
School of Mathematics and Statistics
Amnon Neeman
University of Virginia
KTheory.
Friday 14th May, 121pm, Carslaw 375.
Let R be a commutative ring. Given an nbyn
matrix over R, we can form its determinant. If
the matrix is invertible, the determinant is a
unit. The determinant of the product is the
product of the determinants. The determinants
of all conjugates of a matrix agree. In
other words, the determinant is independent of
the basis.
It turns out to be interesting to ask: can
one define determinants for matrices over
noncommutative rings. This can be done, and
the determinant takes its values in an abelian
group called K_1(R).
Now we can ask how the abelian group K_1(R),
the universal abelian group in which the determinant
makes sense, varies with the ring R. In the talk,
we will review the history of the subject, leading
to the result that K_1(R) depends only on the
derived category of R, at least when R is regular.
