Excision and K theory purity at a smooth closed point

Choose a smooth affine finite-type neighbourhood $V$ of $x$ and shrink it as in [F3] so that the ideal of $x$ in $V$ is finitely generated by a regular sequence. The point is closed in $V$, and $V\setminus \{x\}$ is quasi-compact: it is the finite union of the principal opens where the chosen generators are nonzero. The empty sequence gives the empty complement when $V$ is the point.

If $Z$ is any quasi-compact open of $X$ containing $V$, then $Z\setminus \{x\}$ is quasi-compact. Indeed, cover $Z$ by finitely many affine opens. An affine open $A$ not containing $x$ needs no removal. If $x\in A$, choose a principal neighbourhood $D(f)$ of $x$ contained in $A\cap V$. The closed set $A\setminus D(f)$ is quasi-compact and avoids $x$, so finitely many principal opens in $A$ avoiding $x$ cover it. Inside $D(f)$ the complement of $x$ is quasi-compact: it is the intersection of $D(f)$ and $V\setminus \{x\}$, hence quasi-compact because $X$ is quasi-separated and both are quasi-compact opens of $X$. These finite covers prove the claim. In particular $U$ is quasi-compact by taking $Z=X$. Every such $Z$ is quasi-separated as an open subscheme of $X$.

The displayed scheme square is cartesian because intersections of open subschemes are their fiber products. It is cocartesian because $S\cup U=X$: $x\in S$, and $U$ contains every other point. Morphisms from $S$ and $U$ to any scheme that agree on $S\cap U$ glue uniquely over this open cover.

Let $\mathcal{C}$ be the poset of quasi-compact opens $Z$ of $S$ containing $V$. It is cofinal among quasi-compact opens of $S$, since any such open $B$ is contained in the quasi-compact open $B\cup V$. By step 2.1, each $Z\setminus \{x\}$ is quasi-compact. These complements are cofinal among quasi-compact opens of $S\setminus \{x\}$: a quasi-compact open $B$ there is contained in $(B\cup V)\setminus \{x\}$. Consequently [F1], cofinality of homotopy limits, and preservation of limits by fibers give $K(S\mathrm{/}(S\cap U))\simeq {\mathrm{holim}}_{Z\in {\mathcal{C}}^{\mathrm{o}\mathrm{p}}}\mathrm{fib}(K(Z)\to K(Z\setminus \{x\}))\mathrm{.}$ The comparison is induced by restriction, not merely an abstract equivalence.

For each $Z\in \mathcal{C}$, localization and supported excision in [F2] identify its displayed fiber, naturally in $Z$, with $K_{\{x\}}(V)$. The indexing poset has the least element $V$, so its nerve is contractible; the homotopy limit of this coherent constant diagram is $K_{\{x\}}(V)$. Applying the same localization and excision to $V\subseteq X$, using the quasi-compact complement from step 2.1, identifies $K(X\mathrm{/}U)$ with $K_{\{x\}}(V)$ compatibly with restriction. Thus restriction induces an equivalence $K(X\mathrm{/}U)\simeq K(S\mathrm{/}(S\cap U))$.

The restriction square on $X,S,U,S\cap U$ is coherent. Its horizontal fibers, taken over $K(U)$ and $K(S\cap U)$, are equivalent by step 5.1. In spectra a square with equivalent corresponding fibers is homotopy cartesian: compare its upper-left corner with the homotopy pullback of the other three corners and use the fiber sequences over the common base. This proves the K-theory square, including when $S$ is not quasi-compact.

The scheme $V$ is regular, Noetherian and finite-dimensional, and the closed subscheme $\mathrm{Spec}\kappa (x)$ has underlying set $\{x\}$. By [F3], supported dévissage gives ${\pi }_0{K}_{\{x\}}(V)\cong G_0(\kappa (x))$. Every finite-dimensional vector space over a field is a finite direct sum of copies of the field, and dimension is additive in short exact sequences. Hence dimension gives $G_0(\kappa (x))\cong \mathbb{Z}$, with inverse $1\mapsto [\kappa (x)]$. Combining this with step 5.1 gives the asserted isomorphisms.

The inverse in dévissage sends $[\kappa (x)]$ to the class of the residue sheaf on $V$. Let $E=i_{\ast }\kappa (x)[0]$ be the globally defined residue sheaf in degree zero. Its restriction to $V$ is perfect by the Koszul resolution in [F3], applied only to the smooth finite-type scheme $V$. Its restriction to $U=X\setminus \{x\}$ is zero, hence perfect. Since $V\cup U=X$, the local definition in [F4] makes $E$ perfect on $X$; its cohomology stalks vanish away from $x$, so it belongs to the supported perfect category on $X$. Naturality of localization and restriction in [F2] identifies the supported class of this object with that of its restriction to $V$. Its image under support-to-global is the class of the same perfect object on $X$. Thus it is the inverse image of $1$ under the displayed isomorphisms. The image in global $K_0$ can vanish, so no injectivity is used.