  The purpose of this paper is to revisit the Bianchi identities existing for
the Riemann and Weyl tensors in the combined framework of the formal theory of
systems of partial differential equations (Spencer cohomology, differential
systems, formal integrability) and Algebraic Analysis (homological algebra,
differential modules, duality). In particular, we prove that the n 2 (n 2 --
1)(n -- 2)/24 generating Bianchi identities for the Riemann tensor are first
order and can be easily described by means of the Spencer cohomology of the
first order Killing symbol in arbitrary dimension n $\ge$ 2. Similarly, the n(n
2 -- 1)(n + 2)(n -- 4)/24 generating Bianchi identities for the Weyl tensor are
first order and can be easily described by means of the Spencer cohomology of
the first order conformal Killing symbol in arbitrary dimension n $\ge$ 5. As A
MOST SURPRISING RESULT, the 9 generating Bianchi identities for the Weyl tensor
are of second order in dimension n = 4 while the analogue of the Weyl tensor
has 5 components of third order in the metric with 3 first order generating
Bianchi identities in dimension n = 3. The above results, which could not be
obtained otherwise, are valid for any non-degenerate metric of constant
riemannian curvature and do not depend on any conformal factor. They are
checked in an Appendix produced by Alban Quadrat (INRIA, Lille) by means of
computer algebra. We finally explain why the work of Lanczos and followers is
not coherent with these results and must therefore be also revisited.
