nLab
Z-infinity-module

Idea

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

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

fails because this is not a subring of $ℝ$!

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

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

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

Definition

Let ${\Sigma }_{\mathrm{\infty }}:\text{Set}\to \text{Set}$ be the functor sending a set $S$ to the set

$$\{\stackrel{\rightharpoonup }{v}\in {ℝ}^{(S)}:{\parallel \stackrel{\rightharpoonup }{v}\parallel }_1=\sum _{s\in S}\mid v_s\mid \le 1\}$$\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 $S$-many vertices. The action of ${\Sigma }_{\mathrm{\infty }}$ on maps of sets is the obvious one. Let $\eta :\mathrm{id}\Rightarrow {\Sigma }_{\mathrm{\infty }}$ be the natural transformation given by insertion of generators, and let $\mu :{\Sigma }_{\mathrm{\infty }}{\Sigma }_{\mathrm{\infty }}\Rightarrow {\Sigma }_{\mathrm{\infty }}$ be the natural transformation given by “evaluation” of “octahedral” combinations:

$$\sum _i{\alpha }_i\eta (\sum _j{\beta }_{i,j}\eta (s_j))\mapsto \sum _{i,j}{\alpha }_i{\beta }_{i,j}\eta (s_j)$$\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 $({\Sigma }_{\mathrm{\infty }},\eta ,\mu )$ on $\text{Set}$. A ${\mathbb{Z}}_{\mathrm{\infty }}$-module is defined to be a module for this monad.

Properties

The ${\Sigma }_{\mathrm{\infty }}$ monad is a monad with arities: the category of arities may be taken to be $\text{FinSet}$.

The ${\mathbb{Z}}_{\mathrm{\infty }}$-module structure on a set $M$ is entirely determined by the map ${\alpha }_2:{\Sigma }_{\mathrm{\infty }}(2)\times M^2\to M$ given by $(({\lambda }_1,{\lambda }_2),(x_1,x_2))\mapsto {\lambda }_1{x}_1+{\lambda }_2{x}_2$. Conversely, a set $M$ together with an element ${\alpha }_0$ and a map ${\alpha }_2:{\Sigma }_{\mathrm{\infty }}(2)\times M^2\to M$ satisfying certain equations is a ${\mathbb{Z}}_{\mathrm{\infty }}$-module. A map commuting with ${\alpha }_0$ and ${\alpha }_2$ is a homomorphism of ${\mathbb{Z}}_{\mathrm{\infty }}$-modules, thus the theory of ${\mathbb{Z}}_{\mathrm{\infty }}$-modules is a finitary algebraic theory, with all that this implies.

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

• It is a complete category (with the forgetful functor creating all limits).

• It is a cocomplete category (with the forgetful functor creating all filtered colimits).

• It has a zero object.

• It has a tensor product and an internal hom, making it into a symmetric monoidal closed category.

Examples

Every normed $ℝ$-vector space $V$ induces a flat ${\mathbb{Z}}_{\mathrm{\infty }}$-module $E$ where $E_{ℝ}=V$ and $E_{{\mathbb{Z}}_{\mathrm{\infty }}}=\{\stackrel{\rightharpoonup }{v}\in V:\parallel \stackrel{\rightharpoonup }{v}\parallel \le 1\}$. (In the other direction, every flat ${\mathbb{Z}}_{\mathrm{\infty }}$-module induces a seminorm on the underlying $ℝ$-vector space.)

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

References

• Nikolai Durov, New approach to Arakelov theory. Ph.D. thesis (2007) arXiv:0704.2030.