Isomorphism types of commutative algebras of finite rank over an algebraically closed field

Abstract. 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 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 [Poo07]. 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

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