  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.
