Transchromatic Generalized Character Maps in Equivariant Stable Homotopy Theory

This monograph explores the rigorous construction and homotopical implications of transchromatic generalized character maps within the framework of equivariant stable homotopy theory. Building upon the foundational generalized character theory of Hopkins, Kuhn, and Ravenel, which relates height n cohomology to height 0 rational cohomology, we investigate the intermediate transchromatic shifts introduced by Stapleton. These maps, denoted n,t, provide a bridge between Morava E-theory of height n, En, and E-theories of lower height t < n, by exploiting the moduli of p-divisible groups and Drinfeld level structures. We present a detailed analysis of the algebraic geometry governing these maps, specifically the trivialization of isogenies over transchromatic extensions of the base rings. Furthermore, we establish the isomorphism of the transchromatic character map upon suitable localization, thereby providing a powerful computational tool for analyzing the equivariant cohomology of finite groups En*(BG) via the chromatic filtration.