  Our recent arXiv preprints and published papers on the solution of the
Riemann-Lanczos and Weyl-Lanczos problems have brought our attention on the
importance of revisiting the algebraic structure of the Bianchi identities in
Riemannian geometry. We also discovered in the meantime that, in our first book
of 1978, we had already used a new way for studying the various compatibility
conditions (CC) of an operator that may not be necessarily formally integrable
(FI) in order to construct canonical formally exact differential sequences on
the jet level. The purpose of this paper is to prove that the combination of
these two facts clearly shows the specific importance of the Spencer operator
and the Spencer $\delta$-cohomology, totally absent from mathematical physics
today. The results obtained are unavoidable because they only depend on
elementary combinatorics and diagram chasing. They also provide for the first
time the purely intrinsic interpretation of the respective numbers of
successive first, second, third and higher order generating CC. However, if
they of course agree with the linearized Killing operator over the Minkowski
metric, they largely disagree with recent publications on the respective
numbers of generating CC for the linearized Killing operator over the
Schwarzschild and Kerr metrics. Many similar examples are illustrating these
new techniques, providing in particular the only symbol existing in the
literature which is 2-ayclic witout being of finite type, contrary to the
conformal situation.
