CORRECTION TO ‘ON THE CYCLIC HOMOLOGY OF EXACT CATEGORIES’

BERNHARD KELLER

1. Mistakes pointed out by R.-O. Buchweitz

I thank R.-O. Buchweitz for his letter [1], where he points out and corrects the following mistakes in [2]:

In 1.11 of [2], it should be HH_{∗}(mod k) = k (rather than k[u]).

Paragraph 2.5 [2] ends with a claim about the structure of HH_{∗}(B), where B = k[T] viewed as a DG algebra with T of degree 1 and trivial differential. This claim is false (unless k is of characteristic 2). The correct result is given in R.-O. Buchweitz’ letter [1]: … B has to be viewed not as the symmetric but as the tensor algebra over T, and the correct answer is somewhat smaller. One has, e.g. from Prop. 5.3.10 in Loday’s book [3]

HH_{∗}(k[T]) = (k[T^{2}]/(2T^{2}) ⊕ k[T^{2}]dT) ⊕ (T k[T^{2}] ⊕ Ann_{2}T k[T^{2}]dT)
≅ HH_{0} ⊕ (HH_{1} ⊕ HH_{2})[T^{2}],
HH_{0} = k ⊕ kdT
HH_{1} = kT ⊕ Ann_{2}(kdT)
HH_{2} = (k/2k)T^{2} ⊕ kT^{2}dT

where the brackets in the first expansion correspond to even and odd part resp., as T is of homological degree -1 and dT is of homological degree 0.

Finally, the readers of [2] should be warned about a poor choice of notation in 2.5 and 2.6: In both paragraphs, we consider the differential graded endomorphism algebra Hom_{A}(P, P) of a complex P. In 2.5, the symbol B denotes the homology of this algebra, whereas in 2.6, it denotes the algebra itself (note that in both paragraphs, B denotes a differential graded algebra quasi-isomorphic to the endomorphism algebra; this was in fact the reason for the choice of the notation).

2. Mistake concerning homotopy

In section 3.1 of [2], we consider homotopies between morphisms between DG algebras and erroneously claim that the homotopy relation is an equivalence relation. Instead we should consider the equivalence relation generated by the homotopy relation. The error has no incidence on what follows. Indeed, what we use in the sequel is homotopy between DG functors, treated in 3.3 of [loc. cit.] and in this section, we correctly consider the equivalence relation generated by all ‘one-step homotopies’.

Date: May 26, 2000.

References

[1] R.-O. Buchweitz, Letter to B. Keller, May 23, 2000.

[2] B. Keller, On the cyclic homology of exact categories, JPAA 136 (1999), 1–56.

[3] J.-L. Loday, Cyclic homology, Grundlehren 301, Second edition, Springer-Verlag Berlin Heidelberg, 1998.

UFR de Mathématiques, UMR 7586 du CNRS, Case 7012, Université Paris 7, 2, place Jussieu, 75251 Paris Cedex 05, France

 E-mail address: keller@math.jussieu.fr www.math.jussieu.fr/ ˜ keller