Solid Abelian Groups in Condensed Mathematics

• Solid abelian groups are condensed abelian groups satisfying a universal extension property derived from profinite completions, bridging algebra and topology.
• They exhibit robust categorical properties, being abelian with all small limits and colimits, and feature a symmetric monoidal solidification functor.
• They provide a framework for analytic geometry and homological algebra, integrating classical algebra with modern cohomology theories in condensed mathematics.

Solid abelian groups constitute a central structure within the theory of condensed mathematics, introduced by Clausen and Scholze, and play a critical role in the formalization of completeness for abelian groups in a sheaf-theoretic analytic framework. They are defined as a subcategory of condensed abelian groups, characterized by a certain universal extension property, and exhibit structural parallels with complete topological groups. The well-behaved categorical and homological properties of solid abelian groups equip them as foundational objects for advanced developments in analytic geometry, homological algebra, and modern cohomology theories involving profinite and condensed structures (Tang, 2024, Asgeirsson, 2023).

1. Background: Condensed Mathematics and Abelian Groups

Condensed mathematics replaces the classical notion of topological spaces with condensed sets—sheaves on the site of compact Hausdorff extremally disconnected spaces (CHED) with covers given by finitely many jointly surjective maps. For a topological space $T$, the condensed set $\underline{T}$ is given by $S \mapsto C(S, T)$, the set of continuous maps from $S$ to $T$. This formalism extends naturally to condensed abelian groups (CondAb), rings (CondRing), and modules by using sheaves of abelian groups, rings, or modules, respectively.

CondAb forms a Grothendieck abelian category, possessing all small limits, colimits, and enough projectives. It is equipped with a closed symmetric monoidal structure $(\CondAb, \otimes, \mathbb{Z})$, where tensor products and internal Homs are defined sectionwise and then sheafified: $(M \otimes N)(S) = (M(S) \otimes_\mathbb{Z} N(S))^{\mathrm{sh}}, \qquad \underline{\Hom}(M,N)(S) = \Hom_{\CondAb}(\mathbb{Z}[S] \otimes M, N).$ For a condensed ring $R$, the category of condensed $R$-modules, $\CMod(R)$, is equivalent to the category of $\underline{T}$0-module objects in this monoidal category.

2. Definition and Universal Property of Solid Abelian Groups

A condensed abelian group $\underline{T}$1 is called solid if for every profinite set $\underline{T}$2 (inverse limit of finite discrete spaces), the natural map

$\underline{T}$3

is a bijection, where $\underline{T}$4 is the pro-completion of the free condensed abelian group on $\underline{T}$5.

Equivalently, $\underline{T}$6 is solid if every map of condensed sets $\underline{T}$7 extends uniquely to a morphism of condensed abelian groups $\underline{T}$8 (Tang, 2024, Asgeirsson, 2023). This universal extension property ensures that solid groups model a notion of algebraic completeness analogous to completed topological modules.

Solid abelian groups form a full subcategory $\underline{T}$9. Analogously, a condensed ring $S \mapsto C(S, T)$0 is solid if $S \mapsto C(S, T)$1, and a module $S \mapsto C(S, T)$2 over $S \mapsto C(S, T)$3 is a solid $S \mapsto C(S, T)$4-module if its underlying condensed abelian group is solid.

3. Structural and Categorical Properties

The subcategory $S \mapsto C(S, T)$5 possesses favorable structural and categorical properties (Tang, 2024):

• It is an abelian (Grothendieck) subcategory of $S \mapsto C(S, T)$6, closed under all small limits, colimits, and extensions.
• The free solid groups $S \mapsto C(S, T)$7 are compact projective generators for $S \mapsto C(S, T)$8. Every solid abelian group can be constructed as a colimit of such building blocks.
• The inclusion functor $S \mapsto C(S, T)$9 admits a left adjoint—the solidification functor $S$0, which preserves colimits and is symmetric monoidal.
• There exists a unique closed symmetric monoidal structure on $S$1 making solidification symmetric monoidal: $S$2 and, for free solid groups,

$S$3

For any condensed ring $S$4, the same arguments yield that the category $S$5 of solid $S$6-modules is abelian, closed under limits, colimits, and extensions, and is generated by $S$7 with a left adjoint solidification functor.

4. Generation and Existence: The Role of Nöbeling’s Theorem

The construction of nontrivial solid abelian groups fundamentally relies on Nöbeling’s theorem. For a profinite space $S$8, Nöbeling’s theorem states that the discrete abelian group of continuous maps $S$9 is free as a $T$0-module. In the condensed setting, $T$1, ensuring these completions are massive but nonzero solid abelian groups (Asgeirsson, 2023).

Nöbeling’s theorem ensures that $T$2 for any profinite $T$3, making these the essential atomic objects in the construction of all solid abelian groups via colimits. Without this result, the entire theory would lack nontrivial examples.

The formal proof of Nöbeling’s theorem, recently mechanized in the Lean theorem prover, involves a sophisticated induction on ordinals, with three principal phases:

• Reduction to closed subsets of Boolean cubes.
• Construction of an explicit $T$4-basis indexed by finite strictly decreasing sequences.
• Proof of linear independence by transfinite induction, using exact sequences and directed unions.

5. Solid Structures Associated to Profinite Rings and Modules

Given a profinite ring $T$5, viewed as an inverse limit of finite discrete rings, the associated condensed ring $T$6 is solid and analytic.

