Cohn's theory of universal skew fields of fractions
- Feb 6, 11:30-Feb 18, 11:30, 2018
- Room -101
Jurij Volcic (BGU) will give three lectures on the theory of universal skew fields of fractions due to P.M. Cohn, a beautiful and powerful tool of noncommutative algebra little known outside of a small circle of cognoscenti. Everybody is invited (the lectures should be accessible for graduate students).
Time and place:
Tuesday Feb. 6, Thursday Feb. 15, and Sunday Feb. 18 at 11:30-13:00 in Room -101
As opposed to the commutative setting, localization in noncommutative rings is a much more intricate topic, accompanied with various subtle obstacles and at first sight counter-intuitive phenomena. An exceptional contribution to this field has been made by Paul Moritz Cohn (1924-2006) with the introduction of universal skew fields of fractions in the 70’s, and his subsequent development of this theory. In the self-contained course of three talks I will try to present some of Cohn’s core results with the view towards the universal skew field of fractions of a free algebra.
In the first lecture we will consider the general problem of embedding rings into skew fields, and early difficulties accompanying it. Following Cohn we will then introduce the category of epic skew fields and specializations assigned to a fixed ring. This will allow us to talk about universal skew fields of fractions and to draw parallels with the commutative theory.
In the second lecture we will explain the interplay between specializations, singular kernels, matrix ideals in matrix representation of fractions. Our aim will be to determine necessary and sufficient conditions for a ring to admit a (universal) skew field of fractions.
In the last lecture we will finally look at a concrete family of rings admitting universal skew fields of fractions: free ideal rings (firs), among which are free algebras, and their generalizations. We will trace down their distinguishing property that fulfills the sufficient condition from the previous lecture.