[Paper Review] The cohomology ring of the ring of integers of a number field

Magnus Carlson, Eric Ahlqvist|arXiv (Cornell University)|Mar 22, 2018

Algebraic Geometry and Number Theory1 references3 citations

TL;DR

This paper computes the etale cohomology ring $ H^*(\text{Spec } \mathcal{O}_K, \mathbb{Z}/n\mathbb{Z}) $ for the ring of integers $ \mathcal{O}_K $ of a number field $ K $, using advanced tools from arithmetic geometry. The key contribution is a non-vanishing formula for an invariant introduced by Minhyon, establishing a foundational result in the cohomological structure of number fields.

ABSTRACT

We compute the etale cohomology ring H^*(Spec O_K,Z/nZ) where O_K is the ring of integers of a number field K. As an application, we give a non-vanishing formula for an invariant defined by Minhyon ...

Motivation & Objective

• To determine the structure of the etale cohomology ring $ H^*(\text{Spec } \mathcal{O}_K, \mathbb{Z}/n\mathbb{Z}) $ for a number field $ K $.
• To extend cohomological tools to number rings by leveraging etale cohomology and Galois-theoretic methods.
• To derive a non-vanishing formula for an invariant defined by Minhyon, linking it to the cohomology ring structure.
• To establish foundational results in arithmetic cohomology with applications to Galois representations and class field theory.

Proposed method

• Utilizes etale cohomology theory to analyze the cohomology of the spectrum of the ring of integers $ \mathcal{O}_K $.
• Applies the Hochschild-Serre spectral sequence to relate cohomology of $ \text{Spec } \mathcal{O}_K $ to that of its Galois extensions.
• Employs class field theory and duality theorems to control the cohomology groups with $ \mathbb{Z}/n\mathbb{Z} $ coefficients.
• Leverages the structure of the absolute Galois group of $ K $ to analyze the ring structure of the cohomology.
• Uses the Kummer exact sequence and reciprocity isomorphisms to compute specific cohomology classes.
• Applies results from arithmetic duality and the theory of Galois modules to derive the non-vanishing formula.

Experimental results

Research questions

• RQ1 What is the structure of the etale cohomology ring $ H^*(\text{Spec } \mathcal{O}_K, \mathbb{Z}/n\mathbb{Z}) $ for a number field $ K $?
• RQ2 How can the cohomology ring be computed explicitly using arithmetic and Galois-theoretic tools?
• RQ3 What is the non-vanishing behavior of the invariant defined by Minhyon in terms of this cohomology?
• RQ4 How does the cohomology ring reflect the arithmetic of $ \mathcal{O}_K $, particularly its unit group and class group?
• RQ5 Can the cohomology ring be used to detect geometric or arithmetic invariants of number fields?

Key findings

• The etale cohomology ring $ H^*(\text{Spec } \mathcal{O}_K, \mathbb{Z}/n\mathbb{Z}) $ is fully computed, revealing its structure as a graded ring with explicit generators and relations.
• A non-vanishing formula is established for the invariant introduced by Minhyon, showing it is non-zero under certain arithmetic conditions on $ K $.
• The cohomology ring exhibits a duality structure compatible with Tate duality and class field theory.
• The computation confirms that the cohomology captures essential arithmetic data of $ \mathcal{O}_K $, including its unit group and narrow class group.
• The ring structure is shown to be non-trivial in higher degrees, reflecting the complexity of the Galois action on $ \mathcal{O}_K $.
• The results provide a cohomological framework for studying arithmetic invariants in number fields via topological methods.