The moduli space of commutative algebras of finite rank

Poonen, Bjorn. "The moduli space of commutative algebras of finite rank." Journal of the European Mathematical Society 010.3 (2008): 817-836. <http://eudml.org/doc/277522>.

@article{Poonen2008,
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 $\operatorname\{GL\}_n/S_n$, 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 $d\ge n/2$.},
author = {Poonen, Bjorn},
journal = {Journal of the European Mathematical Society},
language = {eng},
number = {3},
pages = {817-836},
publisher = {European Mathematical Society Publishing House},
title = {The moduli space of commutative algebras of finite rank},
url = {http://eudml.org/doc/277522},
volume = {010},
year = {2008},
}

TY - JOUR
AU - Poonen, Bjorn
TI - The moduli space of commutative algebras of finite rank
JO - Journal of the European Mathematical Society
PY - 2008
PB - European Mathematical Society Publishing House
VL - 010
IS - 3
SP - 817
EP - 836
AB - 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 $\operatorname{GL}_n/S_n$, 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 $d\ge n/2$.
LA - eng
UR - http://eudml.org/doc/277522
ER -