TY - JOUR

T1 - Descent in algebraic K-theory and a conjecture of Ausoni-Rognes

AU - Clausen, Dustin

AU - Mathew, Akhil

AU - Naumann, Niko

AU - Noel, Justin

PY - 2020

Y1 - 2020

N2 - Let A → B be a G-Galois extension of rings, or more generally of E∞-ring spectra in the sense of Rognes. A basic question in algebraic K-theory asks how close the map K(A) → K(B)hG is to being an equivalence, i.e., how close algebraic K-theory is to satisfying Galois descent. An elementary argument with the transfer shows that this equivalence is true rationally in most cases of interest. Motivated by the classical descent theorem of Thomason, one also expects such a result after periodic localization. We formulate and prove a general result which enables one to promote rational descent statements as above into descent statements after periodic localization. This reduces the localized descent problem to establishing an elementary condition on K0(−) ☉ Q. As applications, we prove various descent results in the periodically localized K-theory, TC, THH, etc. of structured ring spectra, and verify several cases of a conjecture of Ausoni and Rognes.

AB - Let A → B be a G-Galois extension of rings, or more generally of E∞-ring spectra in the sense of Rognes. A basic question in algebraic K-theory asks how close the map K(A) → K(B)hG is to being an equivalence, i.e., how close algebraic K-theory is to satisfying Galois descent. An elementary argument with the transfer shows that this equivalence is true rationally in most cases of interest. Motivated by the classical descent theorem of Thomason, one also expects such a result after periodic localization. We formulate and prove a general result which enables one to promote rational descent statements as above into descent statements after periodic localization. This reduces the localized descent problem to establishing an elementary condition on K0(−) ☉ Q. As applications, we prove various descent results in the periodically localized K-theory, TC, THH, etc. of structured ring spectra, and verify several cases of a conjecture of Ausoni and Rognes.

KW - Algebraic K-theory

KW - Chromatic homotopy theory

KW - Descent

KW - Galois extensions

KW - Structured ring spectra

U2 - 10.4171/JEMS/942

DO - 10.4171/JEMS/942

M3 - Journal article

AN - SCOPUS:85086317921

VL - 22

SP - 1149

EP - 1200

JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

SN - 1435-9855

IS - 4

ER -