  We propose a systematic generating procedure to construct free Lagrangians
for massive, massless and partially massless, totally-symmetric tensor fields
on $AdS_{d+1}$ starting from the BRST Lagrangian description of massless fields
in the flat ambient space $\mathbb{R}^{d,2}$. A novelty is that the Lagrangian
is described by a $d+1$ form on $\mathbb{R}^{d,2}$ whose pullback to
$AdS_{d+1}$ gives the genuine Lagrangian defined on anti de Sitter spacetime.
Our derivation uses the triplet formulation originating from the
first-quantized BRST approach, where the action principle is determined by the
BRST operator and the inner product of a first-quantised system. In this way we
build, in a manifestly $so(2,d)$-covariant manner, a unifying action principle
for the three types of fields mentioned above. In particular, our derivation
justifies the form of some actions proposed earlier for massive and massless
fields on AdS. We also give a general setup for ambient Lagrangians in terms of
the respective jet-bundles and variational bi-complexes. In particular we
introduce a suitable ambient-space Euler-Lagrange differential which allows to
derive the equation of motion ambiently, i.e. without the need to explicitly
derive the respective spacetime Lagrangian.
