nLab
Z-infinity-module

Contents

Idea

The idea is that there should be some kind of “complete local ring \mathbb{Z}_\infty corresponding to the archimedean valuation on \mathbb{Q}, by analogy with the (genuine!) complete local rings p\mathbb{Z}_p corresponding to the (non-archimedean) pp-adic valuations on \mathbb{Q}: the p-adic integers. However, the naïve approach taking

={x:1x1}\mathbb{Z}_\infty = \lbrace x \in \mathbb{R} : -1 \le x \le 1 \rbrace

fails because this is not a subring of \mathbb{R}!

Nikolai Durov’s definition of \mathbb{Z}_\infty is inspired by classical Arakelov geometry and starts with the observation that any p\mathbb{Z}_p-lattice Λ\Lambda in a finite-dimensional p\mathbb{Q}_p-vector space EE defines a maximal compact subgroup submonoid M ΛM_\Lambda of End(E)End(E), and M Λ=M ΛM_\Lambda = M_{\Lambda'} if and only if Λ\Lambda and Λ\Lambda' are similar lattices; accordingly, a \mathbb{Z}_\infty-lattice up-to-similarity should correspond to maximal compact submonoids of End(E)End(E) for a finite-dimensional \mathbb{R}-vector space EE, i.e. the monoid of \mathbb{R}-linear endomorphisms of EE that are short with respect to some norm on EE. Thus, Durov defines a \mathbb{Z}_\infty-lattice to be a finite-dimensional normed \mathbb{R}-vector space, and a morphism of \mathbb{Z}_\infty lattices is defined to be a short \mathbb{R}-linear map. This gives a category -Lat\mathbb{Z}_{\infty}\mathbf{\text{-Lat}} with finite limits and colimits. (Note, however, that it is not an additive category!)

Now, notice that every flat p\mathbb{Z}_p-module is the filtered colimit of its finite-dimensional p\mathbb{Z}_p-submodules (which are necessarily free because p\mathbb{Z}_p is a local ring), and in fact the category of flat p\mathbb{Z}_p-modules is equivalent to the category of ind-objects of p-Lat\mathbb{Z}_{p}\mathbf{\text{-Lat}}. So we may define a flat \mathbb{Z}_\infty-module to be an ind-object of -Lat\mathbb{Z}_{\infty}\mathbf{\text{-Lat}}. One may also give the following explicit description: a flat \mathbb{Z}_\infty-module EE is a (possibly infinite-dimensional) \mathbb{R}-vector space E E_{\mathbb{R}} together with a symmetric convex body E E_{\mathbb{Z}_\infty}, and a morphism EEE \to E' is an \mathbb{R}-linear map E E E_{\mathbb{R}} \to E'_{\mathbb{R}} that restricts to a map E E E_{\mathbb{Z}_\infty} \to E'_{\mathbb{Z}_\infty}. This defines a category -FlMod\mathbb{Z}_{\infty}\mathbf{\text{-FlMod}}.

Finally, noting that the forgetful functor U: -FlModSetU : \mathbb{Z}_{\infty}\mathbf{\text{-FlMod}} \to \mathbf{\text{Set}} taking a flat \mathbb{Z}_\infty-module EE to its underlying symmetric convex body E E_{\mathbb{Z}_\infty} has a left adjoint FF, Durov defines a (not necessarily flat) \mathbb{Z}_\infty-module to be a module for the induced monad Σ =UF\Sigma_\infty = U F. The comparison functor embeds -FlMod\mathbb{Z}_{\infty}\mathbf{\text{-FlMod}} as a full subcategory of -Mod\mathbb{Z}_{\infty}\mathbf{\text{-Mod}}.

Definition

Let Σ :SetSet\Sigma_\infty : \mathbf{\text{Set}} \to \mathbf{\text{Set}} be the functor sending a set SS to the set

{v (S):v 1= sS|v s|1}\left\lbrace \vec{v} \in \mathbb{R}^{(S)} : \left\| \vec{v} \right\|_1 = \sum_{s \in S} \left| v_s \right| \le 1 \right\rbrace

i.e. the solid regular cross-polytope with SS-many vertices. The action of Σ \Sigma_\infty on maps of sets is the obvious one. Let η:idΣ \eta : id \Rightarrow \Sigma_\infty be the natural transformation given by insertion of generators, and let μ:Σ Σ Σ \mu : \Sigma_\infty \Sigma_\infty \Rightarrow \Sigma_\infty be the natural transformation given by “evaluation” of “octahedral” combinations:

iα iη( jβ i,jη(s j)) i,jα iβ i,jη(s j)\sum_i \alpha_i \eta \left( \sum_j \beta_{i,j} \eta (s_j) \right) \mapsto \sum_{i, j} \alpha_i \beta_{i, j} \eta (s_j)

One may verify that this defines a monad (Σ ,η,μ)(\Sigma_\infty, \eta, \mu) on Set\mathbf{\text{Set}}. A \mathbb{Z}_\infty-module is defined to be a module for this monad.

Properties

The Σ \Sigma_\infty monad is a monad with arities: the category of arities may be taken to be FinSet\mathbf{\text{FinSet}}.

The \mathbb{Z}_\infty-module structure on a set MM is entirely determined by the map α 2:Σ (2)×M 2M\alpha_2 : \Sigma_\infty (2) \times M^2 \to M given by ((λ 1,λ 2),(x 1,x 2))λ 1x 1+λ 2x 2((\lambda_1, \lambda_2), (x_1, x_2)) \mapsto \lambda_1 x_1 + \lambda_2 x_2. Conversely, a set MM together with an element α 0\alpha_0 and a map α 2:Σ (2)×M 2M\alpha_2 : \Sigma_\infty (2) \times M^2 \to M satisfying certain equations is a \mathbb{Z}_\infty-module. A map commuting with α 0\alpha_0 and α 2\alpha_2 is a homomorphism of \mathbb{Z}_\infty-modules, thus the theory of \mathbb{Z}_\infty-modules is a finitary algebraic theory, with all that this implies.

The category -Mod\mathbb{Z}_{\infty}\mathbf{\text{-Mod}} has the following properties:

Examples

Every normed \mathbb{R}-vector space VV induces a flat \mathbb{Z}_\infty-module EE where E =VE_{\mathbb{R}} = V and E ={vV:v1}E_{\mathbb{Z}_\infty} = \left\lbrace \vec{v} \in V : \left\| \vec{v} \right\| \le 1 \right\rbrace. (In the other direction, every flat \mathbb{Z}_\infty-module induces a seminorm on the underlying \mathbb{R}-vector space.)

A finitely-generated flat \mathbb{Z}-module is the symmetric convex hull of a finite set of vectors.

References

Revised on August 13, 2012 10:45:58 by Urs Schreiber (89.204.137.247)