Bloch’s formula for 0-cycles with modulus and higher-dimensional class field theory

Abstract

We prove Bloch’s formula for the Chow group of 0-cycles with modulus on a smooth quasi-projective surface over a field. We use this formula to give a simple proof of the rank one case of a conjecture of Deligne and Drinfeld on lisse Q ¯ ℓ \overline {\mathbb {Q}}_{\ell } -sheaves. This was originally solved by Kerz and Saito in characteristic ≠ 2 \neq 2 .

Suslin homology via cycles with modulus and applications

“…7.5]). The latter group coincides with the fundamental group with modulus π ab 1 (X, D) used in [8] and [29].…”

“…Combining (1.4) with the main results of [8], [29] and [20,Thm. 1.4], we obtain the following reciprocity theorem for Russell's relative Chow group.…”

“…Kerz-Saito. The goal of geometric class field theory is to describe the abelian fundamental group of a variety (say, defined over a finite field) in terms of certain groups of algebraic cycles. The modern perspective on the problem is given by the work of Kerz and Saito [29] (see also [8]), where the class groups used to describe the abelian fundamental group of a variety X with bounded ramification along a divisor D is precisely the Chow group of zero-cycles with modulus.…”

“…To prove Theorem 1.1, we shall use the following fundamental exact sequence (see [5,Thm. 1.9] when k is perfect, [8,Thm. 2.11] if dim(X) = 2 and [22, Thm.…”

“…3.13]). It is shown in [8,Thm. 7.7] (based on the original result due to Levine) that cyc S X is injective and its image is SK 0 (S X ).…”