A Comparison of Categories of Nori Motivic Sheaves

TL;DR

This work proves that Ivorra--Morel's and Ayoub's constructions of Nori motivic sheaves are canonically equivalent, by leveraging the six-functor formalism and the Tannakian theory of motivic local systems. The authors construct a canonical system of comparison functors and show they induce equivalences at the generic level and over points, culminating in an isomorphism between $Db(\MNori(X))$ and $Dbgeo(X)^{\GmotAy(k)}$ for all $k$-varieties $X$, compatible with the six operations. A Beilinson-type equivalence is established in the equivariant setting, yielding an $\infty$-categorical enhancement of the six-functor formalism for Nori motivic sheaves and new realizations from Voevodsky motives to Nori motives. Additionally, the work proves independence of the complex embedding $\sigma: k \hookrightarrow \C$ in the motivic Galois action and provides a robust framework for motivic local systems and Beilinson-style realizations, with potential applications to Hodge-theoretic variants and further structural understanding of mixed motivic sheaves.

Abstract

We show that two different possible theories of Nori motivic sheaves, introduced by Ivorra--Morel and by Ayoub, respectively, are canonically equivalent. The proof of this result, which exploits the six functor formalism systematically, is based on the Tannakian theory of motivic local systems. As a consequence, we obtain a system of realization functors of Voevodsky motivic sheaves into Nori motivic sheaves compatible with the six operations, previously constructed by Tubach using different methods.

Theorems & Definitions (65)

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• proof : Proof of Proposition \ref{["prop:e_*BtiAlg'"]}
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• proof : Proof of Proposition \ref{['prop:BtiAlg_directsum_embeddings']}
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