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Partial Bloch--Kato Selmer groups of $B$-pairs as delta functors

Rustam Steingart

TL;DR

The paper addresses extending partial Bloch–Kato Selmer groups from degree one to higher cohomological degrees within the $B$-pair framework by developing a cohomological delta-functor structure. It introduces and analyzes $H^i_{g,J}(W)$ for $B$-pairs on the subcategory of $J$-de Rham objects with non-positive Hodge–Tate weights, using twists by rank-one objects $\chi_D$ to obtain Tate duality and an Euler–Poincaré formula. The main contributions include proving that these groups form a delta-functor, establishing a perfect pairing with dual objects, and connecting to both Galois and analytic cohomology via natural specializations; the work also yields explicit dimension formulas in the $K=\mathbb{Q}_p$ case and ties to $D_{cris}$-theoretic invariants. Altogether, the results provide a comprehensive arithmetic description of higher Selmer-type cohomology for $B$-pairs and extend the analytic perspective to a general $E$-linear setting with a robust duality theory.

Abstract

In this article we revisit the partial Selmer groups introduced by Ding in cohomological degree one. On the subcategory of partially de Rham positive $B$-pairs we extend them to higher cohomological degree and show that the resulting groups form a cohomological delta functor satisfying a variant of the Euler--Poincaré characteristic formula and Tate duality.

Partial Bloch--Kato Selmer groups of $B$-pairs as delta functors

TL;DR

The paper addresses extending partial Bloch–Kato Selmer groups from degree one to higher cohomological degrees within the -pair framework by developing a cohomological delta-functor structure. It introduces and analyzes for -pairs on the subcategory of -de Rham objects with non-positive Hodge–Tate weights, using twists by rank-one objects to obtain Tate duality and an Euler–Poincaré formula. The main contributions include proving that these groups form a delta-functor, establishing a perfect pairing with dual objects, and connecting to both Galois and analytic cohomology via natural specializations; the work also yields explicit dimension formulas in the case and ties to -theoretic invariants. Altogether, the results provide a comprehensive arithmetic description of higher Selmer-type cohomology for -pairs and extend the analytic perspective to a general -linear setting with a robust duality theory.

Abstract

In this article we revisit the partial Selmer groups introduced by Ding in cohomological degree one. On the subcategory of partially de Rham positive -pairs we extend them to higher cohomological degree and show that the resulting groups form a cohomological delta functor satisfying a variant of the Euler--Poincaré characteristic formula and Tate duality.

Paper Structure

This paper contains 11 sections, 14 theorems, 43 equations.

Table of Contents

  1. Preliminaries
  2. $B$-pairs
  3. Cyclotomic $(\varphi,\Gamma)$-modules
  4. Lubin--Tate $(\varphi_L,\Gamma_L)$-modules.
  5. Partial Selmer groups
  6. Rank one $B$-pairs and twists
  7. Ding's partial de Rham $B$-pairs
  8. Comparison to the analytic case
  9. Selmer groups as delta functors
  10. Euler--Poincaré formula
  11. Main results

Key Result

Theorem 1

Let $K/\mathbb{Q}_p$ be finite, let $J\subseteq \Sigma_K= \operatorname{Hom}_{\mathbb{Q}_p}(K,\mathbf{B}_{\mathrm{dR}}^+),$ let $E\subseteq \mathbf{B}_{\mathrm{dR}}^+$ be a normal closure and let $\mathbf{n} \in \mathbb{N}^J.$ Choose $\kappa_{\sigma}>n_\sigma$ for every $\sigma \in J$ and let $\delt

Theorems & Definitions (45)

  • Theorem 1
  • Definition 1.1.1
  • Definition 1.1.2
  • Theorem 1.1.3
  • proof
  • Definition 1.2.1
  • Remark 1.2.2
  • proof
  • Theorem 1.2.3
  • proof
  • ...and 35 more