Appears in collections : Arithmetic and Algebraic Geometry: A conference in honor of Ofer Gabber on the occasion of his 60th birthday, Fields medallists - 2018

The original proof of Grothendieck's purity conjecture in étale cohomology (the Thomason-Gabber theorem) relies on results on l-adic K-theory and its relation to étale cohomology when l is invertible. Using recent advances of Clausen-Mathew-Morrow and joint work with Bhatt and Morrow, our understanding in the complementary case of p-adic K-theory of p-adic rings has reached a similar level. In particular, we can express p-adic étale K-theory in terms of the cohomology theories of integral p-adic Hodge theory, such as the prismatic cohomology discussed in Bhatt's talk. Depending on time, I may indicate some possible applications.