Distinguishing smooth-point and motivic filtration notation

PDF p. 14: Remark 6.6

Assume the Axiom of Choice and the size conventions of the smooth-point filtration definition. Let $k$ be a perfect field and let $X$ be a quasi-compact, quasi-separated $k$-scheme of pure finite dimension $d\ge 0$, as in that definition. The subgroup of $K_0(X)$ generated by the classes of the residue sheaves of closed points where $X$ is smooth over $k$ is often written $F^d{K}_0(X)$. The article writes it $F_{\mathrm{s}\mathrm{m}}^d{K}_0(X)$ to distinguish this notation from $F_{\mathrm{m}\mathrm{o}\mathrm{t}}^d{K}_0(X)$.

Here $K_0(X)={\pi }_{0}K(X)$ for the supplied nonconnective perfect-complex $K$-theory spectrum, and $F_{\mathrm{m}\mathrm{o}\mathrm{t}}^d{K}_0(X)$ is the image of ${\pi }_0{F}_{\mathrm{m}\mathrm{o}\mathrm{t}}^{d}K(X)\to K_0(X)$. The stated distinction is a distinction of notation; the subgroup inclusion below does not assert that the two groups are unequal.

Subgroup generated by smooth closed points

1.1 Under the given hypotheses, including the inherited Choice premise in [F3], [F1] identifies the smooth-point subgroup with the group whose alternative notation is described in the article. [F1, F3, given]

2.1 By [F1] and [F2], $F_{\mathrm{s}\mathrm{m}}^d{K}_0(X)\subseteq F_{\mathrm{m}\mathrm{o}\mathrm{t}}^d{K}_0(X)\subseteq K_0(X)$. Writing the two subscripts makes their definitions distinguishable without asserting that the inclusion is strict. This establishes the remark. [F1, F2, step 1.1] ∎