Algebraic cycles and the cycle group of a scheme of finite type over a field

Let $k$ be a field and let $X$ be a scheme locally of finite type over $k$ (Locally finite type and finite type morphisms, Schemes and morphisms over a base). An algebraic cycle on $X$ is a formal $\mathbb{Z}$-linear combination of integral closed subschemes of $X$ (Integral schemes, Closed immersions of schemes); it is finite when the combination is finite. For an integer $d$, the group of $d$-cycles is the free abelian group

$Z_d(X):={⨁}_V\mathbb{Z}\cdot [V],$

Here the dimension of any scheme is the supremum of lengths of strict chains of nonempty irreducible closed subsets of its underlying space, with $\dim \mathrm{\varnothing }=-\mathrm{\infty }$. This extends the Noetherian convention Chain dimension and the empty-space convention without requiring quasi-compactness. The direct sum runs over the integral closed subschemes $V\subseteq X$ of dimension $d$ (Chain dimension and the empty-space convention, Noetherian topological spaces via ACC on opens or DCC on closed subsets, Free abelian group on a set); for finite type $X$ one writes $Z_{\ast }(X)={⨁}_{d\in \mathbb{Z}}{Z}_d(X)$. The support $\mathrm{\mid }\alpha \mathrm{\mid }$ of a cycle $\alpha ={\sum }_i{n}_i[V_i]$ is the union of the $V_i$ with $n_i\mathrm{\ne }0$, a closed subset of $X$; a cycle is effective if all $n_i\ge 0$.

Conventions. (i) A cycle class may be represented by a locally finite sum in the situations below; finite sums suffice for the schemes of finite type over a field used in this pair. (ii) All integral closed subschemes are taken with the reduced structure, and $[V]$ denotes the associated basis element; the fundamental cycle of an integral $X$ of dimension $n$ is $[X]\in Z_n(X)$. (iii) For $X$ equidimensional of pure dimension $n$ one writes $A^p(X):=A_{n-p}(X)$ for the codimension-$p$ Chow group once $A_{\ast }$ is available (def-chow-group-of-cycles-mod-rational-equivalence). (iv) No choice is used in this definition beyond the free abelian group on the set of subvarieties (Free abelian group on a set). For a locally finite type scheme not assumed quasi-compact, the locally finite cycle group consists instead of formal sums whose component supports meet each quasi-compact open in only finitely many terms. Its restriction to every such open is a finite cycle. The displayed free direct sum is the finite-cycle group; throughout assertions on merely locally finite type schemes use the locally finite group. For finite type schemes the two groups coincide. For general locally finite type $X$, $Z_{\ast }(X)$ denotes locally finite sums over all dimensions; it need not be the direct sum of the $Z_d(X)$ when component dimensions are unbounded. Every fixed-dimensional operation below is defined degreewise.