Algebraic K-theory of the fraction field of topological K-theory

Abstract

We compute the algebraic K-theory modulo p and v_1 of the S-algebra ell/p = k(1), using topological cyclic homology. We use this to compute the homotopy cofiber of a transfer map K(L/p) --> K(L_p), which we interpret as the algebraic K-theory of the "fraction field" of the p-complete Adams summand of topological K-theory. The results suggest that there is an arithmetic duality theorem for this fraction field, much like Tate--Poitou duality for p-adic fields.

References in corpus (3)

• Galois extensions of structured ring spectra
• Topological Hochschild homology and cohomology of A_\infty ring spectra
• Uniqueness of Morava K-theory

Cited by in corpus (5)

• Commutative ring objects in pro-categories and generalized Moore spectra
• Logarithmic structures on topological K-theory spectra
• p-adic J-homomorphisms and a product formula
• A construction of some objects in many base cases of an Ausoni-Rognes conjecture
• A computational reduction for many base cases in profinite telescopic algebraic $K$-theory