Finitely presented algebra

Summary

An algebra $B$ over a commutative ring $A$ is finitely presented if it can be described by finitely many generators and finitely many relations. That is, $B \cong A[x_1, \ldots, x_n]/I$ with $I$ a finitely generated ideal. Over a Noetherian ring every finite type algebra is finitely presented, so the distinction matters only over non-Noetherian bases. Finite presentation is the right finiteness condition for limit arguments and for étale and smooth morphisms in algebraic geometry.