Algebra Seminar

Let R be a commutative ring. Given an n-by-n 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 non-commutative 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.