Subgroup generated by smooth closed points

PDF p. 2: Before Theorem C; PDF p. 14: Definition 6.5

Assume the Axiom of Choice and the size conventions of the support construction. Let $k$ be a perfect field, meaning that every algebraic extension of $k$ is separable, and let $X$ be a quasi-compact quasi-separated $k$-scheme of pure finite dimension $d$. Thus $X$ has a finite affine open cover, intersections of quasi-compact opens are quasi-compact, and every irreducible component has dimension $d$; dimension is the supremum of lengths of chains of distinct irreducible closed subsets. Take $d\ge 0$.

Let $S$ be the set of closed points $x\in X$ at which $X\to \mathrm{Spec}k$ is smooth: each such point has an open neighbourhood smooth over $k$. For each $x\in S$, Smooth-point K theory class map gives a homomorphism ${\phi }_x:\mathbb{Z}\to K_0(X)$, with ${\phi }_x(1)=[i_{\ast }\kappa (x)]$, where $i:\mathrm{Spec}\kappa (x)\hookrightarrow X$ is the closed immersion and the bracket denotes the class of the perfect residue sheaf. Here $K_0(X)={\pi }_{0}K(X)$ for the nonconnective $K$-theory spectrum of perfect complexes.

The smooth-point subgroup is $F_{\mathrm{s}\mathrm{m}}^d{K}_0(X):=\mathrm{im}({⨁}_{x\in S}\mathbb{Z}\stackrel{\oplus {\phi }_x}{\to }{F}_{\mathrm{m}\mathrm{o}\mathrm{t}}^d{K}_0(X))\subseteq F_{\mathrm{m}\mathrm{o}\mathrm{t}}^d{K}_0(X)\subseteq K_0(X)\mathrm{.}$ Here the direct sum consists of integer families with finite support, and its displayed map sends $(n_x)$ to the finite sum ${\sum }_x{\phi }_x(n_x)$. The target $F_{\mathrm{m}\mathrm{o}\mathrm{t}}^d{K}_0(X)$ is the image of ${\pi }_0{F}_{\mathrm{m}\mathrm{o}\mathrm{t}}^{d}K(X)\to K_0(X)$ as defined in Motivic filtration and image groups. Equivalently, $F_{\mathrm{s}\mathrm{m}}^d{K}_0(X)$ is the subgroup of $K_0(X)$ generated by the classes $[i_{\ast }\kappa (x)]$ for $x\in S$. If $S$ is empty, it is zero.

1.1 For every $x\in S$, [F1] gives a homomorphism to $K_0(X),$ and [F2] allows its codomain to be restricted to the filtration subgroup. For a finitely supported integer family, the sum of its values is therefore a well-defined element of that subgroup. Finite sums and the homomorphism property of each ${\phi }_x$ make the displayed assignment a homomorphism. [F1, F2, given]

2.1 Its image is a subgroup of the target: zero is the image of the zero family, and sums and inverses of images are images of the corresponding sums and inverses of families. Every image is a finite integer linear combination of the values ${\phi }_x(1)$, and every such combination is the image of a finitely supported family. By [F1] these values are the stated residue-sheaf classes. If $S$ is empty, the direct sum and its image are zero. This establishes the definition and its equivalent generator description. [F1, step 1.1] ∎