  In this article we are defining a refinement of Kool-Thomas invariants of
local surfaces via the equivariant $K$-theoretic invariants proposed by
Nekrasov and Okounkov. Kool and Thomas defined the reduced obstruction theory
for the moduli of stable pairs $\mathcal{P}_{\chi}(X,i_{*}\beta)$ as the degree
of the virtual class $\left[\mathcal{P}_{\chi}(S,\beta)\right]^{red}$ after we
apply $\tau([pt])^{m}\in H^{*}(\mathcal{P}_{\chi}(X,i_{*}\beta),\mathbb{Z})$.
$\tau([pt])$ contain the information of the incidence of a point and a curve
supporting a $(\mathcal{F},s)$.
  $\textbf{Keywords: }$Kool-Thomas invariants, $K$-theoretic invariants,
G\"ottsche Shende invariants
