Hiroyasu Miyazaki (Riken): On Nisnevich topology with modulus

In Voevodsky’s theory of mixed motives, the Nisnevich topology plays an important role. He used the abelian category of sheaves with transfers with respect to this topology as a fundamental building block to construct his category of motives. Recently, an attempt to generalize this theory to a “non-homotopy invariant” version, called the theory of motives with modulus, was initiated by Kahn-Saito-Yamazaki. The idea is to construct the theory of motives which takes into account the behavior of motives in the neighborhood of the boundary (=modulus) of compactifications of varieties. To follow Voevodsky’s argument, we need a reasonable notion of topology which respects such boundary information. In this talk, I will introduce a topology on the category of proper modulus pairs. A proper modulus pair is just a pair of a proper variety and an effective Cartier divisor on it. The divisor can be regarded as the boundary information. This topology enables us to define the category of motives with modulus. Moreover, I will explain that this new topology is compatible with the usual Nisnevich topology, which implies the existence of an embedding of Voevodsky’s category into the category of motives with modulus. This talk is based on the joint work with Bruno Kahn, Shuji Saito and Takao Yamazaki.