On the Beilinson fiber square

Using topological cyclic homology, we give a refinement of Beilinson’s p-adic Goodwillie isomorphism between relative continuous K-theory and cyclic homology. As a result, we generalize results of Bloch–Esnault–Kerz and Beilinson on the p-adic deformations of K-theory classes. Furthermore, we prove structural results for the Bhatt–Morrow–Scholze filtration on TC and identify the graded pieces with the syntomic cohomology of Fontaine–Messing.