Plethystic algebra

By James M. Borger, Benjamin Wieland. 2004

52 citations recorded by the source corpus.

. The notion of a Z -algebra has a non-linear analogue, whose purpose it is to control operations on commutative rings rather than linear opera- tions on abelian groups. These plethories can also be considered non-linear gen-eralizations of cocommutative bialgebras. We establish a number of category- theoretic facts about plethories and their actions, including a Tannaka–Krein-style 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.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/ef1fc5a5fc387e13da29743a1e6c80c6dca1316b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Charles Rezk
2. James M. Borger
3. L. Hesselholt
4. David I. Spivak
5. C. Berger
6. Ieke Moerdijk
7. Kathryn Hess
8. W. Dwyer

In Geometry and Topology

1. Charles Rezk
2. James M. Borger
3. L. Hesselholt
4. Carlos Simpson
5. David I. Spivak
6. C. Berger
7. Emily Riehl
8. Michael Shulman

In Statistical and Nonlinear Physics

1. Charles Rezk
2. James M. Borger

Citations in the corpus

• The congruence criterion for power operations in Morava $E$-theory (2009, background): “the notion of Hopf algebroids (or more precisely, of “affine category schemes”), an observation we will make explicit below (§5.7). I’ve taken the terminology “twisted bialgebra” and some notation from [BW05]. 22 CHARLES REZK 5.1. Bimodules, multimorphisms, and functors. Let Rbe a commutative ring, an”
• Lambda-rings and the field with one element (2009, background): “(See Borger–Wieland [5].)”
• The basic geometry of Witt vectors, I: The affine case (2008, background): “(In the language of schemes, one would say this makes Spec3R,E an R-algebra scheme over R; or in the language of [Borger and Wieland 2005], it makes 3R,E an R-Rbiring.”
• The basic geometry of Witt vectors (2008, background): “Again, this follows from the ghost principle.”