For the category $T$7 of profinite topological $T$8-modules, the compact-open condensed module $T$9 yields a fully faithful and exact functor to $(\CondAb, \otimes, \mathbb{Z})$0. Furthermore, every profinite module maps to a solid module over $(\CondAb, \otimes, \mathbb{Z})$1 (Tang, 2024). The embedding

$(\CondAb, \otimes, \mathbb{Z})$2

is fully faithful and exact, and preserves projectives, implying:

• The $(\CondAb, \otimes, \mathbb{Z})$3 functor is preserved: $(\CondAb, \otimes, \mathbb{Z})$4 for all $(\CondAb, \otimes, \mathbb{Z})$5.
• The completed tensor product $(\CondAb, \otimes, \mathbb{Z})$6 corresponds to the solid tensor product: $(\CondAb, \otimes, \mathbb{Z})$7 in $(\CondAb, \otimes, \mathbb{Z})$8.

Thus, the homological algebra of profinite modules is faithfully embedded into that of solid modules, extending classical theory fully into the condensed framework.

6. Analyticity, Homological Algebra, and Applications

Any condensed ring $(\CondAb, \otimes, \mathbb{Z})$9 equipped with suitable functorial data is called analytic if, for bounded-above complexes $(M \otimes N)(S) = (M(S) \otimes_\mathbb{Z} N(S))^{\mathrm{sh}}, \qquad \underline{\Hom}(M,N)(S) = \Hom_{\CondAb}(\mathbb{Z}[S] \otimes M, N).$0 of sums of objects $(M \otimes N)(S) = (M(S) \otimes_\mathbb{Z} N(S))^{\mathrm{sh}}, \qquad \underline{\Hom}(M,N)(S) = \Hom_{\CondAb}(\mathbb{Z}[S] \otimes M, N).$1, derived Hom over $(M \otimes N)(S) = (M(S) \otimes_\mathbb{Z} N(S))^{\mathrm{sh}}, \qquad \underline{\Hom}(M,N)(S) = \Hom_{\CondAb}(\mathbb{Z}[S] \otimes M, N).$2 and over abelian groups agree: $(M \otimes N)(S) = (M(S) \otimes_\mathbb{Z} N(S))^{\mathrm{sh}}, \qquad \underline{\Hom}(M,N)(S) = \Hom_{\CondAb}(\mathbb{Z}[S] \otimes M, N).$3 Any profinite ring $(M \otimes N)(S) = (M(S) \otimes_\mathbb{Z} N(S))^{\mathrm{sh}}, \qquad \underline{\Hom}(M,N)(S) = \Hom_{\CondAb}(\mathbb{Z}[S] \otimes M, N).$4 gives rise to an analytic condensed ring $(M \otimes N)(S) = (M(S) \otimes_\mathbb{Z} N(S))^{\mathrm{sh}}, \qquad \underline{\Hom}(M,N)(S) = \Hom_{\CondAb}(\mathbb{Z}[S] \otimes M, N).$5 (Tang, 2024), ensuring $(M \otimes N)(S) = (M(S) \otimes_\mathbb{Z} N(S))^{\mathrm{sh}}, \qquad \underline{\Hom}(M,N)(S) = \Hom_{\CondAb}(\mathbb{Z}[S] \otimes M, N).$6 and its derived category are compactly generated closed monoidal Grothendieck categories. Cohomologies such as continuous group cohomology are fully compatible: $(M \otimes N)(S) = (M(S) \otimes_\mathbb{Z} N(S))^{\mathrm{sh}}, \qquad \underline{\Hom}(M,N)(S) = \Hom_{\CondAb}(\mathbb{Z}[S] \otimes M, N).$7 This compatibility situates solid abelian groups as a bridge between classical and modern cohomology of topological groups and sheaf-theoretic contexts.

Solid abelian groups, together with their “liquid” characteristic zero analogues, form the two major realms in the categorical landscape of condensed mathematics, enabling generalizations of analytic geometry, $(M \otimes N)(S) = (M(S) \otimes_\mathbb{Z} N(S))^{\mathrm{sh}}, \qquad \underline{\Hom}(M,N)(S) = \Hom_{\CondAb}(\mathbb{Z}[S] \otimes M, N).$8-adic Hodge theory, and functional analysis without leaving the language of sheaves on the category of profinite spaces.

7. Examples and Further Directions

Concrete examples of solid abelian groups include, for any profinite $(M \otimes N)(S) = (M(S) \otimes_\mathbb{Z} N(S))^{\mathrm{sh}}, \qquad \underline{\Hom}(M,N)(S) = \Hom_{\CondAb}(\mathbb{Z}[S] \otimes M, N).$9,

$R$0

for some indexing set $R$1. For $R$2 (the profinite integers), $R$3 is a highly nontrivial product of copies of $R$4. For $R$5, $R$6 for $R$7 the set of finite strictly decreasing sequences in $R$8: these serve as “universal” solid groups indexed by potentially massive sets (Asgeirsson, 2023).

A key subtlety: for a discrete abelian group $R$9, its solidification $R$0 can vanish even when the profinite completion $R$1 is large. For example, $R$2.

Once situated within the analytic context of $R$3 for a profinite $R$4, one can systematically form derived tensor products, spectral sequences, and the full apparatus of homological algebra.

Solid abelian groups and their associated modules thus provide a robust categorical and homological language for bridging the worlds of algebra, topology, analysis, and sheaf theory in the framework of condensed mathematics (Tang, 2024, Asgeirsson, 2023).