Isomorphism Types of Commutative Algebras of Finite Rank over an Algebraically Closed Field

Let k be an algebraically closed field. We list the finitely many isomorphism types of rank n commutative k-algebras for n ≤ 6. There are infinitely many types for each n ≥ 7. All algebras are assumed to be commutative, associative, and with 1 (except briefly in Remark 1.1). We assume that k is an algebraically closed field, except in Section 2. By the rank of a k-algebra, we mean its dimension as a k-vector space. 1. Local algebras of rank up to 6 Our main goal is to list representatives for the (finitely many) isomorphism classes of rank n k-algebras for n ≤ 6. As we discuss in Section 2, it is known [Sup56] that the number of isomorphism classes is infinite for every n ≥ 7, so it is natural to stop at 6. One purpose of these calculations is to give insight into the moduli space of based rank n algebras for small values of n: see [Poo08]. The geometry of this moduli space seems to be what is behind the parameterization and enumeration of number fields of fixed low degree and bounded discriminant, as in the work of Bhargava [Bha04a,Bha04b,Bha04c,Bha05]. Remark 1.1. Many partial results had been obtained by earlier authors. For example, in the case char k = 0, these algebras were determined implicitly, by classifying nilpotent commutative subalgebras of the algebra of n×n matrices up to conjugacy, by [Cha54] (n ≤ 5) and [Dym66] (n = 6). See also [ST66, §2.8 and §2.9]. The paper [Maz80] classifies nilpotent commutative associative algebras without 1 up to n ≤ 5 for char k 6 = 2, 3; isomorphism types of such algebras of rank n are in bijection with isomorphism types of local commutative associative algebras with 1 of rank n+ 1 (send A to k ⊕ A), so our results are novel only in that they handle the cases of characteristic 2 and 3. Our methods are more elementary than those in [Maz80], which used Hochschild cocyles. Remark 1.2. The classification of finite-rank algebras is useful in the study of regular subgroups of the affine group AGLn(k); see [PTB16]. Classifying the local algebras A is enough since every finite-rank algebra is a product of these. Let m be the maximal ideal of A. For i ≥ 1, let di = dimk(m/m). We classify local algebras A first by n := dimk A, and next by the sequence ~ d := (di)i≥1 (we retain only the finitely many nonzero terms). Each algebra is represented by P/I where P is the polynomial ring in the first d1 of the variables x, y, z, w, v, and I is an ideal in P . Table 1 Date: January 11, 2016 (the print version had an error in the classification of symmetric bilinear forms in characteristic 2; this led to a few redundant entries in the table; these have now been corrected). 2000 Mathematics Subject Classification. Primary 13E10.