JournalsjemsVol. 10 , No. 3DOI 10.4171/jems/131

The moduli space of commutative algebras of finite rank

  • Bjorn Poonen

    Massachusetts Institute of Technology, Cambridge, United States
The moduli space of commutative algebras of finite rank cover

Abstract

The moduli space of rank-nn commutative algebras equipped with an ordered basis is an affine scheme \frakBn\frakB_n of finite type over Z\Z, with geometrically connected fibers. It is smooth if and only if n3n \le 3. It is reducible if n8n \ge 8 (and the converse holds, at least if we remove the fibers above 22 and 33). The relative dimension of \frakBn\frakB_n is 227n3+O(n8/3)\frac{2}{27} n^3 + O(n^{8/3}). The subscheme parameterizing \'etale algebras is isomorphic to \GLn/Sn\GL_n/S_n, which is of dimension only n2n^2. For n8n \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 pnp^n and the dimension of the Hilbert scheme of nn points in dd-space for dn/2d \ge n/2.