Transchromatic twisted character maps

Authors: Nathaniel Stapleton

Publish Date: 2013/06/25

Volume: 10, Issue: 1, Pages: 29-61

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Abstract

In this paper we construct a refinement of the transchromatic generalized character maps by taking into account the torus action on the inertia groupoid also known as the Fix functor The relationship between this construction and the geometry of pdivisible groups is made preciseIn chromatic homotopy theory there is a history of trying to understand height n phenomena in terms of height n1 phenomena associated to the free loop space There is an S1action on the free loop space by rotation This action plays a key role in topological cyclic homology and the redshift conjecture such as in 3 and in Witten’s work on the elliptic genus see 10 11 In generalized character theory the S1action has been traditionally ignored In this work we describe a generalized character theory in which this S1action is accounted for and we explain the relationship between it and the geometry of pdivisible groupsWe also show how to canonically recover the transchromatic generalized character maps of 8 using the canonical map B t oversetlongrightarrow hatC t It should be noted that in personal correspondence Lurie has described a method for building the transchromatic twisted character maps from the transchromatic generalized character maps of 8This construction was inspired by the work of Ando and Morava in 1 Section 5 The idea behind the construction is that the ring B tprime has nt canonical points for the formal group mathbbG 0 = mathbbG L KtE n given by the set barq The elements of barA are invisible to mathbbG 0 in the sense that there are no maps L tx oversetlongrightarrow B tprime such that x mapsto A i for any i We formalize all of this in the next proposition using the language of formal algebraic geometry

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