Tensor structures in the theory of modulus presheaves with transfers

TL;DR

This paper extends the tensor product concept for sheaves with transfers to reciprocity sheaves using modulus presheaves, providing new motivic descriptions of differential forms and infinitesimal neighborhoods.

Contribution

It generalizes Voevodsky's tensor product to reciprocity sheaves and computes new motivic presentations of key algebraic structures.

Findings

New tensor product construction for reciprocity sheaves

Motivic presentations of absolute Kähler differentials

Descriptions of infinitesimal neighborhoods

Abstract

The tensor product of $\mathbb{A}^1$-invariant sheaves with transfers introduced by Voevodsky is generalized to reciprocity sheaves via the theory of modulus presheaves with transfers. We prove several general properties of this construction and compute it in some cases. In particular we obtain new (motivic) presentations of the absolute K\"ahler differentials and the first infinitesimal neighborhood of the diagonal.