Author: Eiko

Time: 2025-03-23 22:06:39 - 2025-03-23 22:06:39 (UTC)

Reference: Arithmetic Jet Spaces

Notes taken when I try to study the paper Arithmetic Jet Spaces

Introduction

• There are the following viewpoints for giving a derivation on a ring $R$

  • A map $\phi :R\to R$ such that $\phi (ab)=\phi (a)b+a\phi (b)$.

  • A map in ${\mathrm{Hom}}_R({\mathrm{\Omega }}_R^1,R)$.

  • A map in ${\mathrm{Hom}}_{\mathbf{Rings}}(R,R[ϵ]/{ϵ}^2)$, i.e. a map of schemes $\mathrm{Spec}(R[ϵ]/{ϵ}^2)\to \mathrm{Spec}R$.

  • A vector field on $\mathrm{Spec}R$.

• One can construct a jet space for each $n$, let $D_n(R)=R[ϵ]/{ϵ}^{n+1}$.