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Let

be a functor, i.e. a presheaf of abelian groups on \(\mathrm{ProFin}\). Assume that \(M\) preserves finite products, and that for any surjective map \(f: T\to S\), the complex

is exact.

Then \(M\) is a condensed abelian group, and for all profinite sets \(S\) and \(i{\gt}0\), one has \(H^i(S,M)=0\) for \(i{\gt}0\).

Consider an inverse system \((X_i)_i\) of compact-Hausdorffly-filtered-pseudonormed abelian groups where all transition maps \(X_i\to X_j\) send \(X_{i,\leq c}\) to \(X_{j,\leq c}\). Then

is compact Hausdorff, and

is naturally a compact-Hausdorffly-filtered pseudonormed abelian group which is the limit of \((X_i)_i\) in the strict category structure.

A pseudo-normed group \(M\) is *CH-filtered* if each of the sets \(M_c\) is endowed with a topological space structure making it a compact Hausdorff space, such that following maps are all continuous:

the inclusion \(M_{c_1} \to M_{c_2}\) (for \(c_1 \le c_2\));

the negation \(M_c \to M_c\);

the addition \(M_{c_1} \times M_{c_2} \to M_{c_1 + c_2}\).

The pseudo-normed group \(M\) is *profinitely filtered* if moreover the filtration sets \(M_c\) are totally disconnected, making them profinite sets.

Consider an exact sequence of abelian groups

such that all of \(X'\), \(X\) and \(X''\) carry the structure of compact-Hausdorffly-filtered-pseudonormed abelian groups. Assume that \(f(X'_{\leq c})\subset X_{\leq c}\) and \(g(X_{\leq c})\subset X''_{\leq c}\). If the sequence is strongly exact with respect to \(r\), then the sequence

of condensed abelian groups is exact.

A *morphism* of CH-filtered pseudo-normed groups \(M \to N\) is a group homomorphism \(f \colon M \to N\) that is

*bounded*: there is a constant \(C\) such that \(x \in M_c\) implies \(f(x) \in N_{Cc}\);*continuous*: for one (or equivalently all) constants \(C\) as above, the induced map \(M_c \to N_{Cc}\) is a morphism of profinite sets, i.e. continuous.

The reason the two definitions of continuity are equivalent is that a continuous injection from a compact space to a Hausdorff space must be a topological embedding.

A morphism \(f \colon M \to N\) is *strict* if \(x \in M_c\) implies \(f(x) \in N_c\) (in other words, if we can take \(C = 1\) in the boundedness condition above).

Let \(r'\) be a positive real number. A CH-filtered pseudo-normed group \(M\) has an *\(r'\)-action of \(T^{-1}\)* if it comes endowed with a distinguished morphism of CH-filtered pseudo-normed groups \(T^{-1} \colon M \to M\) that is bounded by \(r'^{-1}\): if \(x \in M_c\) then \(T^{-1}x \in M_{c/r'}\).

A morphism of CH-filtered pseudo-normed groups with \(r'\)-action of \(T^{-1}\) is a morphism \(f \colon M \to N\) of CH-filtered pseudo-normed groups that commutes with the action of \(T^{-1}\).

There is a natural functor

where \(\underline{M}(S)\) is defined to be collection of functions \(f \colon S \to M\) that factor as continuous through \(M_c\), for some \(c\). In symbols:

Let \(0 {\lt} r {\lt} r' {\lt} 1\) be real numbers. Let \(S\) be a profinite set, and let \(V\) be an \(r\)-normed \(\mathbb Z[T^{\pm 1}]\)-module. Then \(\operatorname{Ext}_{\mathrm{Mod}^{\mathrm{cond}}_{\mathbb Z[T^{-1}]}}^i(\mathcal L_{r'}(S), V) = 0\) for all \(i {\gt} 0\). In other words,

is a bijection for all \(i {\gt} 0\) and a surjection for \(i = 0\).

Let \(0 {\lt} r {\lt} r' {\lt} 1\) be real numbers. Let \(S\) be a profinite set, and let \(V\) be a \(r\)-normed (Banach?) \(\mathbb Z[T^{\pm 1}]\)-module. Then \(\operatorname{Ext}_{\mathrm{Mod}^{\mathrm{cond}}_{\mathbb Z[T^{-1}]}}^i(\overline{\mathcal L}_{r'}(S), V) = 0\) for all \(i \ge 0\). In other words,

is a bijection for all \(i\).

