K-groups of reciprocity functors

TL;DR

This paper introduces reciprocity functors, constructs their K-groups, and computes these groups in various cases, advancing the understanding of reciprocity sheaves and related algebraic structures.

Contribution

It defines reciprocity functors, constructs their K-groups, and provides computations, bridging concepts like algebraic groups and sheaves with transfers.

Findings

K-groups of reciprocity functors are themselves reciprocity functors.

Explicit computations of K-groups in several cases.

Reciprocity functors generalize algebraic groups and sheaves with transfers.

Abstract

In this work we introduce reciprocity functors, construct the associated K-group of a family of reciprocity functors, which itself is a reciprocity functor, and compute it in several different cases. It may be seen as a first attempt to get close to the notion of reciprocity sheaves imagined by B. Kahn. Commutative algebraic groups, homotopy invariant Nisnevich sheaves with transfers, cycle modules or K\"ahler differentials are examples of reciprocity functors. As commutative algebraic groups do, reciprocity functors are equipped with symbols and satisfy a reciprocity law for curves.