# Algebra/Topology seminar

Speaker: Ryomei Iwasa (KU)

Title: On relative algebraic K-theory

Abstract: Let X be a scheme and D a closed subscheme of X. In this talk, I present some results on the relative algebraic K-theory K(X,D), i.e., the homotopy fiber of the canonical morphism K(X) → K(D) between the K-theory spectra.

We divide the talk into three parts:

1. A concrete description of K0(X,D)= π0K(X,D).
2. A relation between K0(X,D) and algebraic cycles with modulus (here, X is regular and D an effective Cartier divisor) — this is a joint work with Wataru Kai.
On higher relative K-theory, we only have a satisfactory result for the homotopy invariant part.
3. A motivic spectral sequence from ''relative'' Chow groups to relative homotopy K-theory KH*(X,D) (here, X is smooth over a field) — this is a joint work with Amalendu Krishna.