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
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}.
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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