# The moduli space of commutative algebras of finite rank

### Bjorn Poonen

Massachusetts Institute of Technology, Cambridge, United States

## Abstract

The moduli space of rank-$n$ commutative algebras equipped with an ordered basis is an affine scheme $B_{n}$ of finite type over $Z$, with geometrically connected fibers. It is smooth if and only if $n≤3$. It is reducible if $n≥8$ (and the converse holds, at least if we remove the fibers above $2$ and $3$). The relative dimension of $B_{n}$ is $272 n_{3}+O(n_{8/3})$. The subscheme parameterizing étale algebras is isomorphic to $GL_{n}/S_{n}$, which is of dimension only $n_{2}$. For $n≥8$, there exist algebras that are not limits of étale algebras. The dimension calculations lead also to new asymptotic formulas for the number of commutative rings of order $p_{n}$ and the dimension of the Hilbert scheme of $n$ points in $d$-space for $d≥n/2$.

## Cite this article

Bjorn Poonen, The moduli space of commutative algebras of finite rank. J. Eur. Math. Soc. 10 (2008), no. 3, pp. 817–836

DOI 10.4171/JEMS/131