Descent in algebraic K-theory and a conjecture of Ausoni-Rognes
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Descent in algebraic K-theory and a conjecture of Ausoni-Rognes. / Clausen, Dustin; Mathew, Akhil; Naumann, Niko; Noel, Justin.
I: Journal of the European Mathematical Society, Bind 22, Nr. 4, 2020, s. 1149-1200.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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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 -
ID: 271819253