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K-theoretic Poitou-Tate duality in higher dimensions: proper case

Oliver Braunling

Abstract

We generalize Blumberg-Mandell's K-theoretic Poitou-Tate duality to arithmetic schemes of arbitrary dimension, smooth and proper over S-integers. As in our earlier papers on the subject, we discuss how to model the compactly supported side via the K-theory of locally compact modules.

K-theoretic Poitou-Tate duality in higher dimensions: proper case

Abstract

We generalize Blumberg-Mandell's K-theoretic Poitou-Tate duality to arithmetic schemes of arbitrary dimension, smooth and proper over S-integers. As in our earlier papers on the subject, we discuss how to model the compactly supported side via the K-theory of locally compact modules.

Paper Structure

This paper contains 23 sections, 40 theorems, 197 equations.

Table of Contents

  1. Setup
  2. Running Conventions
  3. Arguments on the motivic side
  4. Generalities
  5. Locally compact modules over rings of integers
  6. Symmetric monoidal structure over the base $R$
  7. Vector modules
  8. Definition via Lurie tensor product
  9. Arguments on the descent side
  10. Open $!$-pushforward for rings of integers
  11. Duals and lifting the spectral trace morphism
  12. Definition via co-descent
  13. Definition
  14. Rings spanned by units
  15. A vanishing result for $\operatorname*{Ext}^{1}$-classes
  16. ...and 8 more sections

Key Result

Theorem A

Let $S$ be finite such that $\frac{1}{p}\in\mathcal{O}_{S}$. Suppose $\pi\colon\mathcal{X\rightarrow O}_{S}$ is a smooth proper arithmetic scheme. Define the Lurie tensor product Then there is a $!$-pushforward exact functor and a natural pairing as in Eq. lint4,

Theorems & Definitions (108)

  • Theorem A
  • Corollary
  • Theorem B
  • Remark 1.1: MR3607274
  • Remark 1.2
  • Example 1.3
  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Lemma 2.4
  • ...and 98 more