On the non additivity of the trace in derived categories

Daniel Ferrand (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

In this note we provide an example of an endomorphism of a short exact sequence of perfect complexes, with the trace of the middle map not equal to the sum of the traces of the two other ones. The point is that the squares involved are commutative only up to homotopy. In view of this example I have found in 1968, Deligne immediately created his "categories spectrales", and soon afterwards Illusie introduced the "filtered derived categories" where a satisfactory kind of additivity is restored for the trace. This paper, written in French, ends up with a brief chronological comment.

Mots clés: derived categories; traces.

Mots clés: traces

Domaines: Catégories et ensembles [math.CT]

Daniel Ferrand. On the non additivity of the trace in derived categories. 2005. ⟨hal-00005708⟩