Tensor structures in the theory of modulus presheaves with transfers

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.