  In chromatic homotopy theory, there is a well-known conjecture called
blue-shift phenomenon (BSP). In this paper, we propose a general blue-shift
phenomenon (GBSP) which unifies BSP and a new variant of BSP introduced by
Balmer-Sanders under one framework. To explain GBSP, we use the roots of
$p^j$-series of the formal group law of a complex-oriented spectrum $E$ in the
homotopy group of the generalized Tate spectrum of $E$. We also incorporate the
relationship between roots and coefficients of a polynomial in any commutative
ring. With this fresh perspective, we successfully achieve our goal of
explaining GBSP for certain abelian cases. Additionally, we establish that the
generalized Tate construction lowers Bousfield class, along with numerous Tate
vanishing results. These findings strengthen and extend previous theorems of
Balmer-Sanders and Ando-Morava-Sadofsky. While our approach only reproduces a
result of Barthel-Hausmann-Naumann-Nikolaus-Noel-Stapleton, it appears to be
more accessible for dealing with GBSP in non-abelian cases. Furthermore, our
approach simplifies the original proof of a result of
Bonventre-Guillou-Stapleton, indicating that its applications are not limited
to GBSP. As a result, our approach holds significant promise and merits further
study and application.
