affine algebra

Given a field kk an affine algebra is any finitely generated noetherian associative commutative unital kk-algebra without nilpotent elements. This term is limited to a small community of algebraists and may bring confusion if quoted outside of that context. The name is from the fact that the affine algebras are the coordinate rings of affine varieties; equivalently, an affine kk-variety is precisely the maximal spectrum of an affine kk-algebra.

Some people drop the condition of commutativity and talk about noncommutative affine algebras. This is even less standard as in the noncommutative context there is a further hesitation to drop noetherianess as well.

Yet another confusion may arise when people informally say affine algebra for an affine Lie algebra; of course, the latter, full, term is preferred (or “affine Kac-Moody algebra”).

Created on November 13, 2012 at 03:21:43. See the history of this page for a list of all contributions to it.