## 2.4 Condensed abelian groups

For the time being, the following facts will be used without proof in this text. (They have or will be formalized in Lean though.)

There is a natural functor \(\operatorname{Top}\to \operatorname{Cond}(\operatorname{Sets})\).

The category of condensed abelian groups (resp. condensed \(R\)-modules) is an abelian category with enough projectives. For \(S\) an extremally disconnected set, the objects \(\mathbb Z[S]\) (resp. \(R[S]\)) is projective.

We write \(H^i(S, M)\) for \(\text{Ext}^i(\mathbb Z[S], M)\).

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 \(r\).

Pass to cofiltered limits of compact Hausdorff spaces in the statements \(\mathrm{ker}(g)\cap X_{\leq c}\subset f(X'_{\leq r(c)})\), noting that cofiltered limits of surjections of compact Hausdorff spaces are still surjective (by an application of Tychonoff).

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:

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.

We evaluate at \(S\in \mathrm{ExtrDisc}\). Since the sequence is exact with respect to \(r\), we know that for all \(c\) the natural map

is surjective. Therefore, any continuous map from \(S\) to the codomain of \(\phi \) can be lifted; as \(S\) is extremally disconnected. Since every map \(S \to X\) factors over some \(X_{\le c}\), this shows that the kernel of \(g \colon X(S)\to X''(S)\) is in the image of \(f \colon X'(S)\to X(S)\).

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 now, see Lemma 2.1 of [ Sch20 ] .

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 \(i{\gt}0\).

We prove by induction on \(i{\gt}0\) that \(H^i(S,M)=0\) for all profinite sets \(S\), so assume the vanishing of \(\mathrm{Ext}^1,\ldots ,\mathrm{Ext}^i\) for some \(i\geq 0\). (This is vacuous for \(i=0\).) We aim to prove that \(H^{i+1}(S,M)=0\) for all profinite sets \(S\). Pick any profinite set \(S\) and a cover \(T\to S\) with \(T\in \mathrm{ExtrDisc}\). We get a long exact sequence of condensed abelian groups

Indeed, taken as presheaves on \(\mathrm{ExtrDisc}\), this is already true on the level of presheaves, where it reduces to the case of surjections of sets in which case one can write down a contracting homotopy. (Actually, the similar result is true in any topos, where one has to maybe argue a bit more carefully.)

The following argument is making explicit something usually seen through a spectral sequence. Define inductively

etc. One gets exact sequences

for \(n\geq 2\). From the long exact sequence

we see that we have to prove that \(\mathrm{Ext}^i(K_1,M)=0\) (if \(i{\gt}0\), otherwise that \(M(T)\) surjects onto \(\mathrm{Hom}(K_1,M)\)). Assuming \(i{\gt}0\), we can go on, and using the inductive hypothesis applied to the fibre products \(T^{\ast /S}\), we inductively see that

and eventually that this is the same as the cokernel of

But there is an exact sequence

and \(\mathrm{Hom}(K_{i+2},M)\) injects into \(M(T^{(i+2)/S})\). We see that

and we need to see that

is surjective, which is precisely the exactness of the Čech complex.

Let \((M,\| \cdot \| )\) be a complete normed group, regarded as a topological group. Then the corresponding condensed abelian group \(\underline{M}\) sends any profinite set \(S\) to the completion of normed group of locally constant maps \(S\to M\) (with the supremum norm).

This is a standard result. We omit the proof here, but it is formalized in Lean.

Let \((M,\| \cdot \| )\) be a complete normed group, regarded as a topological group. Then for any profinite set \(S\), one has \(H^i(S,\underline{M})=0\) for \(i{\gt}0\).

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 \(i\).

With Proposition 2.3.3, it suffices to prove the following assertion. Pick \(1{\gt}r'{\gt}r{\gt}0\), a profinite \(S\), and some \(r\)-Banach \(\mathbb Z[T^{\pm 1}]\)-module \(V\) as before. Then we want to prove that

for all \(i\geq 0\).

At this point, it is profitable to rewrite this again as the bijectivity of

Now these Ext-groups can be computed! More precisely, recall that \(Q'(\overline{\mathcal M}_{r'}(S))\) is a complex of the form

Termwise, the Ext-groups turn into cohomology groups

Unfortunately, \(\overline{\mathcal M}_{r'}(S)\) itself is not profinite, so we cannot directly apply Proposition 2.4.9. To get around this last cliff, we write \(Q'(\overline{\mathcal M}_{r'}(S))\) as a filtered colimit of complexes

where the constants \(\kappa _0=1,\kappa _1,\ldots \) are positive and chosen so that all differentials are well-defined. (The possibility of choosing such constants has already been formalized; TODO include pointer.) It suffices to prove that

is a pro-isomorphism in \(c\), as then the final result follows by passing to a derived limit over \(c\), see Lemma 2.4.11 below. This final pro-isomorphism assertion can finally be written out, and it unravels to the statement of Theorem 1.7.1.

In passing to the derived limit over \(c\), we use the following lemma.

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.

We have

inducing a resolution

Passing to a corresponding long exact sequence reduces one to checking that the squares

are bicartesian (here, horizontal maps are shift minus identity, and vertical maps are \((T^{-1})_V - (T^{-1})_{\mathcal M}\)). Equivalently, the horizontal maps become isomorphisms on vertical kernels, and vertical cokernels. But the vertical kernels and vertical cokernels induce pro-zero systems of abelian groups, and then the horizontal kernels and cokernels compute \(\varprojlim _n\) and \(\varprojlim _n^1\) of their systems, which vanish.

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

On \(\mathbb Z((T))_{r,\leq c}\), only finitely many nonpositive coefficients can possibly be nonzero, and each of them is bounded. This shows that the nonpositive summand of \(\mathbb Z((T))_r\) is given by \(\mathbb Z[T^{-1}]\). To pass to profinite \(S\), use Proposition 2.4.6.

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 \(i = 0\).

Consider the long exact sequence of Ext-groups arising form the short exact sequence (Lemma 2.4.12)

by applying \(\operatorname{Ext}^*(\_ , V)\).

By Lemma 2.4.7 all groups \(\operatorname{Ext}_{\operatorname{Cond}(\operatorname{Ab})}^i(\mathbb {Z}[S], V)\) vanish for \(i {\gt} 0\). And by Lemma 2.4.10 all groups \(\operatorname{Ext}_{\mathrm{Mod}^{\mathrm{cond}}_{\mathbb Z[T^{-1}]}}^i(\overline{\mathcal L}_{r'}(S), V)\) vanish for \(i \ge 0\). The result follows.

The “In other words” version can be proved without mentioning \(\mathbb Z[T^{-1}]\)-linear Ext groups, by using the same ingredients and the five lemma.

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 \(i \ge 1\).

Recall from Lemma 2.4.14 the short exact sequence

Apply to this \(\text{Ext}^*(\_ , V)\) to obtain a long exact sequence. Note that \(T\) acts on \(V\) via multiplication by \(\tfrac 12\) (by Lemma 2.1.2). Hence we can use Lemma 2.4.13 to obtain isomorphisms between the Ext-groups involving \(\mathcal L_{r'}(S)\), for \(i {\gt} 0\), and a surjection for \(i = 0\). The result follows.