  We study moduli space of holomorphic triples $E_{1}\xrightarrow{\phi} E_{2}$,
composed of torsion-free sheaves $E_{i}, i=1,2$ and a holomorphic mophism
between them, over a smooth complex projective surface $S$. The triples are
equipped with Schmitt stability condition [Alg Rep Th. 6. 1. pp 1-32, 2003]. We
observe that when Schmitt stability parameter $q(m)$ becomes sufficiently
large, the moduli space of triples benefits from having a perfect relative and
absolute deformation-obstruction theory in some cases. We further generalize
our construction by gluing triple moduli spaces, and extend the earlier work of
Gholampour-Sheshmani-Yau [arXiv:1701.08899] where the obstruction theory of
nested Hilbert schemes over the surface was studied. Here we extend the earlier
results to the moduli space of chains$$E_{1}\xrightarrow{\phi_{1}}
E_{2}\xrightarrow{\phi_{2}} \cdots \xrightarrow{\phi_{n-1}} E_{n},$$ where
$\phi_{i}$ are injective morphisms and $rk(E_{i})\geq 1, \forall i$. There is a
connection, by wallcrossing in the master space in the sense of Mochizuki,
between the theory of such higher rank flags, and the theory of Higgs pairs on
the surface, which provides the means to relate the flag invariants to the
local DT invariants of threefold given by a line bundle on the surface, $X
:=Tot(\mathcal{L} \to S)$. The latter, when $\mathcal{L}=\omega_{S}$, provides
the means to compute the contribution of higher rank flag sheaves to partition
function of Vafa-Witten theory on $X$.
