Rational equivalence and the Chow group of cycles

Assume the Axiom of Choice (The Axiom of Choice), inherited from the order function of The order function of a one-dimensional Noetherian local domain; it is used there for the finiteness of the lengths, and here only through that order function. Let $k$ be a field and let $X$ be a scheme locally of finite type over $k$. For $d\in \mathbb{Z}$ let ${\mathrm{Rat}}_d(X)\subseteq Z_d(X)$ be the subgroup generated by all cycles of the form $(i_W{)}_{\ast }\mathrm{div}(r),\mathrm{div}(r)={\sum }_{V\subseteq W}{\mathrm{ord}}_{{\mathcal{O}}_{W,V}}(r)[V],$ where $W\subseteq X$ is an integral closed subscheme of dimension $d+1$ (Integral schemes), $i_W:W\hookrightarrow X$ is the inclusion, $r\in k(W{)}^{\ast }$ is a nonzero rational function on $W$ (Sheaf total quotient rings, The field of fractions $\mathrm{Frac}(D)=(D\setminus \{0\}{)}^{-1}D$ of an integral domain), the sum runs over the integral closed subschemes $V\subseteq W$ of dimension $d$, and ${\mathrm{ord}}_{{\mathcal{O}}_{W,V}}(r)$ is the order function of The order function of a one-dimensional Noetherian local domain for the one-dimensional Noetherian local domain ${\mathcal{O}}_{W,V}$ (A local ring is a nonzero commutative ring with a unique maximal ideal). The sum is locally finite, and finite when $W$ is of finite type, as proved below. The elements of ${\mathrm{Rat}}_d(X)$ are the $d$-cycles rationally equivalent to zero; two cycles are rationally equivalent if their difference is, written $\alpha {\sim }_{\mathrm{r}\mathrm{a}\mathrm{t}}\beta$. The Chow group of $d$-cycles is $A_d(X):=Z_d(X)\mathrm{/}{\mathrm{Rat}}_d(X),A_{\ast }(X)={⨁}_d{A}_d(X)\text{for finite type }X\mathrm{.}$

Basic facts. (i) Rational equivalence is an equivalence relation compatible with the group structure, by definition. (ii) If $W$ is a normal integral closed subscheme, then $\mathrm{div}(r)$ agrees with the principal Weil divisor of $r$ computed with the divisor theory of Order codimension one rational function, Principal weil divisor and class group; in particular for $X$ integral of dimension $n$ and normal (e.g. smooth), $A_n(X)\cong \mathbb{Z}$ generated by $[X]$ and $A_d(X)=0$ for $d>\dim X$. (iii) For $X$ equidimensional of pure dimension $n$, set $A^p(X):=A_{n-p}(X)$; this is the codimension grading used throughout. (iv) Equivalent presentation: $\alpha {\sim }_{\mathrm{r}\mathrm{a}\mathrm{t}}0$ if and only if $\alpha ={\sum }_i(i_{W_i}{)}_{\ast }\mathrm{div}(r_i)$ as above; this is the "Fulton 1.6" definition with supports on closed subschemes of $X\times {\mathbb{P}}^1$ specialized, and it is Stacks' Definition 42.19.1 (tag 02RW). For merely locally finite type $X$, use the locally finite cycle group of Algebraic cycles and the cycle group of a scheme of finite type over a field and locally finite sums of principal divisors supported on locally finite families of the subvarieties $W$. These sums define the rational-equivalence subgroup; finite generation as written above applies when $X$ is of finite type. In the general case the total group $A_{\ast }(X)$ is the quotient of the total locally finite cycle group by locally finite sums of these relations, allowing unbounded dimensions as in Algebraic cycles and the cycle group of a scheme of finite type over a field. Thus pullback along a non-quasi-compact flat map uses locally finite cycles, not a finite-support sum with infinitely many components.

Well-definedness of the divisor sum. Let $W$ be integral of dimension $d+1$ and $0\mathrm{\ne }r\in k(W{)}^{\ast }$. On an affine open chart $\mathrm{Spec}A\subseteq W$ with $W$ of finite type over $k$, write $r=a\mathrm{/}b$ with $a,b\in A\setminus \{0\}$ under the identification $k(W)=\mathrm{Frac}(A)$. For an integral closed subscheme $V\subseteq W$ of dimension $d$ meeting the chart in a height-one prime $\mathfrak{p}$ of $A$, the order ${\mathrm{ord}}_{{\mathcal{O}}_{W,V}}(r)$ of The order function of a one-dimensional Noetherian local domain vanishes unless $\mathfrak{p}$ contains $a$ or $b$, hence unless $\mathfrak{p}$ contains $ab$; then $\mathfrak{p}$ is minimal over the principal ideal $(ab)$, hence has height one by A minimal prime over a principal nonzerodivisor has height one. A Noetherian ring has only finitely many minimal primes over the radical of a principal ideal (A radical ideal in a Noetherian ring is a finite intersection of minimal primes), so only finitely many $V$ meet a fixed affine chart with nonzero coefficient; since $W$ of finite type over a field is quasi-compact, a finite affine cover exhibits $\mathrm{div}(r)$ as a finite sum, and for merely locally finite type $W$ it is a locally finite sum in the sense of Algebraic cycles and the cycle group of a scheme of finite type over a field. Thus the displayed sum defines an element of $Z_d(W)$ in the finite-type case and of the locally finite cycle group in general, and the subgroup ${\mathrm{Rat}}_d(X)$ is well defined.