  We prove a generalized vanishing theorem for certain quasi-coherent sheaves
along the derived blow-ups of quasi-smooth derived Artin stacks. We give four
applications of the generalized vanishing theorem: we prove a $K$-theoretic
version of the generalized vanishing theorem which verified a conjecture of the
author and give a new proof of the $K$-theoretic virtual localization theorem
for quasi-smooth derived schemes through the intrinsic blow-up theory of
Kiem-Li-Savvas; we prove a desingularization theorem for quasi-smooth derived
schemes and give an approximation formula for the virtual fundamental classes;
we give a resolution of the diagonal along the projection map of blow-ups of
smooth varieties, which strengthens the semi-orthogonal decomposition theorem
of Orlov; we illustrate the relation between the generalized vanishing theorem
and weak categorifications of quantum loop and toroidal algebra actions on the
derived category of Nakajima quiver varieties. We also propose several
conjectures related to birational geometry and the $L_{\infty}$-algebroid of
Calaque-C\u{a}ld\u{a}raru-Tu.
