcal{F}) \cap [E] + \ldots \end{align*} by a computation similar to the proof of Chow Homology, Lemma \ref{chow-lemma-compute-koszul}. To prove that \pi_* of this is equal to [Z] it \theta = c_{r - 1}(b^*\alpha^\vee - [\mathcal{O}_{X'}(E)]) \cap [X'] we get the first relation $\theta \cdot [E] = b^![Z]$ for example by Chow Homology, Lemma \ref{chow-lemma-identify-chow-for-smooth}. For the second relation observe that $$j^!\theta = j^!(c_{r - 1}(b^*\alpha^\vee - [\mathcal{O}_{X'}(E)]) \cap [X']) = c_{r - 1}(\mathcal{F}^\vee) \cap j^![X'] = c_{r - 1}(\mathcal{F}^\vee) \cap [E]$$ in the chow groups of $E$. To prove that $\pi_*$ of this is equal to $[Z]$ it suffices to prove that the degree of the codimension $r - 1$ cycle $(-1)^{r - 1}c_{r - 1}(\mathcal{F}) \cap [E]$ on the fibres of $\pi$ is $1$. This is a computation we omit.

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