nLab analytic ring

Definition

Definition

(ScholzeLCM, Def. 7.1)

In condensed mathematics, a pre-analytic ring is a condensed ring $\underline{A}$ together with a functor $S\mapsto\mathcal{A}[S]$ from extremally disconnected topological spaces to $\underline{\mathcal{A}}$-modules in condensed abelian groups taking finite disjoint unions to products, and a natural transformation $S \to\mathcal{A}[S]$.

Definition

(ScholzeLCM, Def. 7.4) An analytic ring is a pre-analytic ring $\mathcal{A}$ such that for any chain complex $C$ of $\underline{\mathcal{A}}$-modules in condensed abelian groups such that all $C_{i}$ are direct sums of objects of the form $\mathcal{A}[T]$ for varying extremally disconnected $T$, the map

$$R\underline{\Hom}_{\underline{\mathcal{A}}}(\mathcal{A}[S],C)\to R\underline{\Hom}_{\underline{\mathcal{A}}}(\mathcal{A}[S],C)$$

of condensed abelian groups is an isomorphism for all extremally disconnected $S$.

Examples

The following examples come from ScholzeLCM, Examples 7.3, as examples of pre-analytic rings. They are shown to be analytic rings as well later on in the same reference.

• The analytic ring $\mathbb{Z}_{\square}$ is given by $\underline{\mathcal{A}}=\mathbb{Z}$ and $S\mapsto \mathbb{Z}[S]^{\square}$ (in the notation of solid abelian group).

• For $A$ a discrete ring, the analytic ring $(A,\mathbb{Z})_{\square}$ is given by $\underline{\mathcal{A}}=A$ as a condensed ring and $S\mapsto \mathbb{Z}_{\square}[S]\otimes_{\mathbb{Z}} A$.

• For $A$ a finitely generated $\mathbb{Z}$-algebra, the condensed ring $A_{\square}$ is given by $\underline{\mathcal{A}}=A$ as a condensed ring and $S\mapsto A_{\square}[S] \;\coloneqq\; \underset {\underset{i}{\leftarrow}} {\lim} \; A\big[S_{i}\big] \,$.

Related concepts

• condensed mathematics

• solid module

References

• Peter Scholze, Lectures on condensed mathematics, pdf