The moduli space of commutative algebras of finite rank

Abstract

The moduli space of rank-$n$ commutative algebras equipped with an ordered basis is an affine scheme ${\mathfrak{B}}_n$ of finite type over $\mathbb{Z}$, with geometrically connected fibers. It is smooth if and only if $n\le 3$. It is reducible if $n\ge 8$ (and the converse holds, at least if we remove the fibers above $2$ and $3$). The relative dimension of ${\mathfrak{B}}_n$ is $\frac{2}{27}{n}^3+O(n^{8/3})$. The subscheme parameterizing étale algebras is isomorphic to ${\mathrm{GL}}_n/S_n$, which is of dimension only $n^2$. For $n\ge 8$, 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 $d\ge n/2$.