\(K\)-theory of log-schemes. II: Log-syntomic \(K\)-theory (Q436121)

It is well-known that there exists interesting relationship between 1-adic \(K\)-theory and 1-adic étale cohomology. In this paper the author ask that whether there exists (and, if so, what from it takes) a similar relationship between \(p\)-adic \(K\)-theory and syntomic cohomology. The author proved that suitably truncated topological log-syntomic-étale homotopy \(K\)-theory of proper semistable schemes in mixed characteristic surjects onto the Kummer log-étale \(p\)-adic \(K\)-theory of the log-syntomic-étale cohomology of homotopy \(K\)-theory shcaves. The proofs use \(p\)-adic Hodge theory computations of \(p\)-adic nearby cycles.