Assume that in each degree \(i\), the map

is a pro-isomorphism in \(c\) (i.e., pro-systems of kernels, and of cokernels, are pro-zero). Then

is an isomorphism.

Let \(\mathsf{BD}= (n,f,h)\) be a Breen–Deligne package, and let \(\kappa = (\kappa _0, \kappa _1, \kappa _2, \dots )\) be a sequence of constants in \(\mathbb R_{\ge 0}\) that is \(\mathsf{BD}\)-suitable. Fix radii \(1{\gt}r'{\gt}r{\gt}0\). For any \(m\) there is some \(k\) and \(c_0\) such that for all profinite sets \(S\) and all \(r\)-normed \(\mathbb Z[T^{\pm 1}]\)-modules \(V\), the system of complexes

is \(\leq k\)-exact in degrees \(\leq m\) for \(c\geq c_0\).

Let \(S = \varprojlim S_i\) be a profinite set. Then \(\mathbb Z[S]\) is naturally a profinitely filtered pseudo-normed group, via \(\mathbb Z[S]_{\le c} = \varprojlim \mathbb Z[S_i]_{\le c}\), where \(\mathbb Z[S_i]_{\le c}\) is the set \(\{ \sum _{s \in S_i} n_s[s] \mid \sum _s |n_s| \le c\} \).

There is a natural isomorphism between the free condensed abelian group \(\mathbb Z[S]\) and the colimit \(\varinjlim _c \mathbb Z[S]_{\le c}\) of condensed sets.

For any \(i\geq 0\), the functor \(A\mapsto H_i(Q'(A))\) has the following properties:

It is additive, i.e.

\[ H_i(Q'(A\oplus B))\cong H_i(Q'(A))\oplus H_i(Q'(B)). \]It commutes with filtered colimits, i.e. for a filtered inductive system \(A_i\),

\[ \varinjlim _i H_i(Q'(A))\cong H_i(Q'(\varinjlim _i A_i)). \]

In particular, for torsion-free abelian groups \(A\), there is a functorial isomorphism

Let \(0 {\lt} r' {\lt} 1\) be a real number, and let \(S\) be a profinite set with a presentation \(S=\varprojlim S_i\), where all \(S_i\) are finite. For all \(c{\gt}0\), let \(\mathcal L_{r'}(S)_{\leq c}\) denote the projective limit

endowed with the projective limit topology, and set

Let \(0 {\lt} r' {\lt} 1\) be a real number, and let \(S\) be a finite set. Then \(\mathcal L_{r'}(S)\) denotes the group

Let \(0 {\lt} r {\lt} 1\) be a real number, and let \(S\) be a profinite set. Decomposing \(\mathbb Z((T))_r\) into positive and nonpositive coefficients yields a direct sum decomposition

This extends to a decomposition of spaces of measures

where \(\mathcal M(S,\mathbb Z[T^{-1}]) = \mathbb Z[T^{-1}][S]\) is the free condensed \(\mathbb Z[T^{-1}]\)-module on \(S\). Letting \(\overline{\mathcal L}_r(S)= \mathcal L(S,T\mathbb Z[[T]]_r)\), we get a short exact sequence of condensed \(\mathbb Z[T^{-1}]\)-modules

Let \(0 {\lt} p' {\lt} 1\) be a real number, let \(S\) be a profinite set, and let \(r'\) denote \((\tfrac 12)^{p'}\). There is a short exact sequence of condensed \(\mathbb Z[T^{-1}]\)-modules

where the first map is multiplication by \(2T - 1\), and the second is evaluation at \(T = \tfrac 12\).

Let \(0 {\lt} p' {\lt} p \le 1\) be real numbers, let \(S\) be a profinite set, and let \(V\) be a \(p\)-Banach space. Let \(\mathcal M_{p'}(S)\) be the space of real \(p'\)-measures on \(S\). Then

for \(i \ge 1\).

We will consider the following categories:

\(\text{CHPNG}\) the category of CH-filtered pseudo-normed groups with bounded morphisms.

\(\text{CHPNG}_1\) the category of exhaustive CH-filtered pseudo-normed groups with strict morphisms.

\(\text{ProfinPNG}\) the category of profinitely filtered pseudo-normed groups with bounded morphisms.

\(\text{ProfinPNG}_1\) the category of exhaustive profinitely filtered pseudo-normed groups with strict morphisms.

