The moduli space of rank- commutative algebras equipped with an ordered basis is an affine scheme \( \frakB_n \) of finite type over , with geometrically connected fibers. It is smooth if and only if . It is reducible if (and the converse holds, at least if we remove the fibers above and ). The relative dimension of \( \frakB_n \) is . The subscheme parameterizing \'etale algebras is isomorphic to \( \GL_n/S_n \), which is of dimension only . For , 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 and the dimension of the Hilbert scheme of points in -space for .
Cite this article
Bjorn Poonen, The moduli space of commutative algebras of finite rank. J. Eur. Math. Soc. 10 (2008), no. 3, pp. 817–836DOI 10.4171/JEMS/131