  The search for generating compatibility conditions (CC) for a given operator
is a very recent problem met in General Relativity in order to study the
Killing operator for various standard useful metrics (Minkowski, Schwarschild
and Kerr). In this paper, we prove that the link existing between the lack of
formal exactness of an operator sequence on the jet level, the lack of formal
exactness of its corresponding symbol sequence and the lack of formal
integrability (FI) of the initial operator is of a purely homological nature as
it is based on the long exact connecting sequence provided by the so-called
snake lemma. It is therefore quite difficult to grasp it in general and even
more difficult to use it on explicit examples. It does not seem that any one of
the results presented in this paper is known as most of the other authors who
studied the above problem of computing the total number of generating CC are
confusing this number with a kind of differential transcendence degree, also
called degree of generality by A. Einstein in his 1930 letters to E. Cartan.
The motivating examples that we provide are among the rare ones known in the
literature and could be used as testing examples for future applications of
computer algebra.
