The moduli space of commutative algebras of finite rank
By Bjorn Poonen. 2006
63 citations recorded by the source corpus.
The moduli space of rank-$n$ commutative algebras equipped with an ordered basis is an affine scheme $\frakB_n$ of finite type over $\Z$, 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 $\frakB_n$ is $\frac{2}{27} n^3 + O(n^{8/3})$. The subscheme parameterizing \'etale algebras is isomorphic to $\GL_n/S_n$, which is of dimension only $n^2$. For $n \ge 8$, there exist algebras that are not limits of \'etale 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 $d \ge n/2$.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/2ff109c39afbfebf635fd6a14220a3af70eb6a36
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In Discrete Mathematics and Combinatorics
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- ISOMORPHISM TYPES OF COMMUTATIVE ALGEBRAS OF FINITE RANK OVER AN ALGEBRAICALLY CLOSED FIELD (2016, background): “Remark 1.8. Let Bn;k be the moduli space of rank-n algebras equipped with a basis: see [ Poo07 ].”