Let Wm be the Witt ring of length m, ψ0 a character of Wm(Fp) into Q of exact order pm, and ψ the lift of ψ0 to Wm(Fq). Let J be a subset of {1, · · · , n}, f a nonzero morphism over Fq from n ∏ i=1,i6∈J Gm × ∏ i∈J Ga to Wm, and ∆ its Newton polyhedron. Let L(f, T ) be the Lfunction of the exponential sum associated to f and ψ. Suppose that f is commode and regular with respect to ∆. Then (i) t...

We describe an alternate construction of some of the basic rings introduced by Fontaine in p-adic Hodge theory. In our construction, the central role is played by the ring of p-typical Witt vectors over a p-adic valuation ring, rather than theWitt vectors over a ring of positive characteristic. This suggests the possibility of forming a meaningful global analogue of p-adic Hodge theory.

Let D be a division ring with an involution. Assuming that D admits Baer orderings, we can study the Witt group of hermitian froms over D by observing its image in the ring of continuous functions on the space of orderings. We are led to define a new class of rings which, when viewed in an abstract setting, provide a natural generalization of the spaces of orderings and real spectra studied in ...

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 theor...

In a previous paper, we stated a general almost purity theorem in the style of Faltings: if R is a ring for which the Frobenius maps on finite p-typical Witt vectors over R are surjective, then the integral closure of R in a finite étale extension of R[p−1] is “almost” finite étale over R. Here, we use almost purity to lift the finite étale extension of R[p−1] to a finite étale extension of rin...

Abstract For a not-necessarily commutative ring $R$ we define an abelian group $W(R;M)$ of Witt vectors with coefficients in -bimodule $M$ . These groups generalize the usual big rings and prove that they have analogous formal properties structure. One main result is $W(R) := W(R;R)$ Morita invariant -linear endomorphism $f$ finitely generated projective -module characteristic element $\chi _f ...

The p-typical Witt vectors are a ubiquitous object in algebra and number theory. They arise as a functorial construction that takes perfect fields k of prime characteristic p > 0 to p-adically complete discrete valuation rings of characteristic 0 with residue field k and are universal in that sense. A. Dress and C. Siebeneicher generalized this construction by producing a functor WG attached to...

We define the universal thickening of the field of real numbers. This construction is performed in three steps which parallel the universal perfection, the Witt construction and a completion process. We show that the transposition of the perfection process at the real archimedean place is identical to the “dequantization” process and yields Viro’s tropical real hyperfield R[. Then, we prove tha...