# stacks/stacks-project

K_0(Coh) otimes Q equals chow

First case we can prove directly using localized chern classes
 @@ -10672,26 +10672,28 @@ \section{Calculating some classes} \xrightarrow{\pi} X $$Then p^* \circ \pi_* \circ c_r(i_*\mathcal{O}_X) = (-1)^{r - 1}(r - 1)! j^* Then c_t(i_*\mathcal{O}_X) = 0 for t = 1, \ldots, r - 1 and in A^0(C \to E) we have$$ p^* \circ \pi_* \circ c_r(i_*\mathcal{O}_X) = (-1)^{r - 1}(r - 1)! j^* $$where j : C \to E and p : C \to X are the inclusion and structure morphism of the vector bundle C = \underline{\Spec}(\text{Sym}^*(\mathcal{C})). \end{lemma} \begin{proof} To prove the equality it suffices to assume that X is integral and prove that both sides give the same result when capping with [E], see Lemma \ref{lemma-bivariant-zero}. The canonical map \pi^*\mathcal{C} \to \mathcal{O}_E(1) vanishes exactly along i(X). Hence the Koszul complex on the map$$ \pi^*\mathcal{C} \otimes \mathcal{O}_E(-1) \to \mathcal{O}_E $$is a resolution of i_*\mathcal{O}_X. In particular we see that i_*\mathcal{O}_X is a perfect object of D(\mathcal{O}_E) whose chern classes are defined. By Lemma \ref{lemma-compute-koszul} we conclude whose chern classes are defined. The vanishing of c_t(i_*\mathcal{O}_X) for t = 1, \ldots, t - 1 follows from Lemma \ref{lemma-compute-koszul}. This lemma also gives$$ c_r(i_*\mathcal{O}_X) = - (r - 1)! c_r(\pi^*\mathcal{C} \otimes \mathcal{O}_E(-1)) @@ -10701,10 +10703,30 @@ \section{Calculating some classes} c_r(\pi^*\mathcal{C} \otimes \mathcal{O}_E(-1)) = (-1)^r c_r(\pi^*\mathcal{C}^\vee \otimes \mathcal{O}_E(1)) $$and \pi^*\mathcal{C}^\vee \otimes \mathcal{O}_E(1) has a section vanishing exactly along i(X). Thus c_r(\pi^*\mathcal{C}^\vee \otimes \mathcal{O}_E(1)) \cap [E] is [i(X)] by Lemma \ref{lemma-top-chern-class}. and \pi^*\mathcal{C}^\vee \otimes \mathcal{O}_E(1) has a section s vanishing exactly along i(X). \medskip\noindent After replacing X by a scheme locally of finite type over X, it suffices to prove that both sides of the equality have the same effect on an element \alpha \in A_*(E). Since C \to X is a vector bundle, every cycle class on C is of the form p^*\beta for some \beta \in A_*(X) (Lemma \ref{lemma-vectorbundle}). Hence by Lemma \ref{lemma-restrict-to-open} we can write \alpha = \pi^*\beta + \gamma where \gamma is supported on E \setminus C. Using the equalities above it suffices to show that$$ p^*(\pi_*(c_r(\pi^*\mathcal{C}^\vee \otimes \mathcal{O}_E(1)) \cap [W])) = j^*[W] $$when W \subset E is an integral closed subscheme which is either (a) disjoint from C or (b) is of the form W = \pi^{-1}Y for some integral closed subscheme Y \subset X. Using the section s and Lemma \ref{lemma-top-chern-class} we find in case (a) c_r(\pi^*\mathcal{C}^\vee \otimes \mathcal{O}_E(1)) \cap [W] = 0 and in case (b) c_r(\pi^*\mathcal{C}^\vee \otimes \mathcal{O}_E(1)) \cap [W] = [i(Y)]. The result follows easily from this; details omitted. \end{proof} @@ -10715,10 +10737,14 @@ \section{Calculating some classes} between schemes locally of finite type over S. Let \mathcal{N} = \mathcal{C}_{Z/X}^\vee be the normal sheaf. If X is quasi-compact and has the resolution property, then c_t(Z \to X, i_*\mathcal{O}_Z) = 0 for t = 1, \ldots, r - 1 and$$ c_r(Z \to X, i_*\mathcal{O}_Z) = (-1)^{r - 1} (r - 1)! c(Z \to X, \mathcal{N}) \quad\text{in}\quad A^r(Z \to X) $$in A^r(Z \to X). The left hand side is the localized chern class where c_t(Z \to X, i_*\mathcal{O}_Z) is the localized chern class of Definition \ref{definition-localized-chern}. \end{lemma} @@ -10762,15 +10788,22 @@ \section{Calculating