K-theory of singularities

Christian Haesemeyer (UCLA/Melbourne)

Abstract

Algebraic K-theory is, fundamentally, an approach to studying linear algebra over general rings. For example, K-theory tells us that any idempotent matrix whose entries are polynomials with coefficients in the ring of germs of functions at a smooth point on an algebraic variety is stably diagonalisable. This is false if we use germs of functions at a singular point instead. In this talk I will try to explain how methods from abstract homotopy theory, commutative algebra and algebraic geometry (developed to study K-theory in a joint project with G. Cortinas, M. Walker, C. Weibel and in part M. Schlichting) can be used to measure this failure in terms of the algebra and geometry of the singularity in question.