[Paper Review] Lambda actions of rings of integers

James Borger, Bart de Smit|arXiv (Cornell University)|May 24, 2011

Algebraic Geometry and Number Theory2 references6 citations

TL;DR

This paper introduces $Λ_{\mathcal{O}}$-ring structures on torsion-free ${\mathcal{O}}$-algebras for the ring of integers ${\mathcal{O}}$ of a number field $K$, generalizing $\lambda$-ring structures. It establishes a criterion for a finite étale $K$-algebra with such a structure to admit an integral model over ${\mathcal{O}}$, using the Deligne-Ribet monoid and ray class groups, with the key result linking the existence of such models to the action factoring through a ray class group via the monoid $I({\mathcal{O}}) \times G_K$. The result extends earlier work on $\lambda$-rings over $\mathbb{Z}$ to number fields using class field theory.

ABSTRACT

Let O be the ring of integers of a number field K. For an O-algebra R which is torsion free as an O-module we define what we mean by a Lambda_O-ring structure on R. We can determine whether a finite etale K-algebra E with Lambda_O-ring structure has an integral model in terms of a Deligne-Ribet monoid of K. This a commutative monoid whose invertible elements form a ray class group.

Motivation & Objective

• To generalize $\lambda$-ring structures over $\mathbb{Z}$ to rings of integers ${\mathcal{O}}$ of number fields via $\Lambda_{\mathcal{O}}$-ring structures.
• To determine when a finite étale $K$-algebra with a $\Lambda_{\mathcal{O}}$-structure admits an integral model over ${\mathcal{O}}$ that is finite flat as an ${\mathcal{O}}$-module.
• To establish a cohomological criterion for the existence of such integral models using the action of the monoid $I({\mathcal{O}}) \times G_K$ on the set of $K$-algebra homomorphisms to a separable closure.
• To connect the existence of integral models to ray class groups via the Deligne-Ribet monoid, extending the cyclotomic class field theory approach used in the $\mathbb{Z}$-case.

Proposed method

• Define a $\Lambda_{\mathcal{O}}$-structure on a torsion-free ${\mathcal{O}}$-algebra $E$ as a commuting family of Frobenius lifts $\psi_{\mathfrak{p}}$ at each prime ${\mathfrak{p}}$ of ${\mathcal{O}}$.
• Translate the $\Lambda_{\mathcal{O}}$-structure into a monoid map $I({\mathcal{O}}) \to \mathrm{Map}_{G_K}(S,S)$, where $S$ is the set of $K$-algebra homomorphisms $E \to K^{\mathrm{sep}}$.
• Use Grothendieck's Galois theory to identify finite étale $K$-algebras with finite discrete $G_K$-sets, and interpret $\Lambda_{\mathcal{O}}$-structures as actions of $I({\mathcal{O}}) \times G_K$.
• Introduce the Deligne-Ribet monoid $\mathrm{DR}({\mathfrak{f}})$ associated to a cycle ${\mathfrak{f}}$, whose invertible elements form a ray class group.
• Construct a cycle ${\mathfrak{f}}$ from the ramification data of the $G_K$-action on $S$, using the least common multiple of conductors of subactions.
• Prove that the $I({\mathcal{O}}) \times G_K$-action factors through $\mathrm{DR}({\mathfrak{f}})$ if and only if an integral $\Lambda_{\mathcal{O}}$-model exists, using local-global principles and the Artin reciprocity law.

Experimental results

Research questions

• RQ1 When does a finite étale $K$-algebra with a $\Lambda_{\mathcal{O}}$-structure admit a finite flat ${\mathcal{O}}$-algebra integral model?
• RQ2 How can the existence of such an integral model be characterized in terms of Galois-theoretic and arithmetic data?
• RQ3 What is the role of the Deligne-Ribet monoid in classifying $\Lambda_{\mathcal{O}}$-structures on finite étale $K$-algebras?
• RQ4 How does the $\Lambda_{\mathcal{O}}$-structure relate to ray class field theory and the Artin symbol?
• RQ5 Can the criterion for integral models over $\mathcal{O}$ be reduced to a condition on the action of $I({\mathcal{O}}) \times G_K$ factoring through a ray class group?

Key findings

• A finite étale $K$-algebra $E$ with a $\Lambda_{\mathcal{O}}$-structure admits an integral $\Lambda_{\mathcal{O}}$-model if and only if the action of $I({\mathcal{O}}) \times G_K$ on the $K$-algebra homomorphisms $E \to K^{\mathrm{sep}}$ factors through the Deligne-Ribert monoid $\mathrm{DR}({\mathfrak{f}})$ for some cycle ${\mathfrak{f}}$.
• The cycle ${\mathfrak{f}}$ is constructed as the least common multiple of the conductors of the $G_K$-subactions on the $I({\mathcal{O}})$-orbits of $S$, ensuring that the action factors through $\mathrm{Cl}({\mathfrak{f}})$.
• For primes ${\mathfrak{p}}$ unramified in the extension, the action of ${\mathfrak{p}}$ on $S$ coincides with the Artin symbol $[{\ olimits}{\mathfrak{p}}] \in \mathrm{Cl}({\mathfrak{f}})$, linking the $\Lambda_{\mathcal{O}}$-structure to class field theory.
• The existence of a local $\Lambda_{{\mathcal{O}}_{\mathfrak{p}}}$-model at each prime ${\mathfrak{p}}$ is guaranteed by Theorem 1.1 when the inertia group acts trivially on the unramified part and the Frobenius element matches the action on that part.
• The intersection of the local integral models over all ${\mathfrak{p}}$ yields a finite-index ${\mathcal{O}}$-subalgebra $A \subset R$ (where $R$ is the integral closure of ${\mathcal{O}}$ in $E$) that is stable under all $\psi_{\mathfrak{p}}$ and thus forms a global integral $\Lambda_{\mathcal{O}}$-model.
• The criterion is sharp: if the $I({\mathcal{O}}) \times G_K$-action factors through $\mathrm{DR}({\mathfrak{f}})$, then such a model exists, and conversely, the existence of a model implies the action factors through $\mathrm{DR}({\mathfrak{f}})$ for some ${\mathfrak{f}}$.