Mathematics Seminar Date: Thursday, 07 September 2023 Time: 2:00 PM Venue: Lecture Hall 6 On Gerstentype conjecture for mod p etale motivic cohomology in mixed characteristic (0, p) Makoto Sakagaito IIT Gandhinagar. 070923 Abstract Let X be a smooth scheme over the spectrum of a regular local ring A. Let Z(n)^X be Bloch’s cycle complex for etale topology and Z/m(n)^X := Z(n)^X ⊗Z/mZ. Then GeisserLevine proved that Z/m(n)^X is quasiisomorphic to a shifted logarithmic de RhamWitt sheaf W_rΩ^n_{X, \mathrm{log}}[−n]) of X, in the case where A is a field of positive characteristic p > 0 and m = p^r. Moreover, GeisserLevine also proved that Z/m(n) is quasiisomorphic to the sheaf μ_m of mth roots of unity, in the case where A is a field and m is prime to the characteristic of A. As an analogy of Gersten’s conjecture for algebraic Ktheory, GrosSuwa and Bloch Ogus proved that there is an exact sequence for etale hypercohomology of a local ring O_{X,x} of X at a point x with values in Z/m(n) (this etale hypercohomology is called mod m etale motivic cohomology) in the above cases. In this talk, we prove that such a Gerstentype conjecture holds for mod p etale ́ motivic cohomology of the henselization of a local ring O_{X,x} in the case where A is a discrete valuation ring of mixed characteristic (0, p) and A contains pth roots of unity.
