adequate equivalence relation

An algebraic cycle on a scheme XX of finite type over a field kk is a finite linear combination i=1 rn iZ i\sum_{i=1}^r n_i Z_i of integral closed subschemes Z iXZ_i\subset X with integral coefficients n in_i. The algebraic cycles form a group 𝒵=𝒵 *\mathcal{Z}= \mathcal{Z}_* of algebraic cycles on XX which is graded by the dimensions of the cycles. Sometimes (for equidimensional XX) one looks at the grading by codimension 𝒵 *\mathcal{Z}_*.

Let SmProj k\mathrm{SmProj}_k be the category of smooth projective varieties over kk. A rule giving an equivalence relation 𝒵 *(X)\mathcal{Z}^*(X) for every XX in SmProj k\mathrm{SmProj}_k, and which is compatible with grading is an adequate equivalence relation such that

  1. For every pair of cycles a,b𝒵 *(X)a,b\in \mathcal{Z}^*(X), there exists aaa'\sim a such that aa' is transversal to bb.

  2. Consider a product X×YX\times Y in SmProj k\mathrm{SmProj}_k, denote by p X,p Yp_X,p_Y its projections. Consider cycles a𝒵 *(X)a\in \mathcal{Z}^*(X) and b𝒵 *(X×Y)b\in \mathcal{Z}^*(X\times Y) such that bb and p X *(a)p_X^*(a) intersect properly. Then a0a\sim 0 implies p Y*(p X *(a)b)0p_Y_*(p_X^*(a)\cdot b) \sim 0 where p X *(a)bp_X^*(a)\cdot b denotes the intersection product.

The intersection product, which is associative but only partially defined on 𝒵 *\mathcal{Z}^*, then becomes globally defined on 𝒵 */\mathcal{Z}^*/{\sim}.

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.

Revised on May 28, 2009 21:46:43 by Toby Bartels (