Levine–Weibel zero cycles

PDF p. 3: Levine–Weibel cycle group

Assume the Axiom of Choice. Let $k$ be a perfect field and let $X$ be a reduced equidimensional quasi-projective $k$-scheme of dimension $d\ge 1$. Here reduced means that the local rings have no nonzero nilpotents, equidimensional means that all irreducible components have dimension $d$, and quasi-projective means that $X$ is an open subscheme of a projective $k$-scheme. A perfect field is one whose algebraic extensions are separable. Write $U=X_{\mathrm{r}\mathrm{e}\mathrm{g}}$ for the smooth locus of $X$ over $k$ and $Y=X_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}=X\setminus U$ with its reduced closed subscheme structure. Let $Z_0(X,Y)$ be the free abelian group on the closed points of $U$.

A Cartier curve relative to $Y$ is a reduced, purely one-dimensional closed subscheme $C\hookrightarrow X$ such that no irreducible component of $C$ is contained in $Y$, and, at every point $x\in C\cap Y$, the ideal of $C$ in ${\mathcal{O}}_{X,x}$ is generated by a regular sequence. Purely one-dimensional means that every irreducible component has dimension one. A regular sequence $a_1,\dots ,a_r$ in a local ring means that $(a_1,\dots ,a_r)$ is proper and each $a_i$ is a non-zero-divisor modulo the preceding elements; the empty sequence is allowed.

For such a curve let $C_1,\dots ,C_r$ be its irreducible components with generic points ${\eta }_i$, and put $S=(C\cap Y)\cup \{{\eta }_1,\dots ,{\eta }_r\}$. Let , where $V$ runs through open neighborhoods of $S$. For $f\in {\mathcal{O}}_{C,S}^{\times }$, let $f_i\in k(C_i{)}^{\times }$ be its restriction to ${\eta }_i$, where $k(C_i)$ is the function field of the integral curve $C_i$. Form the finite zero-cycle ${\mathrm{div}}_C(f)={\sum }_{i=1}^r{\mathrm{div}}_{C_i}(f_i),$ viewed on $X$ through the closed immersions $C_i\hookrightarrow X$. For an integral curve $D$ and $g\in k(D{)}^{\times }$, its principal divisor is ${\sum }_{x\in D\text{ closed}}{\mathrm{ord}}_{{\mathcal{O}}_{D,x}}(g)[x]$: writing $g=a\mathrm{/}b$ with nonzero $a,b\in {\mathcal{O}}_{D,x}$, the coefficient is $\mathrm{length}({\mathcal{O}}_{D,x}\mathrm{/}(a))-\mathrm{length}({\mathcal{O}}_{D,x}\mathrm{/}(b))$.

Let $R_0^{LW}(X,Y)$ be the subgroup of $Z_0(X,Y)$ generated by all ${\mathrm{div}}_C(f)$ for these curves and units. The Levine–Weibel Chow group of zero cycles is ${\mathrm{CH}}_0^{LW}(X)=Z_0(X,Y)\mathrm{/}{R}_0^{LW}(X,Y)\mathrm{.}$ This construction uses regular-point generators and the specified Cartier-curve relations; it is not the definition of the unrestricted ordinary Chow group on a singular scheme.

1.1 The noetherian curve $C$ has finitely many irreducible components. Since none is contained in the closed subset $Y$, the closed subset $C\cap Y$ has dimension zero: any irreducible closed subset of it lies properly inside a one-dimensional component of $C$. A zero-dimensional noetherian space has finitely many points, all closed. Thus $S$ is finite. Each component with its reduced induced structure is an integral finite-type curve. [given, construct]

2.1 A unit in the displayed colimit is represented, after shrinking its open neighborhood of $S$, by a regular function with a regular inverse there. Its restriction to each generic point is therefore a nonzero rational function. By [F1], each component divisor is a well-defined finite zero-cycle, so their finite sum is a finite zero-cycle on $X$. [F1, step 1.1, construct]

3.1 At every $x\in C\cap Y$, the representatives of $f$ and its inverse map to inverse elements of ${\mathcal{O}}_{C_i,x}$ for each component through $x$. Their order in that local domain is zero: in the length formula both quotients by these units have length zero. Hence the component divisors have no coefficient at any point of $Y$, and their sum belongs to $Z_0(X,Y)$. [F1, step 2.1]

4.1 The set of these cycles generates a subgroup of the free abelian group $Z_0(X,Y)$, and the quotient by that subgroup is an abelian group. The definition of the generators is independent of representatives by steps 2.1 and 3.1, so the displayed quotient is well defined. [step 2.1, step 3.1, construct] ∎