Torsion and divisibility for reciprocity sheaves and 0-cycles with modulus

Abstract

The notion of modulus is a striking feature of Rosenlicht-Serre's theory of generalized Jacobian varieties of curves. It was carried over to algebraic cycles on general varieties by Bloch-Esnault, Park, Rülling, Krishna-Levine. Recently, Kerz-Saito introduced a notion of Chow group of 0-cycles with modulus in connection with geometric class field theory with wild ramification for varieties over finite fields. We study the non-homotopy invariant part of the Chow group of 0-cycles with modulus and show their torsion and divisibility properties.Modulus is being brought to sheaf theory by Kahn-Saito-Yamazaki in their attempt to construct a generalization of Voevodsky-Suslin-Friedlander's theory of homotopy invariant presheaves with transfers. We prove parallel results about torsion and divisibility properties for them.

Torsion 0-cycles with modulus on affine varieties

“…In fact, in the proofs we essentially reproduce and adapt Levine's arguments to our context. This is a continuation of our study of torsion 0-cycles in [3].…”

“…Let T X|D denote the subgroup of the group of degree zero 0-cycle CH 0 (X|D) 0 that is generated by the images of torsion 0-cycles on proper smooth curves mapping to X (and having image not contained in D). In [3], Section 2, we proved that the operation n −1 as defined in 3.2.1 satisfies two important properties (the second implied by the first) i) given two smooth curves C 1 and C 2 in X such that α ∈ C 1 ∩ C 2 , one has the equality…”

“…Replacing X with X affine, we have to modify slightly the arguments using the same strategy as in the proof of 3.2, making use of the nice compactification provided by Lemma 3.1. Again, we closely follow [15] (or [14], as we did in [3]) to prove the following Lemma. Lemma 3.7.…”

Cube invariance of higher Chow groups with modulus

“…This result is closely related to [3,Theorem 2.7], which states that if X is proper over k, then the "unipotent part" (=non-homotopcally invariant part) of CH 0 (X ) = CH dim X (X , 0) is p-primary torsion when char(k) = p > 0 and that it is divisible when char(k) = 0. The group CH 0 (X ) = CH 0 (X|D) is called the Chow group of zero cycles with modulus, and it is used by M. Kerz and S. Saito in [11] to give a cycle-theoretic description of theétale fundamental group of the interior X \ |D|.…”

Rigidity for relative 0-cycles

“…This statement can be made precise in the context of the theory of motives with modulus, as recently developed by Kahn Saito and Yamazaki. See [20,Corollary 4.2.6 and Remark 4.2.7 b)] (using some results in [3]). We therefore conjecture that the cycle class map…”