[[Schur's lemma]]
# Quillen's lemma

Let $A$ be a [[K-monoid]] over $\mathbb{K}$ and $V$ be a [[Simple module|simple]] $A$-[[Module over a unital associative algebra|module]].
If $A$ has a filtration $\{ F_{i}A \}_{i=1}^\infty$ such that $1 \in F_{0}A$ and the [[associated graded algebra]] is a finitely-generated commutative $\mathbb{K}$-[[K-algebra|algebra]],
then every $A$-module endomorphism $\vartheta \in \lMod A(V,V)$ is an [[algebraic element]] over $\mathbb{K}$.[^1969]
#m/thm/module 

> [!missing]- Proof
> #missing/proof


  [^1969]: 1969\. [[Sources/@quillenEndomorphismRingSimple1969|On the endomorphism ring of a simple module over an enveloping algebra]], p. 171

## Corollaries

The following algebras fulfil the hypothesis:

1. The [[Universal enveloping algebra]] of a finite-dimensional [[Lie algebra]], since by [[Poincaré-Birkhoff-Witt theorem]] $G_{\bullet}U(\mathfrak{g}) = S^\bullet \mathfrak{g}$.

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