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Tame proper base change for discretely ringed adic spaces

Katharina Hübner

TL;DR

The paper proves a proper base change isomorphism for the tame site in discretely ringed adic spaces: for a proper morphism f and a locally closed immersion i: S' -> S, a p-torsion sheaf on X_t satisfies i^* Rf_* F ≅ Rf'_ * i'^* F. The authors first establish a base change theorem for the strongly étale topology and then transfer the result to the tame site by exploiting the agreement of tame and strongly étale cohomology for p-torsion. Central to the argument are Riemann-Zariski points, the maximal RZ open, and a comparison framework with Temkin’s RZ spaces, together with a colimit description of cohomology via X-modifications Y that reduces to scheme-level base change. This provides a robust method to compute tame cohomology on discretely ringed adic spaces and lays groundwork for extending the results to broader classes of adic spaces in future work.

Abstract

We consider a proper morphism $X \to S$ and a locally closed immersion $S' \to S$ of discretely ringed adic spaces and prove proper base change for the tame topology in this setting. More precisely, we show that for an abelian $p$-torsion sheaf ($p = char^+(S)$) on the tame site of $X$ that the base change homomorphism for the derived pushforward along $X \to S$ with the pullback along $S' \to S$ is an isomorphism.

Tame proper base change for discretely ringed adic spaces

TL;DR

The paper proves a proper base change isomorphism for the tame site in discretely ringed adic spaces: for a proper morphism f and a locally closed immersion i: S' -> S, a p-torsion sheaf on X_t satisfies i^* Rf_* F ≅ Rf'_ * i'^* F. The authors first establish a base change theorem for the strongly étale topology and then transfer the result to the tame site by exploiting the agreement of tame and strongly étale cohomology for p-torsion. Central to the argument are Riemann-Zariski points, the maximal RZ open, and a comparison framework with Temkin’s RZ spaces, together with a colimit description of cohomology via X-modifications Y that reduces to scheme-level base change. This provides a robust method to compute tame cohomology on discretely ringed adic spaces and lays groundwork for extending the results to broader classes of adic spaces in future work.

Abstract

We consider a proper morphism and a locally closed immersion of discretely ringed adic spaces and prove proper base change for the tame topology in this setting. More precisely, we show that for an abelian -torsion sheaf () on the tame site of that the base change homomorphism for the derived pushforward along with the pullback along is an isomorphism.

Paper Structure

This paper contains 11 sections, 50 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Setup and notation
  3. Riemann-Zariski points and morphisms
  4. Specialization and generalization of continuous valuations
  5. Riemann-Zariski points
  6. Riemann-Zariski morphisms
  7. The maximal Riemann-Zariski open
  8. Comparison with Riemann-Zariski spaces
  9. Specialization
  10. Proper morphisms of discretely ringed adic spaces
  11. The proper base change theorem

Key Result

Theorem 1.1

Let $f \colon {\mathscr X} \to {\mathscr S}$ be a proper morphism of discretely ringed pseudoadic spaces of specialization characteristic $\mathop{\mathrm{char}}\nolimits^+({\mathscr S}) = p > 0$. We consider the cartesian square \begin{tikzcd} \cX' \ar[r,"\iota_\cX"] \ar[d,"f'"'] & \cX \ar[d,"f" is an isomorphism.

Theorems & Definitions (104)

  • Theorem 1.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Definition 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • ...and 94 more