### Plethystic algebra

J. Borger, B. Wieland, * Plethystic algebra*,
Advances in Mathematics **194/2** (2005), pp 246-283

**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 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.

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