Smooth-point K theory class map

PDF p. 7: Equation (2.8)

Assume the Axiom of Choice and the size conventions of the support construction. Let $k$ be a perfect field, $X$ a quasi-compact quasi-separated $k$-scheme, and $x$ a closed point at which $X\to \mathrm{Spec}k$ is smooth. A perfect field is a field all of whose algebraic extensions are separable; smooth at $x$ means that $x$ has an open neighbourhood on which the structure morphism is smooth. Put $U=X\setminus \{x\}$.

Write $K(X\mathrm{/}U)$ for the fiber of the restriction map $K(X)\to K(U)$, and $K_0(X)={\pi }_{0}K(X)$. The support-image group $K_0(\{x\};X)$ is the image of ${\pi }_{0}K(X\mathrm{/}U)\to K_0(X)$. Let $p_x:{\pi }_{0}K(X\mathrm{/}U)\stackrel{\sim }{\to }\mathbb{Z}$ be the canonical isomorphism of Excision and K theory purity at a smooth closed point, normalized so that the supported class of the residue sheaf $i_{\ast }\kappa (x)$ maps to $1$, where $i:\mathrm{Spec}\kappa (x)\hookrightarrow X$ is the closed immersion. Define

${\phi }_x:\mathbb{Z}⟶K_0(\{x\};X)\subseteq K_0(X)$

as $p_x^{-1}$ followed by the support-to-global homomorphism, with its codomain restricted to its image. In particular, ${\phi }_x(n)=n[i_{\ast }\kappa (x)]$ in $K_0(X)$, where the bracket is the class of the perfect residue sheaf. The map need not be injective.

1.1 By [F1], $p_x$ is an isomorphism of abelian groups, so its inverse is a well-defined homomorphism from $\mathbb{Z}$ to ${\pi }_{0}K(X\mathrm{/}U)$. Composing it with support-to-global gives a homomorphism to $K_0(X)$. [A1, F1, given]

2.1 By [F2], every value of this composite lies in $K_0(\{x\};X)$, so restricting its codomain defines the displayed homomorphism. Since $p_x^{-1}$ is surjective, its composite has image exactly $K_0(\{x\};X)$. [A1, F2, step 1.1]

3.1 By [F1], the value at $1$ is $[i_{\ast }\kappa (x)]$. Additivity gives the formula for every integer $n$. The construction makes no injectivity assertion about support-to-global; thus it does not assert that ${\phi }_x$ is injective. This establishes the definition and its stated properties. [A1, F1, F2, step 2.1] ∎