Motivic sheaf

In mathematics, a motivic sheaf is a motivic-cohomology counterpart of an l-adic sheaf. It was first introduced by Morel and Voevodsky^[1] and was later developed by J. Ayoub,^[2] Deniz-Charles Cisinski, F. Déglise, F. Morel, and others.^[3] For Nori motives, the first construction is due to D. Arapura.^[4] In practice, a motivic sheaf is sometimes used instead of an l-adic sheaf because the former’s cycle-theoretic nature may be important. In the language of Ayoub,^[3]^[2]

ℓ-adic sheaves are a “transcendental” invariant: they have strong finiteness properties, behave well in families, and are relatively computable; but their relationship to algebraic cycles is tenuous (highly conjectural at best). By contrast, motivic cohomology is what Ayoub calls an “algebro-geometric invariant”, which is built directly out of objects of interest in algebraic geometry (e.g., algebraic cycles), but behaves “chaotically”: it does not have good finiteness properties, it varies violently in families, and it is not amenable to computation.

References

^ Vladimir Voevodsky, A1-homotopy theory, Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), no. Extra Vol. I, 1998, pp. 579–604.
^ ^a ^b Joseph Ayoub, A guide to (étale) motivic sheaves, Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. II, Kyung Moon Sa, Seoul, 2014, pp. 1101–1124
^ ^a ^b § 1.4.1. of Feng, Tony; Khan, Adeel A. (2024). "Modularity of higher theta series II: Chow group of the generic fiber". arxiv (arXiv:2403.19711 [math.NT]). arXiv:2403.19711.
^ Donu Arapura, An Abelian category of Motivic sheaves (arXiv:0801.0261)

References

Further reading