some classes}$$ c_p(Z \to X, F) = c'_p(Q) = (E \to Z)_* \circ c'_p(Q|_E) \circ C $$On the other hand, by construction of c(Z \to X, \mathcal{N}) and because \mathcal{C}_{Z/X} = \mathcal{N}^\vee is locally free we have for all p \geq 1. Observe that Q|_E is equal to the pushforward of the structure sheaf of Z via the morphism Z \to E which is the base change of i' by \infty. Thus the vanishing of c_t(Z \to X, F) for 1 \leq t \leq r - 1 by Lemma \ref{lemma-compute-section} applied to E \to Z. Because \mathcal{C}_{Z/X} = \mathcal{N}^\vee is locally free the bivariant class c(Z \to X, \mathcal{N}) is characterized by the relation$$ j^* \circ C = p^* \circ c(Z \to X, \mathcal{N}) $$where j : C_ZX \to W_\infty and p : C_ZX \to Z are the given maps. Thus the relationship follows from Lemma \ref{lemma-compute-section} applied to E \to Z. (Recall C \in A^0(W_\infty \to X) is the class of Lemma \ref{lemma-gysin-at-infty}.) Thus the displayed equation in the statement of the lemma follows from the corresponding equation in Lemma \ref{lemma-compute-section}. \medskip\noindent Proof of the claim. Let A and f_1, \ldots, f_r be as above. @@ -10798,6 +10831,155 @@ \section{Calculating some classes} (and Lemma \ref{lemma-localized-chern-pre}) are immediately verified. \end{proof} \begin{lemma} \label{lemma-actual-computation} In the situation of Lemma \ref{lemma-agreement-with-loc-chern} say \dim_\delta(X) = n. Then we have \begin{enumerate} \item c_t(Z \to X, i_*\mathcal{O}_Z) \cap [X]_n = 0 for t = 1, \ldots, r - 1, \item c_r(Z \to X, i_*\mathcal{O}_Z) \cap [X]_n = (-1)^{r - 1}(r - 1)![Z]_{n - r}, \item ch_t(Z \to X, i_*\mathcal{O}_Z) \cap [X]_n = 0 for t = 0, \ldots, r - 1, and \item ch_r(Z \to X, i_*\mathcal{O}_Z) \cap [X]_n = [Z]_{n - r}. \end{enumerate} \end{lemma} \begin{proof} Parts (1) and (2) follow immediately from Lemma \ref{lemma-agreement-with-loc-chern} combined with Lemma \ref{lemma-gysin-fundamental}. Then we deduce parts (3) and (4) using the relationship between ch_p = (1/p!)P_p and c_p given in Lemma \ref{lemma-loc-chern-character}. (Namely, (-1)^{r - 1}(r - 1)!ch_r = c_r provided c_1 = c_2 = \ldots = c_{r - 1} = 0.) \end{proof} \section{Chow groups and K-groups revisited} \label{section-chow-and-K-II} \noindent This section is the continuation of Section \ref{section-chow-and-K}. Let (S, \delta) be as in Situation \ref{situation-setup}. Let X be locally of finite type over S. The K-group K_0(\textit{Coh}(X)) has a canonical increasing filtration$$ F_kK_0(\textit{Coh}(X)) = \Im(K_0(\textit{Coh}_{\leq k}(X)) \to K_0(\textit{Coh}(X))) $$This is called the filtration by dimension of supports. Observe that$$ \text{gr}_k K_0(\textit{Coh}(X)) \subset K_0(\textit{Coh}(X))/F_{k - 1}K_0(\textit{Coh}(X)) = K_0(\textit{Coh}(X)/\textit{Coh}_{\leq k - 1}(X)) $$where the equality holds by Homology, Lemma \ref{homology-lemma-serre-subcategory-K-groups}. The discussion in Remark \ref{remark-good-cases-K-A} shows that there are canonical maps$$ A_k(X) \longrightarrow \text{gr}_k K_0(\textit{Coh}(X)) $$defined by sending the class of an integral closed subscheme Z \subset X of \delta-dimension k to the class of [\mathcal{O}_Z] on the right hand side. \begin{proposition} \label{proposition-K-tensor-Q} Let (S, \delta) be as in Situation \ref{situation-setup}. Assume given a closed immersion i : X \to Y of schemes locally of finite type over S with Y regular, quasi-compact, affine diagonal, and \delta_{Y/S} : Y \to \mathbf{Z} bounded. Then the map$$ A_k(X) \otimes \mathbf{Q} \longrightarrow \text{gr}_k K_0(\textit{Coh}(X)) \otimes \mathbf{Q} $$(see above) is