The Stacks project

42.57 Chow groups and K-groups revisited

This section is the continuation of Section 42.23. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. The K-group $K'_0(X) = K_0(\textit{Coh}(X))$ of coherent sheaves on $X$ has a canonical increasing filtration

\[ F_ kK'_0(X) = \mathop{\mathrm{Im}}\Big(K_0(\textit{Coh}_{\leq k}(X)) \to K_0(\textit{Coh}(X)\Big) \]

This is called the filtration by dimension of supports. Observe that

\[ \text{gr}_ k K'_0(X) \subset K'_0(X)/F_{k - 1}K'_0(X) = K_0(\textit{Coh}(X)/\textit{Coh}_{\leq k - 1}(X)) \]

where the equality holds by Homology, Lemma 12.11.3. The discussion in Remark 42.23.5 shows that there are canonical maps

\[ \mathop{\mathrm{CH}}\nolimits _ k(X) \longrightarrow \text{gr}_ k K'_0(X) \]

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.

Proposition 42.57.1. Let $(S, \delta )$ be as in Situation 42.7.1. Assume given a closed immersion $X \to Y$ of schemes locally of finite type over $S$ with $Y$ regular and quasi-compact. Then the composition

\[ K'_0(X) \to K_0(D_{X, perf}(\mathcal{O}_ Y)) \to A^*(X \to Y) \otimes \mathbf{Q} \to \mathop{\mathrm{CH}}\nolimits _*(X) \otimes \mathbf{Q} \]

of the map $\mathcal{F} \mapsto \mathcal{F}[0]$ from Remark 42.56.8, the map $ch(X \to Y, -)$ from Remark 42.56.11, and the map $c \mapsto c \cap [Y]$ induces an isomorphism

\[ K'_0(X) \otimes \mathbf{Q} \longrightarrow \mathop{\mathrm{CH}}\nolimits _*(X) \otimes \mathbf{Q} \]

which depends on the choice of $Y$. Moreover, the canonical map

\[ \mathop{\mathrm{CH}}\nolimits _ k(X) \otimes \mathbf{Q} \longrightarrow \text{gr}_ k K'_0(X) \otimes \mathbf{Q} \]

(see above) is an isomorphism of $\mathbf{Q}$-vector spaces for all $k \in \mathbf{Z}$.

Proof. Since $Y$ is regular, the construction in Remark 42.56.8 applies. Since $Y$ is quasi-compact, the construction in Remark 42.56.11 applies. We have that $Y$ is locally equidimensional (Lemma 42.42.1) and thus the “fundamental cycle” $[Y]$ is defined as an element of $\mathop{\mathrm{CH}}\nolimits _*(Y)$, see Remark 42.42.2. Combining this with the map $\mathop{\mathrm{CH}}\nolimits _ k(X) \to \text{gr}_ kK'_0(X)$ constructed above we see that it suffices to prove

  1. If $\mathcal{F}$ is a coherent $\mathcal{O}_ X$-module whose support has $\delta $-dimension $\leq k$, then the composition above sends $[\mathcal{F}]$ into $\bigoplus _{k' \leq k} \mathop{\mathrm{CH}}\nolimits _{k'}(X) \otimes \mathbf{Q}$.

  2. If $Z \subset X$ is an integral closed subscheme of $\delta $-dimension $k$, then the composition above sends $[\mathcal{O}_ Z]$ to an element whose degree $k$ part is the class of $[Z]$ in $\mathop{\mathrm{CH}}\nolimits _ k(X) \otimes \mathbf{Q}$.

Namely, if this holds, then our maps induce maps $\text{gr}_ kK'_0(X) \otimes \mathbf{Q} \to CH_ k(X) \otimes \mathbf{Q}$ which are inverse to the canonical maps $\mathop{\mathrm{CH}}\nolimits _ k(X) \otimes \mathbf{Q} \to \text{gr}_ k K'_0(X) \otimes \mathbf{Q}$ given above the proposition.

Given a coherent $\mathcal{O}_ X$-module $\mathcal{F}$ the composition above sends $[\mathcal{F}]$ to

\[ ch(X \to Y, \mathcal{F}[0]) \cap [Y] \in \mathop{\mathrm{CH}}\nolimits _*(X) \otimes \mathbf{Q} \]

If $\mathcal{F}$ is (set theoretically) supported on a closed subscheme $Z \subset X$, then we have

\[ ch(X \to Y, \mathcal{F}[0]) = (Z \to X)_* \circ ch(Z \to Y, \mathcal{F}[0]) \]

by Lemma 42.50.8. We conclude that in this case we end up in the image of $\mathop{\mathrm{CH}}\nolimits _*(Z) \to \mathop{\mathrm{CH}}\nolimits _*(X)$. Hence we get condition (1).

Let $Z \subset X$ be an integral closed subscheme of $\delta $-dimension $k$. The composition above sends $[\mathcal{O}_ Z]$ to the element

\[ ch(X \to Y, \mathcal{O}_ Z[0]) \cap [Y] = (Z \to X)_* ch(Z \to Y, \mathcal{O}_ Z[0]) \cap [Y] \]

by the same argument as above. Thus it suffices to prove that the degree $k$ part of $ch(Z \to Y, \mathcal{O}_ Z[0]) \cap [Y] \in \mathop{\mathrm{CH}}\nolimits _*(Z) \otimes \mathbf{Q}$ is $[Z]$. Since $\mathop{\mathrm{CH}}\nolimits _ k(Z) = \mathbf{Z}$, 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 10.106.3) we may assume the ideal of $Z$ is generated by a regular sequence, see Divisors, Lemma 31.20.8. Thus we deduce the result from Lemma 42.55.4. $\square$

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