Milnor K-theory of p-adic rings

Morten Lüders; Matthew Morrow

doi:10.1515/crelle-2022-0079

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Milnor K-theory of p-adic rings

Morten Lüders and Matthew Morrow

Published/Copyright: December 9, 2022

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2023 Issue 796

Abstract

We study the mod p r Milnor K-groups of p-adically complete and p-henselian rings, establishing in particular a Nesterenko–Suslin-style description in terms of the Milnor range of syntomic cohomology. In the case of smooth schemes over complete discrete valuation rings we prove the mod p r Gersten conjecture for Milnor K-theory locally in the Nisnevich topology. In characteristic p we show that the Bloch–Kato–Gabber theorem remains true for valuation rings, and for regular formal schemes in a pro sense.

Funding statement: The first author was supported by the DFG Research Fellowship LU 2418/1-1. The second author was partly supported by the ANR JCJC project Périodes en Géométrie Arithmétique et Motivique (ANR-18-CE40-0017).

A A Gersten injectivity result of Gabber

The following injectivity result of Gabber was required in the proof of Theorem 2.17:

Theorem A.1 (Gabber).

Let V be a mixed characteristic discrete valuation ring and R a local, ind-smooth p-henselian V-algebra; let R p ⁢ R h be the henselisation of the discrete valuation ring R p ⁢ R , and F := R p ⁢ R h ⁢ [ 1 p ] its field of fractions. Let F be a torsion étale sheaf on Spec ⁡ R ⁢ [ 1 p ] which is pulled back from Spec ⁡ V ⁢ [ 1 p ] . Then the canonical map

H ét n ⁢ ( R ⁢ [ 1 p ] , ℱ ) → H ét n ⁢ ( F , ℱ )

is injective for all n ≥ 0 .

Proof.

Since both sides commute with filtered colimits in R, we may assume that R is the henselisation along 𝔭 ⁢ S 𝔮 of S 𝔮 , where S is a smooth V-algebra and 𝔮 ⊆ S is some prime ideal containing 𝔭 ⁢ S ; note that then R 𝔭 ⁢ R h = S 𝔭 ⁢ S h , where the latter denotes the henselisation of the discrete valuation ring S 𝔭 ⁢ S .

The beginning of Gabber’s Gersten resolution [16, equation ( ∗ )], at the point 𝔮 of Spec ⁡ S (Gabber’s scheme M), asserts that the map

H ét n ⁢ ( Spec ⁡ S 𝔮 / 𝔭 ⁢ S 𝔮 , i * ⁢ R ⁢ j * ⁢ ℱ ) → H ét n ⁢ ( Spec ⁡ S 𝔭 ⁢ A / 𝔭 ⁢ S 𝔭 ⁢ A , i * ⁢ R ⁢ j * ⁢ ℱ )

is injective, where i , j are the usual closed and open inclusions

Spec ⁡ S / 𝔭 ⁢ S ⁢ ↪ 𝑖 ⁢ Spec ⁡ S ← 𝑗 Spec ⁡ S ⁢ [ 1 p ] ,

and we suppress the additional pullbacks along

Spec ⁡ S 𝔭 ⁢ R / 𝔭 ⁢ S 𝔭 ⁢ S → Spec ⁡ S 𝔮 / 𝔭 ⁢ S 𝔮 → Spec ⁡ S / 𝔭 ⁢ S

from the notation.

Noting that R / 𝔭 ⁢ R = S 𝔮 / 𝔭 ⁢ S 𝔮 , Gabber’s affine analogue of the proper base change theorem [15] implies that the canonical map

H ét n ⁢ ( Spec ⁡ R ⁢ [ 1 p ] , ℱ ) = H ét n ⁢ ( Spec ⁡ R , R ⁢ j * ⁢ ℱ ) → H ét n ⁢ ( Spec ⁡ S 𝔮 / 𝔭 ⁢ S 𝔮 , i * ⁢ R ⁢ j * ⁢ ℱ )

is an isomorphism. The analogous assertion is equally true for the henselian surjection

R 𝔭 ⁢ R h → R 𝔭 ⁢ A / 𝔭 ⁢ R 𝔭 ⁢ A = S 𝔭 ⁢ A / 𝔭 ⁢ S 𝔭 ⁢ A ,

thereby completing the proof. ∎

Acknowledgements

We thank Bhargav Bhatt, Dustin Clausen, and Akhil Mathew for discussions and Shuji Saito for related correspondence. We are grateful to Elden Elmanto for comments on the paper, and to the anonymous referees for various suggestions and improvements.

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Received: 2021-09-03

Revised: 2022-09-16

Published Online: 2022-12-09

Published in Print: 2023-03-01

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