nLab
proxi-Banach ring

Contents

Contents

Idea and motivations

The notion of proxi-Banach ring is a generalization of the notion of Banach ring, essentially obtained by replacing the usual triangular inequality |a+b||a|+|b||a+b|\leq |a|+|b| that appears in the theory of metric spaces by the more natural (in a categorical sense) weak triangular inequality |a+b|Cmax(|a|,|b|)|a+b|\leq C\cdot\max(|a|,|b|) that appears in the theory of proxi-metric spaces.

Their use allows the definition of an + *\mathbb{R}_+^*-action on various categories of global analytic spaces.

They are also the natural objects that appear as π 0\pi_0 of spectral rings in spectral global analytic geometry.

Definition

A pseudo-Banach ring is a ring object in the rig category ( + Sets, , m)(\mathbb{R}_{+\leq}^\mathrm{Sets},\oplus_\infty,\otimes_m) of +\mathbb{R}_+-graded sets (with bounded maps between them). This means a set RR with a norm || R:R +|\cdot|_R:R\to \mathbb{R}_+ such that there exists CC, DD and EE, FF with

  • |a+b|Cmax(|a|,|b|)|a+b|\leq C\cdot \max(|a|,|b|),

  • |ab|D|a||b||ab|\leq D\cdot|a|\cdot|b|,

  • |a|E|a||-a|\leq E|a|,

  • |0|=0|0|=0,

  • |1|F1|1|\leq F\cdot 1 (one often supposes additionally that |1|=1|1|=1).

The free module

If XX is an +\mathbb{R}_+-graded set, and RR is a pseudo-Banach ring, one may define an associated pseudo-normed free module R (X)R^{(X)} by putting on the usual free module the following convenient grading (not given by the usual 1\ell^1 grading): if a x{x}\sum a_x\{x\} is an element with support supp(a)supp(a), one may write it a i{x i}\sum a_i\{x_i\}, and parenthesize it in binary terms. In the case of a support of cardinal three, we may write for example

P(a 1{x 1}+a 2{x 2}+a 3{x 3})=((a 1{x 1}+a 2{x 2})+a 3{x 3}).P(a_1\{x_1\}+a_2\{x_2\}+a_3\{x_3\})=((a_1\{x_1\}+a_2\{x_2\})+a_3\{x_3\}).

Each parenthesis contains only two terms, and there is a finite set of choices for the position of parenthesis, denoted 𝒫a(supp(a))\mathcal{P}a(supp(a)). We then define by induction (CC denotes the norm, i.e., infimum of the CC constants for the addition map), for PP an element of this set,

|(u)+(v)| C,P:=Cmax(|u| C,Pu,|v| C,Pv).|(u)+(v)|_{C,P}:=C\cdot \max(|u|_{C,Pu},|v|_{C,Pv}).

One then takes the maximum of all these expressions to define a natural pseudo-norm on R (X)R^{(X)} by

|a x{x}| ,C:=max P𝒫a(supp(a))|P(a x{x})| C,P. \left|\sum a_x\{x\}\right|_{\infty,C}:=\max_{P\in \mathcal{P}a(supp(a))}\left|P(\sum a_x\{x\})\right|_{C,P}.

If C=1C=1, i.e., in the non-archimedean case, this gives back the usual non-archimedean (maximum) seminorm on the free module.

Modules and their operations

Using the free module, and the coproduct and product of +\mathbb{R}_+-graded sets, one defines direct sums and tensor products of modules, and throught the symmetric algebra construction, the free +\mathbb{R}_+-graded algebra (algebra of convergent “power series” with coefficients in RR and polyradii in XX).

One may also work with the category of ind-pseudo-Banach ring, and of ind-pseudo-Banach modules over them, to develop an overconvergent version of the theory.

A natural flow

The action of + *\mathbb{R}_+^* on all these objects is simply given by acting on the grading through

|||| t:=e tln||.|\cdot|\mapsto |\cdot|^t:=e^{t\mathrm{ln}|\cdot|}.

global analytic geometry

overconvergent global analytic geometry

spectral global analytic geometry

Reference

For the category of +\mathbb{R}_+-graded sets:

Last revised on February 5, 2016 at 03:40:02. See the history of this page for a list of all contributions to it.