---
tag: 856B6C7EE4
kind: statement
role: theorem
name: Nichols–Zoeller theorem
msc: [16T05]
uses: [7XFYP0N84H, 0DZ9DKAKZR, 0DTXBQPA2S, 3C53TD]
about: [7XFYP0N84H]
sources:
  - url: https://en.wikipedia.org/wiki/Hopf_algebra
---

## Statement

Let $K$ be a field and let $H$ be a finite-dimensional [[Hopf algebra|7XFYP0N84H]] over $K$.
Let $B \subseteq H$ be a Hopf subalgebra, that is a subalgebra with $\Delta(B) \subseteq B \otimes B$ and $S(B) \subseteq B$.
Then $H$ is a [[free|0DZ9DKAKZR]] left $B$-module and a free right $B$-module.
In particular, $\dim_K B$ divides $\dim_K H$.

## Remarks

The proof views $H$ as a relative Hopf module: a right $B$-module under multiplication with the compatible right $H$-coaction $\Delta$.
One shows that a direct sum of finitely many copies of such a module is free over $B$.
A Krull–Schmidt argument for finite-dimensional modules then shows that $H$ itself is free.
Example: for a finite group $G$ and a subgroup $L$, the group algebra $KG$ is a free right $KL$-module with basis a set of representatives of the left cosets $gL$.
Then $\dim KL = \abs{L}$ divides $\dim KG = \abs{G}$, which is [[Lagrange's theorem|3C53TD]].
The divisibility statement follows from freeness: if $H \cong B^n$ as $B$-modules, then $\dim H = n \dim B$.

## Explanations

### L1 human

Lagrange's theorem says that the size of a subgroup divides the size of a finite group.
The Nichols–Zoeller theorem is the version for finite-dimensional Hopf algebras: the dimension of a Hopf subalgebra divides the dimension of the whole algebra.
The stronger statement is that the big algebra is a free module over the small one, just as a group is a disjoint union of cosets of a subgroup.

### L2 human

The theorem is a basic tool in the classification of finite-dimensional Hopf algebras of small dimension.
For example, the group-like elements of $H$ span a Hopf subalgebra $KG(H)$, so the order of the group of group-likes divides $\dim H$.
The [[index|0DTXBQPA2S]] analogue is the integer $\dim H / \dim B$, the rank of $H$ as a free $B$-module.