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Reducing an OG-lattice modulo the maximal ideal gives a finite-dimensional kG-module

Statement

If $L$ is an $O G$ -lattice, then $\overline{L} = L / m L$ is a finite-dimensional $k G$ -module.

Facts & Assumptions

Given: A splitting $p$ -modular system $(K, O, k)$ for a finite group $G$ and an $O G$ -lattice $L$ .

[F1]

An $O G$ -lattice is finite free over $O$ , and its reduction modulo $m$ is $L / m L ≅ k \otimes_{O} L$ with induced $G$ -action (An OG-lattice is a finite free module over the valuation ring with G-action, and reduction modulo the maximal ideal produces a kG-module).

Proof

technique · direct

1.1

By [F1], choose an $O$ -basis of $L$ of size $r < \infty$ . Tensoring with $k = O / m$ sends that basis to a $k$ -basis of $k \otimes_{O} L ≅ L / m L$ , so $\dim_{k} \overline{L} = r$ .

F1givenchoosealgebra

2.1

The $G$ -action on $L$ is $O$ -linear, so $m L$ is $G$ -stable and the quotient action on $L / m L$ is well defined. Thus $\overline{L}$ is a finite-dimensional $k G$ -module.

F1step 1.1algebra∎

Depends on

An OG-lattice is a finite free module over the valuation ring with G-action, and reduction modulo the maximal ideal produces a kG-module

Used by

Reducing a standard integral lattice for S3 modulo 3 produces a reducible kS3-module Example

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Peter Webb, A Course in Finite Group Representation Theory (23 Feb 2016 draft) (standard reference, not scraped)