This talk will explain the theorem, by Ausoni–Rognes, that V (2)∗TR(l) is bounded for primes p > 3. (The speaker may assume that p > 5, so that V (2) exists as a homotopy commutative and associative ring, or alternatively may make use of the motivic spectral sequence.) Here, l = BP⟨1⟩ is the Adams summand of p-local complex K-theory. This was the first example of a higher height Lichtenbaum–Quillen theorem, and the speaker will also explain how to deduce chromatic redshift.