We develop a theory of modulus sheaves with transfers, which generalizes Voevodsky's theory of sheaves with transfers. This paper and its sequel are foundational for the theory of motives with modulus, which is developed in [KMSY20]. Comment: 64 pages

We develop a theory of sheaves and cohomology on the category of proper modulus pairs. This complements [KMSY21], where a theory of sheaves and cohomology on the category of non-proper modulus pairs has been developed. Comment: 31 pages

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The higher Chow group with modulus was introduced by Binda-Saito as a common generalization of Bloch's higher Chow group and the additive higher Chow group. In this paper, we study invariance properties of the higher Chow group with modulus. First, we formulate and prove "cube invariance," which generalizes A 1 -homotopy invariance of Bloch's higher Chow group. Next, we introduce the nilpotent higher Chow group with modulus, as an analogue of the nilpotent algebraic K-group, and define a module structure on it over the big Witt ring of the base field. We deduce from the module structure that the higher Chow group with modulus with appropriate coefficients satisfies A 1 -homotopy invariance. We also prove that A 1 -homotopy invariance implies independence from the multiplicity of the modulus divisors.

We study relationships between the Nisnevich topology on smooth schemes and certain Grothendieck topologies on proper and not necessarily proper modulus pairs, which were introduced in previous papers. Our results play an important role in the theory of sheaves with transfers on proper modulus pairs.