Perfect residue classes at smooth closed points

By smoothness at $x$, choose a smooth affine neighbourhood $U=\mathrm{Spec}A$ of $x$; it is of finite type over $k$. The point corresponds to a maximal ideal $\mathfrak{m}$ because it is closed. By [F1], $A$ is noetherian and $R=A_{\mathfrak{m}}={\mathcal{O}}_{X,x}$ is regular local. Choose a regular system of parameters $t_1,\dots ,t_d$ of $R$. Since $\mathfrak{m}R$ is generated by images of elements of $\mathfrak{m}$ and denominators outside $\mathfrak{m}$ are units, after multiplying each $t_i$ by a unit we may represent it by an element $f_i\in \mathfrak{m}$; the resulting tuple remains a regular system of parameters. When $d=0$ take the empty tuple.

The ideal $\mathfrak{m}$ is finitely generated, and its quotient by $(f_1,\dots ,f_d)$ localizes to zero at $\mathfrak{m}$. Thus some $h_0\mathrm{\notin }\mathfrak{m}$ kills that quotient: choose an annihilating denominator for each of its finitely many generators and multiply them. For each $j$, let $M_j=A\mathrm{/}(f_1,\dots ,f_{j-1})$. The kernel of multiplication by $f_j$ on $M_j$ is a finite $A$-module because $A$ is noetherian. Its localization at $\mathfrak{m}$ is zero by [F2] and exactness of localization, so some $h_j\mathrm{\notin }\mathfrak{m}$ kills it. Set $h=h_0{h}_1\cdots h_d$ and $V=D(h)$. On $V$ the $f_i$ generate $\mathfrak{m}{A}_h$ and each multiplication by $f_j$ on $A_h\mathrm{/}(f_1,\dots ,f_{j-1})$ is injective. The final quotient is $A_h\mathrm{/}\mathfrak{m}{A}_h\cong A\mathrm{/}\mathfrak{m}\mathrm{\ne }0$. Hence the sequence is regular on $A_h$, including the empty-sequence case.

The closed point $x$ in $V$ is cut out by $\mathfrak{m}{A}_h=(f_1,\dots ,f_d)$. Applying the Koszul resolution statement in [F2] locally at $x$ and using the same noetherian denominator argument, or equivalently applying its regular-sequence resolution assertion to the sequence on $A_h$, gives a finite free resolution of $A_h\mathrm{/}\mathfrak{m}{A}_h$. Its associated complex of finite free sheaves resolves $i_{\ast }\kappa (x){\mathrm{\mid }}_V$. Since the ideal is generated by a regular sequence on a neighbourhood of the only point of the closed subscheme, $i$ is a regular immersion.

The stalks of $i_{\ast }\kappa (x)$ vanish away from $x$: because $\{x\}$ is closed, each other point has a neighbourhood disjoint from it. On $V$ step 3.1 gives a bounded finite free resolution; on $X\setminus \{x\}$ the sheaf is zero, represented by the zero complex. These opens cover $X$, so [F3] makes $i_{\ast }\kappa (x)$ perfect with support contained in $\{x\}$. Its nonzero stalk at $x$ shows that its support is exactly $\{x\}$. This argument does not extend a locally free resolution by zero: the globally defined object is the residue sheaf itself.

A finite-type scheme over a field is qcqs, so [F4] applies. The object of step 4.1 gives its object class in $K_{0,\{x\}}(X)$; naturality for the exact inclusion sends it to the class of the same object in $K_0(X)$. Every choice of parameters and neighbourhood in steps 1.1 through 3.1 resolves the same globally defined sheaf, so the supported object and its class are independent of those choices. AC is used through the stated suppliers.