Related papers: The basic geometry of Witt vectors, II: Spaces
We give a concrete description of the category of etale algebras over the ring of Witt vectors of a given finite length with entries in an arbitrary ring. We do this not only for the classical p-typical and big Witt vector functors but also…
We give a $K$-theoretic account of the basic properties of Witt vectors. Along the way we re-prove basic properties of the little-known Witt vector norm, give a characterization of Witt vectors in terms of algebraic $K$-theory, and a…
This is the first part of a 2 part survey on the functor of the big and p-adic Witt vectors.
The classical 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…
For any $\pi$-formal group scheme $G$, the Frobenius morphism between arithmetic jet spaces restricts to generalized kernels of the projection map. Using the functorial properties of such kernels of arithmetic jet spaces, we show that this…
The ring of Witt vectors $\mathbb{W} R$ over a base ring $R$ is an important tool in algebraic number theory and lies at the foundations of modern $p$-adic Hodge theory. $\mathbb{W} R$ has the interesting property that it constructs a ring…
The explicit formulas of operations, in particular addition and multiplication, of $p $-adic integers are presented. As applications of the results, at first the explicit formulas of operations of Witt vectors with coefficients in…
We show that various flavors of Witt vectors are functorial with respect to multiplicative polynomial laws of finite degree. We then deduce that the $p$-typical Witt vectors are functorial in multiplicative polynomial maps of degree at most…
This work provides a unified formalism for studying difference and (Hasse-) differential algebraic geometry, by introducing a theory of "iterative Hasse rings and schemes". As an application, Hasse jet spaces are constructed generally,…
The purpose of this article is to prove some results on the Witt vectors of perfect $\mathbf{F}_p$-algebras. Let $A$ be a perfect $\mathbf{F}_p$-algebra for a prime integer $p$ and assume that $A$ has the property $\mathbf{P}$. Then does…
We extend the big and $p$-typical Witt vector functors from commutative rings to commutative semirings. In the case of the big Witt vectors, this is a repackaging of some standard facts about monomial and Schur positivity in the…
An introductory overview of vector spaces, algebras, and linear geometries over an arbitrary commutative field is given. Quotient spaces are emphasized and used in constructing the exterior and the symmetric algebras of a vector space.…
Since the results here have been superseded by another paper cowritten by the author, this article is available for reference purposes only. Fix a Dedekind domain $\mathcal{O}$ and a non-zero prime $\mathfrak{p}$ in it along with a…
Let R be a perfect F_p-algebra, equipped with the trivial norm. Let W(R) be the ring of p-typical Witt vectors over R, equipped with the p-adic norm. At the level of nonarchimedean analytic spaces (in the sense of Berkovich), we demonstrate…
We give a direct construction of the ring spectrum of spherical Witt vectors of a perfect $\mathbb{F}_p$-algebra R as the completion of the spherical monoid algebra $\mathbb{S}[R]$ of the multiplicative monoid $(R,\cdot)$ at the ideal $I =…
This is an introduction to the theory of Witt vectors. It includes a construction of the Witt rings, the Frobenius and Verschiebung endomorphisms, the canonical map from W to W^2 (its lambda-algebra structure), the relation to strict…
The first part of the paper develops the theory of $m$-shifted $\pi$-typical Witt vectors which can be viewed as subobjects of the usual $\pi$-typical Witt vectors. We show that the shifted Witt vectors admit a delta structure that satisfy…
The rings of $p$-typical Witt vectors are interpreted as spaces of vanishing cycles for some perverse sheaves over a disc. This allows to "localize"\ an isomorphism emerging in Drinfeld's theory of prismatization [Dr], Prop. 3.5.1, namely…
In this paper, we collect the fundamental basic properties of jet modules in algebraic geometry and related properties of differential operators. We claim no originality but we want to provide a reference work for own research and the…
For every commutative ring $A$, one has a functorial commutative ring $W(A)$ of $p$-typical Witt vectors of $A$, an iterated extension of $A$ by itself. If $A$ is not commutative, it has been known since the pioneering work of L. Hesselholt…