Commutation of K-theory with Inverse Limits in Adic Spaces

Explore a detailed mathematical seminar talk examining the comparison of two approaches to K-theory of adic spaces, focusing specifically on affine formal schemes. Delve into the relationship between classical continuous K-theory spectrum definition for Spf(R) and the recently developed category of nuclear modules (Nuc(R)) by Clausen and Scholze in condensed mathematics. Learn about the equivalence between K^{cont}(Nuc(R)) and the limit of K(R/I^n), while understanding how the category Nuc(R) connects to the inverse limit of derived categories D(R/I^n). Master the concept of Mittag-Leffler inverse sequence of dualizable categories and discover how continuous K-theory commutes with limits for such sequences, with particular attention to the Mittag-Leffler property of the inverse sequence (D(R/I^n))_n.