an isomorphism of \mathbf{Q}-vector spaces. \end{proposition} \begin{proof} We have the resolution property for Y by Derived Categories of Schemes, Lemma \ref{perfect-lemma-regular-resolution-property}. Thus every perfect object E of D(\mathcal{O}_Y) can be represented by a bounded complex of finite locally free modules, see Derived Categories of Schemes, Lemma \ref{perfect-lemma-resolution-property-perfect-complex}. Hence the chern classes of E are defined (Definition \ref{definition-defined-on-perfect}). Finally, because \delta : Y \to \mathbf{Z} is bounded, the perfect objects of D(\mathcal{O}_Y) are exactly given by the objects of D^b_{\textit{Coh}}(\mathcal{O}_Y), see Derived Categories of Schemes, Lemma \ref{perfect-lemma-perfect-on-regular}. Please keep these facts in mind below. \medskip\noindent Given a coherent \mathcal{O}_X-module \mathcal{F} we can consider the localized chern class$$ ch(X \to Y, i_*\mathcal{F}) \in A^*(X \to Y) \otimes \mathbf{Q} $$See Definition \ref{definition-localized-chern} (and comments in the first paragraph). This construction is additive in short exact sequences, see Lemma \ref{lemma-additivity-loc-chern-P}. Hence we obtain$$ \Psi : K_0(\textit{Coh}(X)) \longrightarrow A_*(X) \otimes \mathbf{Q},\quad [\mathcal{F}] \longmapsto ch(X \to Y, i_*\mathcal{F}) \cap [Y] $$where [Y] = \sum [Y_i] \in A_*(Y) is the sum of the classes of the irreducible components of Y. If \mathcal{F} is (set theoretically) supported on a closed subscheme Z \subset X, then we have$$ ch(X \to Y, i_*\mathcal{F}) = (Z \to X)_* \circ ch(Z \to Y, i_*\mathcal{F}) $$by Lemma \ref{lemma-loc-chern-shrink-Z}. We conclude that \Psi sends F_kK_0(\textit{Coh}(X)) into \bigoplus_{k' \leq k} A_{k'}(X) \otimes \mathbf{Q}. This already defines a map$$ \Psi_k : \text{gr}_k K_0(\textit{Coh}(X)) \longrightarrow A_k(X) \otimes \mathbf{Q} $$for every k \in \mathbf{Z}. Thus to finish the proof, it suffices to show that given an integral closed subscheme Z \subset X of \delta-dimension k we have \Psi_k([\mathcal{O}_Z]) = [Z] in A_k(X). By the above, it suffices to show that$$ ch(Z \to Y, i_*\mathcal{O}_Z) \cap [Y] = [Z] \quad\text{in}\quad A_k(Z) \otimes \mathbf{Q} $$Since A_k(Z) = \mathbf{Z}, in order to prove this we may replace Y by an open neighbourhood of the generic point \xi of Z. Since the maximal ideal of the regular local ring \mathcal{O}_{X, \xi} is generated by a regular sequence (Algebra, Lemma \ref{algebra-lemma-regular-ring-CM}) we may assume the ideal of Z is generated by a regular sequence, see Divisors, Lemma \ref{divisors-lemma-Noetherian-scheme-regular-ideal}. Thus we deduce the result from Lemma \ref{lemma-actual-computation}. \end{proof} \begin{remark} \label{remark-K-tensor-Q} In the situation of Proposition \ref{proposition-K-tensor-Q} the map \Psi of the proof defines an isomorphism$$ K_0(\textit{Coh}(X)) \otimes \mathbf{Q} \cong A_*(X) \otimes \mathbf{Q}  This isomorphism is not canonical: it depends on the choice of the embedding $X \to Y$ and not just on $X$ itself. \end{remark}
 @@ -2015,6 +2015,18 @@ \section{Derived category of coherent modules} \ref{lemma-direct-image-coherent-bdd-below}. \end{proof} \begin{lemma} \label{lemma-perfect-on-regular} Let $X$ be a Noetherian regular scheme of finite dimension. Then every object of $D^b_{\textit{Coh}}(\mathcal{O}_X)$ is perfect and conversely every perfect object of $D(\mathcal{O}_X)$ is in $D^b_{\textit{Coh}}(\mathcal{O}_X)$. \end{lemma} \begin{proof} Combine More on Algebra, Lemma \ref{more-algebra-lemma-regular-perfect} with Lemma \ref{lemma-perfect-affine}. \end{proof}