Rings and their spectrum
Abstract:
Noncommutative geometry is a geometric approach to noncommutative
algebra. The main motivation of noncommutative geometry is to extend
various functors between spaces and functions to the noncommutative
setting. Spaces, which are geometric in nature, can be related to
numerical functions on them, which in general form a commutative ring.
Thus we have functors F:{spaces}->{commutative rings} and G:{commutative
rings}->{spaces}, for instance the contravariant functor
Spec:{commutative rings}->{(spectral) topological spaces}. It is
tempting to hope that one could extend the spectrum to the
noncommutative setting in order to construct the “underlying set of a
noncommutative space.” We will try to discuss these things in a language
understandable to everybody (i.e., to any mathematician...)
References:
(1) M. Reyes, Obstructing extensions of the functor Spec to
noncommutative rings, Israel J. Math. 192 (2012), 667-698.
(2) A. Facchini and L. Heidari Zadeh, On a partially ordered set
associated to ring morphisms, J. Algebra 535 (2019), 456-479.
(3) A. A. Bosi and A. Facchini, A natural fibration for rings, submitted
for publication, 2019.
Pierre-Emmanuel Caprace
