The moduli space of commutative algebras of finite rank

2006/08/20 by Bjorn Poonen · 1 voice · 3 citations
Mathematics · #Algebraic structures and combinatorial models #Finite Group Theory Research #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.4171/jems/131

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 ≤ 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 \mathfrak B_n is (2)/(27) n³ + O(n^8/3) . The subscheme parameterizing étale algebras is isomorphic to GL_n/S_n , which is of dimension only n² . For n ≥ 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 pⁿ and the dimension of the Hilbert scheme of n points in d -space for d ≥ n/2 .

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• Geometry of Deformations via Incidence Varieties
• Local Equations for Hilbert Schemes of Points
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Discussions

• A crazy open problem in algebra: Conjecture (Poonen): 100% (asymptotic density) of finite commutative rings have characteristic EXACTLY 8, i.e. 4 ≠ 0 and 8 = 0 both hold. From this amazing paper: arxi [bsky, 23 points, 7 comments]

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