  When ${\cal{D}}$ is a linear partial differential operator of any order, a
direct problem is to look for an operator ${\cal{D}}_1$ generating the
compatibility conditions (CC) ${\cal{D}}_1\eta=0$ of ${\cal{D}}\xi=\eta$. We
may thus construct a differential sequence with successive operators
${\cal{D}},{\cal{D}}_1,{\cal{D}}_2, ...$, where each operator is generating the
CC of the previous one. Introducing the formal adjoint $ad( )$, we have
${\cal{D}}_i\circ {\cal{D}}_{i-1}=0 \Rightarrow ad({\cal{D}}_{i-1}) \circ
ad({\cal{D}}_i)=0$ but $ad({\cal{D}}_{i-1})$ may not generate all the CC of
$ad({\cal{D}}_i)$. When $D=K[d_1,...,d_n]=K[d]$ is the (non-commutative) ring
of differential operators with coefficients in a differential field $K$, it
gives rise by residue to a differential module $M$ over $D$. The homological
extension modules $ext^i(M)=ext^i_D(M,D)$ with $ext^0(M)=hom_D(M,D)$ only
depend on $M$ and are measuring the above gaps, independently of the previous
differential sequence.The purpose of this rather technical paper is to compute
them for certain Lie operators involved in the formal theory of Lie
pseudogroups in arbitrary dimension $n$. In particular, we prove that the
extension modules highly depend on the Vessiot structure constants $c$. When
one is dealing with a Lie group of transformations or, equivalently, when
${\cal{D}}$ is a Lie operator of finite type, then we shall prove that
$ext^i(M)=0, \forall 0\leq i \leq n-1$. It will follow that the Riemann-Lanczos
and Weyl-Lanczos problems just amount to prove such a result for $i=2$ and
arbitrary $n$ when ${\cal{D}}$ is the Killing or conformal Killing operator. We
finally prove that ${ext}^i(M)=0, \forall i\geq 1$ for the Lie operator of
infinitesimal contact transformations with arbitrary $n=2p+1$. Most of these
new results have been checked by means of computer algebra.
