Relation to Beilinson-Lichtenbaum

This chapter show how the Beilinson–Lichtenbaum condition is equivalent to the assertion that the norm residue is an isomorphism. A diagram featuring this proof is also included. The chapter first proves the reductions introduced in the previous chapter, BL(n) and H90(n), arguing that if BL(n) holds then BL(n − 1) holds and that H90(n) implies H90(n − 1). Next, the chapter turns to the cohomology of singular varieties, cohomology with supports, and rationally contractible presheaves. From there, it argues that Bloch–Kato implies Beilinson–Lichtenbaum and concludes the theorems by proving that H90(n) implies BL(n). The chapter concludes with some historical notes.