nLab Archimedean ordered Artinian local ring

Context

Analysis

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Algebra

Formal geometry

Contents

Idea

Since every ordered Artinian local ring RR has characteristic zero, the positive integers +\mathbb{Z}_+ are a subset of RR, with injection i: +Ri:\mathbb{Z}_+ \hookrightarrow R. An Archimedean ordered Artinian local ring is an ordered Artinian local ring which satisfies the archimedean property: for all elements aRa \in R and bRb \in R, if 0<a0 \lt a and 0<b0 \lt b, then there exists a positive integer n +n \in \mathbb{Z}_+ such that such that a<i(n)ba \lt i(n) \cdot b.

Archimedean ordered Artinian local rings are important because they are the ordered local rings with nilpotent infinitesimals but no infinite elements or invertible infinitesimals, and thus play an important role in synthetic approaches to analysis and differential geometry.

Archimedean ordered fields are the Archimedean ordered Artinian local rings in which the nilradical is the trivial ideal, or equivalently, in which every non-invertible element is equal to zero.

Properties

Purely real and purely infinitesimal elements

Suppose that KK is an Archimedean ordered field and AA is an Archimedean ordered Artinian local KK-algebra. Since AA is a local ring, the quotient of AA by its ideal of non-invertible elements DD is the residue field KK itself, and the canonical function used in defining the quotient is the function :AK\Re:A \to K which takes a number aAa \in A to its purely real component (a)K\Re(a) \in K. Since AA is an ordered KK-algebra, there is a strictly monotone ring homomorphism h:KAh:K \to A. An element aAa \in A is purely real if h((a))=ah(\Re(a)) = a, and an element aAa \in A is purely infinitesimal if it is in the fiber of \Re at 0K0 \in K. Zero is the only element in AA which is both purely real and purely infinitesimal.

Prelattice structure

Now, suppose that the Archimedean ordered field KK has lattice structure min:K×KK\min:K \times K \to K and max:K×KK\max:K \times K \to K. Then the Archimedean ordered local KK-algebra AA has a prelattice structure given by functions ()():A×AA(-)\wedge(-):A \times A \to A and ()():A×AA(-)\vee(-):A \times A \to A, defined by

abh(min((a),(b)))a \wedge b \coloneqq h(\min(\Re(a), \Re(b)))

and

abh(max((a),(b)))a \vee b \coloneqq h(\max(\Re(a), \Re(b)))

Pseudometric and seminorm structure

The distance function on AA is given by the function ρ:A×AK\rho:A \times A \to K, defined as

ρ(a,b)max((a),(b))min((a),(b))\rho(a, b) \coloneqq \max(\Re(a), \Re(b)) - \min(\Re(a), \Re(b))

and the absolute value on AA is given by the function ||:A×AK\vert-\vert:A \times A \to K, defined as

|a|ρ(a,0)\vert a \vert \coloneqq \rho(a, 0)

The distance function and absolute value are pseudometrics and multiplicative seminorms, because every Archimedean ordered field KK embeds into the real numbers \mathbb{R}, and since min(a,b)max(a,b)\min(a, b) \leq \max(a, b), the pseudometric and seminorm are always non-negative. This also implies that in every Archimedean ordered field with lattice structure, the pseudometric defined above is a metric.

Smooth and differentiable function structure

Suppose that KK is an Archimedean ordered field with lattice structure, and AA is an Archimedean ordered Artinian local K K -algebra. Then continuous functions, differentiable functions, and smooth functions are each definable on KK using the algebraic, order, and metric structure on KK.

The ring homomorphism h:KAh:K \to A preserves smooth functions: given a natural number nn \in \mathbb{N} and a purely infinitesimal element ϵD\epsilon \in D such that ϵ n+1=0\epsilon^{n + 1} = 0, then for every smooth function fC ω(K)f \in C^\omega(K), there is a function f A:AAf_A:A \to A such that for all elements xKx \in K, f A(h(x))=h(f(x))f_A(h(x)) = h(f(x)) and

f A(h(x)+ϵ)= i=0 n1i!h(d ifdx i(x))ϵ if_A(h(x) + \epsilon) = \sum_{i = 0}^{n} \frac{1}{i!} h\left(\frac{d^i f}{d x^i}(x)\right) \epsilon^i

If we restrict to Archimedean ordered Artinian local KK-algebras AA where every element of the nilradical DD is a nilsquare element, where for all ϵD\epsilon \in D, ϵ 2=0\epsilon^2 = 0, then the ring homomorphism h:KAh:K \to A preserves differentiable functions; for every differentiable function f:KKf:K \to K with given derivative f:KKf':K \to K, there is a function f A:AAf_A:A \to A such that for all elements xKx \in K and nilpotent elements ϵD\epsilon \in D, f A(h(x))=h(f(x))f_A(h(x)) = h(f(x)) and

f A(h(x)+ϵ)=h(f(x))+h(f(x))ϵf_A(h(x) + \epsilon) = h(f(x)) + h(f'(x)) \epsilon

In particular, every polynomial function p:KKp:K \to K lifts to a polynomial function p A:AAp_A:A \to A.

