# nLab pure motive

A category of pure motives is informally a certain linearization or abelianization of a category of (say smooth projective) algebraic varieties over a field.

Given an adequate equivalence relation for algebraic cycles (the typical 4 choices are the rational, numerical, algebraic and homological equivalences), one constructs a version of Chow rings, or what is axiomatically called a Weyl cohomology theory? for varieties; the treatment of adequate relations typically involves a nontrivial tool, the so called Chow moving lemma?. To define the category of effective pure motives, one starts with the category of smooth varieties over a field $k$, then enlarges the hom-sets between any two objects $X,Y$ by including a linear version of correspondences, which are roughly the Chow cohomology classes of cycles on $X×Y$. This gives an additive category; its Karoubian envelope with formally inverted Lefschetz motive? is the category of effective pure motives over $k$. The Lefschetz motive is inverted in order to make the tensor category of motives rigid.