Plethystic algebra

Abstract

The notion of a Z-algebra has a non-linear analogue, whose purpose it is to control operations on commutative rings rather than linear operations on abelian groups. These plethories can also be considered non-linear generalizations of cocommutative bialgebras. We establish a number of categorytheoretic facts about plethories and their actions, including a Tannaka-Kreinstyle reconstruction theorem. We show that the classical ring of Witt vectors, with all its concomitant structure, can be understood in a formula-free way in terms of a plethystic version of an affine blow-up applied to the plethory generated by the Frobenius map. We also discuss the linear and infinitesimal structure of plethories and explain how this gives Bloch's Frobenius operator on the de Rham-Witt complex.

The hunting of the Hopf ring

“…The case originally considered by Tall and Wraith in [TW70], under the name biring triple, was for V the category of commutative unital rings. Recently, Borger and Weiland [BW05] rediscovered this and extended it to the case where V is the category of commutative unital k-algebras, for a commutative unital ring k. They adopted the term plethory in that situation; thus a plethory is that-which-acts-on-algebras. This is clearly relevant to our purposes as unstable cohomology operations of multiplicative cohomology theories act on the cohomology algebras.…”

Arithmetic differential equations in several variables

“…also the first author's suggestion, in the "one prime case," in the Introduction of [9]. For the theory of lambda rings and the related theory of Witt rings we refer to [19,21,30,2,3].…”

STANDORT Info

“…Denote by k Alg l the category of formal algebra schemes and by k Alg This is a pro-version of what is called a k-l-biring in [TW70,BW05], but that terminology suggests a similarity with bialgebras, which is something completely different, so we will stick to our terminology.…”

“…Plethories were first introduced in [TW70] and were extensively studied in [BW05] from an algebraic point of view. These plethories, however, do not carry filtrations or pro-structures such as needed for topological applications.…”

“…which is equipped with natural transformations F → id and F → F ○ F such that coassociativity and counitality conditions are satisfied. The algebra of plethories was first studied by Tall and Wraith [TW70] and then extended by Borger and Wieland [BW05]. The aim of this paper is to extend the theory of plethories to the setting of graded formal schemes and to study linearizations of them.…”