Hidden constructions in abstract algebra, Krull Dimension, Going Up, Going Down

Abstract

We present constructive versions of Krull's dimension theory for commutative rings and distributive lattices. The foundations of these constructive versions are due to Joyal, Espan\~ol and the authors. We show that this gives a constructive version of basic classical theorems (dimension of finitely presented algebras, Going up and Going down theorem, \ldots), and hence that we get an explicit computational content where these abstract results are used to show the existence of concrete elements. This can be seen as a partial realisation of Hilbert's program for classical abstract commutative algebra.