INVARIANCE AND LOCALIZATION FOR CYCLIC HOMOLOGY
OF DG ALGEBRAS
Bernhard Keller
February 11, 1996
Journal of Pure and Applied Algebra, 123 (1998), 223-273.
Abstract. We show that two flat differential graded algebras
whose derived categories are equivalent by a derived functor
have isomorphic cyclic homology. In particular, `ordinary' algebras
over a field which are derived equivalent (J. Rickard) share their
cyclic homology, and iterated tilting (Brenner-Butler, Happel-Ringel,
Bongartz) preserves cyclic homology. This completes results of
J. Rickard's and D. Happel's. It also extends well known results
on preservation of cyclic homology under Morita equivalence
due to A. Connes, Loday-Quillen, Chr. Kassel, and R. McCarthy.
We then show that under suitable flatness hypotheses,
an exact sequence of derived categories of DG algebras
yields a long exact sequence in cyclic homology.
This may be viewed as an analogue of
Thomason-Trobaugh's and Yao's
localization theorems in K-theory (cf. also Weibel-Yao).
I thank the referee for his very careful reading of the manuscript.
http://www.math.jussieu.fr/~keller/publ/ilcabs.html
Bernhard Keller, le 11 février, 1996.
keller@math.jussieu.fr
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