Poonen's abstract: The moduli space of rank-n commutative algebras equipped with an ordered basis is an affine scheme B_n of finite type over Z, with geometrically connected fibers. It is smooth if and only if n <= 3. It is reducible if n >= 8 (and the converse holds, at least if we remove the fibers above 2 and 3).
The relative dimension of B_n is (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 >= 8, there exist algebras that are not limits of etale algebras. The dimension calculations lead also to 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 >= n/2.
Digits := 100: for d from 1 to 10 do for n from 1 to d do k := d-n+1 ; A := floor(ithprime(k)^(2*n^3/27)) ; printf("%d, ", A) ; od: od: # R. J. Mathar, Jan 22 2009
