THE OSU-OU Ring Theory Seminar


Speaker:  Hans Schoutens
          Ohio State University - Columbus


Title:  Singularity theory beyond the Noetherian realm


Day, Date and Time:  Friday, April 11, 4:30


Room:  MW 154 (OSU-Columbus campus)



Abstract: A local ring R with (unique) maximal ideal M is called 'regular'
in commutative algebra, if M can be generated by d elements, where d is
the 'dimension' of R.  The proper notion of dimension for a Noetherian ring is
its Krull dimension, but this is most of the time infinite if the ring is
no longer Noetherian.  I will discuss classes of non-Noetherian rings for
which we can still find a good notion of dimension in such a way that the
resulting notion of 'regularity' gives a satisfactory generalization.  Of
course I have to assume that the maximal ideal is finitely generated (but
there may be many (prime) ideals which are NOT finitely generated).  I
will briefly discuss an application of this theory to homological
conjectures in mixed characteristic.  In the particular case needed for
this application the notion of regularity coincides with the notion of
regularity for coherent rings studied by Glaz, Vasconcelos, et al.


Commutative algebra also has several notions of 'good' singularities (e.g.
Gorenstein, Cohen-Macaulay) and a similar treatment can be given in this
non-Noetherian setting.