\(\text{ProfinPNGTinv}_1^{r'}\) the category of exhaustive profinitely filtered pseudo-normed groups with \(r'\)-action of \(T^{-1}\) and strict morphisms.

A *pseudo-normed group* is an abelian group \((M,+)\), together with an increasing filtration \(M_c \subseteq M\) of subsets \(M_c\) indexed by \(\mathbb R_{\ge 0}\), such that each \(M_c\) contains \(0\), is closed under negation, and \(M_{c_1} + M_{c_2} \subseteq M_{c_1 + c_2}\). An example would be \(M=\mathbb {R}\) or \(M=\mathbb {Q}_p\) with \(M_c :=\{ x\, :\, |x|\leq c\} \).

A pseudo-normed group \(M\) is *exhaustive* if \(\varinjlim _c M_c = M\).

Let \(M\) and \(N\) be condensed abelian groups with endomorphisms \(f_M\), \(f_N\). Assume that \(M\) is torsion-free (over \(\mathbb Z\)). Then

for all \(i\geq 0\) if and only if

for all \(i\geq 0\). More precisely, the first vanishes for \(0\leq i\leq j\) if and only if the second vanishes for \(0\leq i\leq j\).

Let \(0 {\lt} p' {\lt} 1\) be a real number, and let \(S\) be a finite set. Then \(\mathcal M_{p'}(S)\) denotes the real vector space

endowed with the \(\ell ^{p'}\)-norm \(\Vert \sum _{s\in S}a_s[s]\Vert _{\ell ^{p'}}=\sum _{s\in S}\lvert a_s\rvert ^{p'}\).

For every finite set \(S\), the space \(\mathcal M_{p'}(S)\) can be written as the colimit (or simply the union, in this case)

where \(\mathcal M_{p'}(S)_{\leq c} = \left\{ F \in \mathcal M_{p'}(S) \text{ such that } \Vert F \Vert _{\ell ^{p'}}\leq c\right\} \). Now, given a profinite set \(S=\varprojlim S_i\), where all \(S_i\)’s are finite sets, and a positive real number \(c{\gt}0\), define \(\mathcal M_{p'}(S)_{\leq c}\) as

endowed with the projective limit topology. Finally, set \(\mathcal M_{p'}(S)=\varinjlim _{c}\mathcal M_{p'}(S)_{\leq c}\).

Consider an exact sequence of abelian groups

such that all of \(X'\), \(X\) and \(X''\) carry the structure of compact-Hausdorffly-filtered-pseudonormed abelian groups. Assume that the maps are strict, i.e., \(f(X'_{\leq c})\subset X_{\leq c}\) and \(g(X_{\leq c})\subset X''_{\leq c}\). Let \(r \colon \mathbb {R}_{\ge 0} \to \mathbb {R}_{\ge 0}\) be a function satisfying \(c \le r(c)\) for all \(c\). We say that the sequence is *strongly exact with respect to \(r\)* if \(\mathrm{ker}(g)\cap X_{\leq c}\subset f(X'_{\leq r(c)})\).

Consider an inverse system

of exact sequences that are strongly exact with respect to \(r\) (independent of \(i\)). Moreover, assume that the transition maps \(X'_i\to X'_j\), \(X_i\to X_j\) and \(X''_i\to X''_j\) are strict, and let \(X'\), \(X\) and \(X''\) be their limits. Then

is strongly exact with respect to \(r\).

Fix a finite set \(S\), real numbers \(0{\lt}\xi ,p'{\lt}1\) and set \(r'=(\xi )^{p'}\). The map \(\vartheta _\xi \) of Definition 2.2.4 is continuous when endowing \(\mathcal L_{r'}(S)\) with the topology defined in Definition 2.2.2 and \(\mathcal M_{p'}(S)\) with the topology defined in Definition 2.2.1

Let \(0{\lt}p'{\lt}1\) be a real number, let \(S\) be a finite set, and let \(r'\) denote \(\left(\frac{1}{2}\right)^{p'}\). The sequence

where the first map is multiplication by \(2T-1\), is exact. In particular, the kernel of \(\theta _S\) is principal, generated by \(2T-1\).

For all \(x \in \mathbb {R}\) and \(n\in \mathbb {N}\), set

where \(\lfloor \ast \rfloor \) denotes the floor of \(\ast \), namely the greatest integer that is less or equal than \(\ast \).