Mathematics Seminar
Date: Thursday, 07 September 2023
Time: 2:00 PM
Venue: Lecture Hall 6
On Gersten-type conjecture for mod p etale motivic cohomology in mixed characteristic (0, p)
Makoto Sakagaito
IIT Gandhinagar.
07-09-23

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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 Geisser-Levine proved that Z/m(n)^X is quasi-isomorphic to a shifted logarithmic de Rham-Witt 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, Geisser-Levine also proved that Z/m(n) is quasi-isomorphic to the sheaf μ_m of m-th 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 K-theory, Gros-Suwa 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 Gersten-type 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 p-th roots of unity.