We give an explicit formula for the zeroth A 1 -homology sheaf of a smooth proper variety. We also provide a simple proof of a theorem of Kahn-Sujatha which describes hom sets in the birational localization of the category of smooth varieties.

We generalize Kahn-Saito-Yamazaki's theory of reciprocity sheaves over a field to noetherian base schemes. We also prove an analogue of the Hasse-Arf theorem for reciprocity sheaves. As an application, we provide a new construction of the de Rham-Witt complex via reciprocity sheaves.

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We prove that any commutative group scheme separated over a noetherian normal scheme admits a canonical structure of a presheaf with transfers, which is characterized by a simple condition on radicial transfers.