Alternatively, one could use this property to define differentiable and smooth functions in KK, such as the exponential function, natural logarithm, sine function, and cosine function.

One could also work with partial functions instead. Given a predicate PP on the real numbers \mathbb{R}, let II denote the set of all elements in \mathbb{R} for which PP holds. A partial function f:f:\mathbb{R} \to \mathbb{R} is equivalently a function f:If:I \to \mathbb{R} for any such predicate PP and set II.

A function f:If:I \to \mathbb{R} is smooth at a subset SIS \subseteq I with injection j:Sj:S \hookrightarrow \mathbb{R} if it has a function d fdx :×S\frac{d^{-} f}{d x^{-}}:\mathbb{N} \times S \to \mathbb{R} with (D 0j)(a)=a(D^0 j)(a) = a for all aSa \in S, such that for all Archimedean ordered Artinian local \mathbb{R}-algebras AA with ring homomorphism h A:Ah_A:\mathbb{R} \to A, natural numbers nn \in \mathbb{N}, and purely infinitesimal elements ϵD\epsilon \in D such that ϵ n+1=0\epsilon^{n + 1} = 0

f A(h A(j(a))+ϵ)= i=0 n1i!h A(d ifdx i(a))ϵ if_A(h_A(j(a)) + \epsilon) = \sum_{i = 0}^{n} \frac{1}{i!} h_A\left(\frac{d^i f}{d x^i}\left(a\right)\right) \epsilon^i

A function f:If:I \to \mathbb{R} is smooth at an element aIa \in I if it is smooth at the singleton subset {a}\{a\}, and a function f:If:I \to \mathbb{R} is smooth if it is smooth at the improper subset of II.

A function f:If:I \to \mathbb{R} is differentiable at a subset SIS \subseteq I with injection j:Sj:S \hookrightarrow \mathbb{R} if it has a function dfdx:S\frac{d f}{d x}:S \to \mathbb{R} such that for all Archimedean ordered Artinian local KK-algebras AA with ring homomorphism h:KAh:K \to A such that for all ϵD\epsilon \in D, ϵ 2=0\epsilon^2 = 0, for all nilpotent elements ϵD\epsilon \in D,

f A(h(j(a))+ϵ)=h(j(a))+h(dfdx(a))ϵf_A(h(j(a)) + \epsilon) = h(j(a)) + h\left(\frac{d f}{d x}(a)\right) \epsilon

A function f:If:I \to \mathbb{R} is differentiable at an element aIa \in I if it is differentiable at the singleton subset {a}\{a\}, and a function f:If:I \to \mathbb{R} is differentiable if it is differentiable at the improper subset of II.

Square roots and Euclidean pseudometric structure

Now, assume that KK is an Euclidean field as well, in addition to being an Archimedean ordered field. While KK has a principal square root function :[0,)[0,)\sqrt{-}:[0, \infty) \to [0, \infty), not every Archimedean ordered local KK-algebra AA has a principal square root function :[0,)[0,)\sqrt{-}:[0, \infty) \to [0, \infty), because purely infinitesimal elements in AA are not guaranteed to have square roots. An Archimedean ordered Artinian local KK-algebra is Euclidean if every nilpotent element has a square root.

However, given an Archimedean ordered Artinian local KK-algebra AA, every rank nn AA-module VV with basis v:Fin(n)Vv:\mathrm{Fin}(n) \to V has a Euclidean pseudometric ρ V:V×VK\rho_V:V \times V \to K, given by

ρ V(a,b) iFin(n)ρ(a i,b i) 2\rho_V(a, b) \coloneqq \sqrt{\sum_{i \in \mathrm{Fin}(n)} \rho(a_i, b_i)^2}

for module elements aVa \in V and bVb \in V and scalars a iAa_i \in A and b iAb_i \in A for index iFin(n)i \in \mathrm{Fin}(n), where

a= iFin(n)a iv ib= iFin(n)b iv ia = \sum_{i \in \mathrm{Fin}(n)} a_i v_i \quad b = \sum_{i \in \mathrm{Fin}(n)} b_i v_i

If AA is an ordered field, then the Euclidean pseudometric on VV is a metric.

See also

Last revised on January 13, 2023 at 05:38:36. See the history of this page for a list of all contributions to it.