A General Blue-Shift Phenomenon

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, which provides the first example of Tate blue-shift with height-shifting at arbitrary positive integer in this setting. 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, and reproduce a result of Barthel--Hausmann--Naumann--Nikolaus--Noel--Stapleton. Furthermore, our approach simplifies the original proof of a result of Bonventre--Guillou--Stapleton, indicating that its applications are not limited to GBSP. Our work pioneers the use of commutative algebra to explain the chromatic height-shifting behavior in the blue-shift phenomenon.