By a theorem of Suslin, a Tor-unital (not necessarily unital) ring satisfies excision in algebraic $K$-theory. We give a new and direct proof of Suslin’s result based on an exact sequence of categories of perfect modules. In fact, we prove a more general descent result for a pullback square of ring spectra and any localizing invariant. Our descent theorem contains not only Suslin’s result, but also Nisnevich descent of algebraic $K$-theory for affine schemes as special cases. Moreover, the role of the Tor-unitality condition becomes very transparent.

Let X be a locally compact metrizable space. We show that the Paschke dual construction, which associates to a representation of C0(X) its commutant modulo locally compact operators, can be sheafified. We use this observation to simplify several constructions in analytic K-homology.

This paper studies ‘pro-excision’ for the $K$-theory of one-dimensional, usually semi-local, rings and its various applications. In particular, we prove Geller’s conjecture for equal characteristic rings over a perfect field of finite characteristic, give results towards Geller’s conjecture in mixed characteristic, and we establish various finiteness results for the $K$-groups of singularities, covering both orders in number fields and singular curves over finite fields.

We survey Quillen's contributions to the area of cyclic homology, apart from his very first result in [19].

Fracture of the clavicle as a late complication following radical neck dissection is rare, with an incidence of approximately 0.4-0.5 per cent. We report a case where fracture occurred early following a selective neck dissection.