V. Voevodsky in [Triangulated categories of motives over a field. Cycles, transfers, and motivic homology theories. Princeton, NJ: Princeton University Press. Ann. Math. Stud. 143, 188–238 (2000; Zbl 1019.14009)] has constructed the triangulated category of geometric mixed motives \(DM_{gm}(k)\) over a perfect field \(k\). The main purpose of the theory of generic motives developed by the author is to show that every mixed motive defines canonically a cycle module in the sense of M. Rost [Doc. Math., J. DMV 1, 319–393 (1996; Zbl 0864.14002)]. Let \({\mathcal E}_k\) be the category of function fields, i.e of finite extensions of the field \(k\). For a function field \(E\) the generic motive of \(EM_{gm}(E)\) is the pro-object given by all smooth schemes whose function field is \(E\). The class of all models of \(E\) admits a final object given by all \(\sup k\)-algebras of finite type \(A\) of \(E\) such that \(\text{Spec\,}A\) is smooth. Let’s denote it by \(M^{sm}(E/k)\). If \(E\) is a a function field and \(n\) an integer one defines \(M_{gm}(E)\{n\}\) as the pro-object in \(DM_{gm}(k)\), \[ M_{gm(E)}\{n\}= \lim_{A\in M^{sm}(E/k)} M_{gm}(\text{Spec\,}A)\{n\}, \] where, for a motivic complex \(M\), \(M\{n\}= \mathbb{Z}(n)[n]\otimes M\).
Another interpretation of generic motives is to define the evaluation of a Nisnevich sheaf with transfer \(F\), invariant by homotopy, on a function field \(E\): if \(X\) is a model for \(E\) this evaluation is the fiber of the sheaf \(F\) at the generic point of \(X_{Nis}\). The evaluation of \(F\) on a generic motive \(M_{gm}(E)\{n\}\) for \(n> 0\) yields an exact functor which commutes with arbitrary direct sums, hence a fibered functor.
Then the main consequence of the theory developed by the author is the following result
Theorem 1. Let \(\phi:DM^{op}_{gm}\to{\mathcal A}\) be an additive functor, where \({\mathcal A}\) denotes the category of Abelian groups. For all couples \((i,r)\), with \(i,r\in\mathbb{Z}\), and all function fields \(E\) define \[ K^{\phi,r}_i(E)= \lim_{A\in M^{sm}(E/k)} \phi(M_{gm}(\text{Spec\,}A)\{-i\}(-r). \] Then, for all \(r\in\mathbb{Z}\), \(K^{\phi,r}_*\) has the canonical structure of a cycle module.
In the case \(\phi\) is the functor induced by a homotopic sheaf \(F\) one gets a functor from homotopic sheaves to cycle modules. The author announces that, according to a further result of his, this functor induces an equivalence of categories between the stable category of homotopic sheaves and the category of cycle modules.
If a cohomological theory \(H^*:{\mathcal L}^{op}_k\to{\mathcal A}\), where \({\mathcal L}\) is the additive category of smooth schemes over \(k\) with finite correspondences, extends to a realization functor \(R_H: DM^{op}_{gm}(k)\to{\mathcal A}\) in such a way that \(H^i(X)= R_H(M_{gm}(X)[-i])\), then, according to Theorem 1, it induces a cycle module. Due to a result by A. Huber this is in particular the case for De Rham cohomology: the author announces that the above result also yields the existence of cycle module structure for the rigid cohomology of function fields.