An algebraic cycle on a scheme of finite type over a field is a finite linear combination of integral closed subschemes with integral coefficients . The algebraic cycles form a group of algebraic cycles on which is graded by the dimensions of the cycles. Sometimes (for equidimensional ) one looks at the grading by codimension .
For every pair of cycles , there exists such that is transversal to .
Consider a product in , denote by its projections. Consider cycles and such that and intersect properly. Then implies where denotes the intersection product.
The intersection product, which is associative but only partially defined on , then becomes globally defined on .
Typical choices are rational, algebraic and numerical adequate equivalence relations. The rational is the finest and the numerical is the coarsest nonzero adequate equivalence relation.