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Local duality theorems for commutative algebraic groups

Cristian D. Gonzalez-Aviles

Abstract

If k is an arbitrary field, we construct a category of k-1-motives in which every commutative algebraic k-group G has a dual object $G^{\vee}$. When k is a local field of arbitrary characteristic, we establish Pontryagin duality theorems that relate the fppf cohomology groups of G to the hypercohomology groups of the k-1-motive $G^{\vee}$. We also obtain a duality theorem for the second cohomology group of an arbitrary k-1-motive. These results have applications (to be discussed elsewhere) to certain extensions of Lichtenbaum-van Hamel duality to a class of non-smooth proper k-varieties.

Local duality theorems for commutative algebraic groups

Abstract

If k is an arbitrary field, we construct a category of k-1-motives in which every commutative algebraic k-group G has a dual object . When k is a local field of arbitrary characteristic, we establish Pontryagin duality theorems that relate the fppf cohomology groups of G to the hypercohomology groups of the k-1-motive . We also obtain a duality theorem for the second cohomology group of an arbitrary k-1-motive. These results have applications (to be discussed elsewhere) to certain extensions of Lichtenbaum-van Hamel duality to a class of non-smooth proper k-varieties.
Paper Structure (7 sections, 32 theorems, 74 equations)

This paper contains 7 sections, 32 theorems, 74 equations.

Table of Contents

  1. Preliminaries
  2. Generalities
  3. Commutative algebraic groups
  4. Topological preliminaries
  5. The Nagao topologies
  6. The category of $k$-$1$-motives
  7. Proof of the main theorem

Key Result

Theorem 1

Let $k$ be a non-archimedean local field, let $G$ be a commutative algebraic $k$-group given as an extension $\mathcal{E}=\mathcal{E}(L,\iota, G,\pi, A)$ with associated $k$-$1$-motive $(G,\mathcal{E})=(0,0,\mathcal{E}(L,\iota, G,\pi, A))$ and let $G^{\vee}=[ L^{\! D}\!\overset{\!v}{\to}\! A^{t}]$ b

Theorems & Definitions (77)

  • Theorem 1
  • Theorem 2
  • Remark 1.1
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • proof
  • Definition 1.4
  • Example 1.5
  • Lemma 1.6
  • ...and 67 more