Nori $$1$$ 1 -motives

Abstract

Abstract. Let EHM be Nori's category of effective homological mixed motives. In this paper, we consider the thick abelian subcategory EHM 1 ⊂ EHM generated by the i-th relative homology of pairs of varieties for i ∈ {0, 1}. We show that EHM 1 is naturally equivalent to the abelian category t M 1 of 1-motives with torsion; this is our main theorem. Along the way, we obtain several interesting results. Firstly, we realize t M 1 as the universal abelian category obtained, using Nori's formalism, from the Betti representation of an explicit diagram of curves. Secondly, we obtain a conceptual proof of a theorem of Vologodsky on realizations of 1-motives. Thirdly, we verify a conjecture of Deligne on extensions of 1-motives in the category of mixed realizations for those extensions that are effective in Nori's sense.

Nori Motives of Curves With Modulus and Laumon 1-motives

“…Theorem 1 generalizes the equivalence between (a) and (b) recalled in §1.2 and proved by J. Ayoub and L. Barbieri-Viale in [1,Theorem 5.2]. Note that we do not provide any definition for a non homotopy invariant analog of the full category of Nori's motives of varieties (of arbitrary dimension) with modulus.…”

“…Such a description is not possible integrally for the extension of the theory of 1-motives introduced by G. Laumon in [14] and studied in [4,3,15,20]. Indeed the category of Laumon 1-motives with torsion t M a 1 of [4] contains the category of infinitesimal formal k-groups as a full subcategory 1 . In particular not all Hom groups in t M a 1 are finitely generated Abelian groups and therefore there cannot exist a quiver D and a representation T : D → mod(Z) such that t M a 1 is equivalent to comod(C T ).…”

“…Note that we do not provide any definition for a non homotopy invariant analog of the full category of Nori's motives of varieties (of arbitrary dimension) with modulus. Moreover in [1] the main theorems are valid over any field of characteristic zero embedded into the complex numbers and admit also integral coefficient variants. Here we are not able to provide such generality.…”

The filtered Ogus realisation of motives

“…Let j FOg : P 2 FOg (X)(1) → End FOg (H 1 FOg (A)) 4 The compatiblity of the crystalline and de Rham cycle class has been generalized to the rigid setting in [11,13].…”

Syntactic categories for Nori motives

“…J. Ayoub and L. Barbieri-Viale have proved in [4] that the abelian category constructed in this way from the restriction of the diagram D to triples (X, Y, i) with i at most 1 is equivalent to the abelian category of Deligne 1-motives with torsion (for i = 0 to Artin motives).…”