Motives and Deligne’s Conjecture, Part 2: Examples

We will see that this illustrates a case of Deligne’s conjecture. We’ll start with the following simple task: describe Artin motives (“motives of dimension 0”) over a number field K. We will see that a reasonable category of dimension-0 motives is equivalent to a category of finite-dimensional Galois representations. We will be interested in the determinant of the period matrix of such objects. We construct the category of Artin motives over K as follows. Consider first the category C whose objects are schemes X, smooth of dimension 0 over K. Any such X is a finite disjoint union of spectra of fields finite overK. Given two such objects X and Y one may consider correspondences from X to Y , defined as closed subschemes of the product X × Y . Since X × Y is a finite disjoint union of points the study of correspondences is purely combinatorial; in particular we do not need to quotient out by numerical equivalence or any such equivalence, and the entire theory (such as it is) will be developed without appeal to unproven conjectures. In any case, one forms the rational correspondence group in the obvious way, and takes this rational correspondence group to be Hom(X,Y ) in C. Then one forms the category M of Artin motives from this C by formally introducing elements (X, p) for every idempotent p ∈ Hom(X,X). We aim to prove the following result.