A remark on the tensor product of SC-reciprocity sheaves

TL;DR

This paper characterizes the tensor product of SC-reciprocity sheaves using $K$-groups, explores the tensor product of additive and multiplicative groups, and relates these to Chow groups and differential forms.

Contribution

It provides a new description of tensor products of SC-reciprocity sheaves via $K$-groups and analyzes specific tensor products involving $G_a$ and $G_m$.

Findings

Tensor product of SC-reciprocity sheaves described via $K$-groups.

Tensor product of $G_a$ and $G_m$ is isomorphic to the sheaf of Kähler differentials.

Application to Chow groups of 0-cycles with modulus.

Abstract

We give a description of the tensor product of SC-reciprocity presheaves with transfers in terms of $K$-group of geometric type, and we study a structure of the tensor product of $\mathbb{G}_a$ and $\mathbb{G}_a$. We apply our description to give a description of Chow group of 0-cycles with modulus of products of curves. We also show that the tensor product of $\mathbb{G}_a$ and $\mathbb{G}_m$ is isomorphic to the sheaf $\Omega^1$ of K\"ahler differential forms as reciprocity sheaves over characteristic zero.