Rational equivalence is an equivalence relation on algebraic cycles of a fixed dimension
on an algebraic variety 
. A cycle is rationally equivalent to zero when it is a finite integer linear
combination of differences of fibers of cycles on 
, over two points of the projective
line, with the intersections taken properly and with their multiplicities. Two
cycles are rationally equivalent iff their difference is
rationally equivalent to zero. The resulting quotient is the Chow
group 
of 
-dimensional
cycles. For a smooth variety, intersection products
give the graded Chow ring (Stacks Project 2026).
Rational Equivalence
See also
Algebraic Cycle, Algebraic Variety, Bloch Conjecture, Chow Group, Chow Ring, Decomposition of the DiagonalExplore with Wolfram|Alpha
References
The Stacks Project Authors. "Rational Equivalence." §43.8 in The Stacks Project, Tag 0AZG, 2026. https://stacks.math.columbia.edu/tag/0AZG.Cite this as:
Weisstein, Eric W. "Rational Equivalence." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RationalEquivalence.html