# The moduli space of commutative algebras of finite rank

Journal of the European Mathematical Society (2008)

- Volume: 010, Issue: 3, page 817-836
- ISSN: 1435-9855

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topPoonen, 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 -

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