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, Español 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,
...),
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.
Key words :
Krull dimension, Going Up, Going Down,
Constructive Mathematics.
See also
Hidden constructions in
abstract algebra (3)
Krull dimension of distributive lattices and commutative rings.
Henri Lombardi, Thierry Coquand.
Commutative ring theory and applications. Eds: Fontana M.,
Kabbaj S.-E., Wiegand S.
Lecture notes in pure and applied mathematics vol 131. M. Dekker.
Proceedings for the Fourth International Conference on Commutative Ring Theory and
Applications held June 7 - 11 , 2001 in Fez, Morocco
(2002) pp. 477-499.