In algebraic geometry, the Chow groups (named after Wei-Liang Chow by Claude Chevalley (1958)) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties (so-called algebraic cycles) in a similar way to how simplicial or cellular homology groups are formed out of subcomplexes. When the variety is smooth, the Chow groups can be interpreted as cohomology groups (compare Poincaré duality) and have a multiplication called the intersection product. The Chow groups carry rich information about an algebraic variety, and they are correspondingly hard to compute in general.

For what follows, define a variety over a field ${\displaystyle k}$ to be an integral scheme of finite type over ${\displaystyle k}$. For any scheme ${\displaystyle X}$ of finite type over ${\displaystyle k}$, an algebraic cycle on ${\displaystyle X}$ means a finite linear combination of subvarieties of ${\displaystyle X}$ with integer coefficients. (Here and below, subvarieties are understood to be closed in ${\displaystyle X}$, unless stated otherwise.) For a natural number ${\displaystyle i}$, the group ${\displaystyle Z_{i}(X)}$ of ${\displaystyle i}$-dimensional cycles (or ${\displaystyle i}$-cycles, for short) on ${\displaystyle X}$ is the free abelian group on the set of ${\displaystyle i}$-dimensional subvarieties of ${\displaystyle X}$.

For a variety ${\displaystyle W}$ of dimension ${\displaystyle i+1}$ and any rational function ${\displaystyle f}$ on ${\displaystyle W}$ which is not identically zero, the divisor of ${\displaystyle f}$ is the ${\displaystyle i}$-cycle

where the sum runs over all ${\displaystyle i}$-dimensional subvarieties ${\displaystyle Z}$ of ${\displaystyle W}$ and the integer ${\displaystyle \operatorname {ord} _{Z}(f)}$ denotes the order of vanishing of ${\displaystyle f}$ along ${\displaystyle Z}$. (Thus ${\displaystyle \operatorname {ord} _{Z}(f)}$ is negative if ${\displaystyle f}$ has a pole along ${\displaystyle Z}$.) The definition of the order of vanishing requires some care for ${\displaystyle W}$ singular.

For a scheme ${\displaystyle X}$ of finite type over ${\displaystyle k}$, the group of ${\displaystyle i}$-cycles rationally equivalent to zero is the subgroup of ${\displaystyle Z_{i}(X)}$ generated by the cycles ${\displaystyle (f)}$ for all ${\displaystyle (i+1)}$-dimensional subvarieties ${\displaystyle W}$ of ${\displaystyle X}$ and all nonzero rational functions ${\displaystyle f}$ on ${\displaystyle W}$. The Chow group ${\displaystyle CH_{i}(X)}$ of ${\displaystyle i}$-dimensional cycles on ${\displaystyle X}$ is the quotient group of ${\displaystyle Z_{i}(X)}$ by the subgroup of cycles rationally equivalent to zero. Sometimes one writes ${\displaystyle [Z]}$ for the class of a subvariety ${\displaystyle Z}$ in the Chow group, and if two subvarieties ${\displaystyle Z}$ and ${\displaystyle W}$ have ${\displaystyle [Z]=[W]}$, then ${\displaystyle Z}$ and ${\displaystyle W}$ are said to be rationally equivalent.

For example, when ${\displaystyle X}$ is a variety of dimension ${\displaystyle n}$, the Chow group ${\displaystyle CH_{n-1}(X)}$ is the divisor class group of ${\displaystyle X}$. When ${\displaystyle X}$ is smooth over ${\displaystyle k}$ (or more generally, a locally Noetherian normal factorial scheme ), this is isomorphic to the Picard group of line bundles on ${\displaystyle X}$.

Rationally equivalent cycles defined by hypersurfaces are easy to construct on projective space because they can all be constructed as the vanishing loci of the same vector bundle. For example, given two homogeneous polynomials of degree ${\displaystyle d}$, so ${\displaystyle f,g\in H^{0}(\mathbb {P} ^{n},{\mathcal {O}}(d))}$, we can construct a family of hypersurfaces defined as the vanishing locus of ${\displaystyle sf+tg}$. Schematically, this can be constructed as

using the projection ${\displaystyle \pi _{1}:X\to \mathbb {P} ^{1}}$ we can see the fiber over a point ${\displaystyle [s_{0}:t_{0}]}$ is the projective hypersurface defined by ${\displaystyle s_{0}f+t_{0}g}$. This can be used to show that the cycle class of every hypersurface of degree ${\displaystyle d}$ is rationally equivalent to ${\displaystyle d[\mathbb {P} ^{n-1}]}$, since ${\displaystyle sf+tx_{0}^{d}}$ can be used to establish a rational equivalence. Notice that the locus of ${\displaystyle x_{0}^{d}=0}$ is ${\displaystyle \mathbb {P} ^{n-1}}$ and it has multiplicity ${\displaystyle d}$, which is the coefficient of its cycle class.