Condensed cohomology

Section 8 Condensed cohomology

In this section, we explain some links between condensed abelian groups and classical notions of algebraic topology, e.g., the singular cohomology of CW complexes.

Reference.

This section is based on [8], Lecture 4, except that we take advantage of the material from [8], Lecture 5 to reverse the flow of some information. See also [6], Lecture 4 (with the same proviso).

Subsection 8.1 Condensed cohomology on compact spaces

Definition 8.1.1.

For \(X \in \CSet\) and \(M\) a topological abelian group, we define the condensed cohomology groups of \(X\) with values in \(M\) to be

\begin{equation*} H^i(X, M) := \Ext^i_{\CAb}(\ZZ[X], \underline{M}). \end{equation*}

When \(X \in \Top\) we may also write \(H^i_{\cond}(X, M)\) to mean \(H^i(\underline{X}, M)\text{.}\)

Lemma 8.1.2.

For \(X \in \Prof\) and \(M \in \TopAb \times_{\CAb} \CAb_\solid\text{,}\) \(\Ext^i_{\CAb}(\ZZ[\underline{X}], \underline{M}) = 0\) for all \(i \gt 0\text{.}\)

Proof.

This is included in Lemma 7.3.7: we have \(\ZZ[\underline{X}] \cong \prod_i \underline{\ZZ} \cong P_\solid\) and the latter is internally projective in \(\CAb_\solid\) by Proposition 5.4.8.

Corollary 8.1.3.

For \(X \in \Prof\) and \(M \in \Ab\text{,}\) \(H^i_{\cond}(X, M) = 0\) for all \(i \gt 0\text{.}\)

Proof.

This follows from Lemma 8.1.2 because discrete abelian groups are solid.

Theorem 8.1.4.

For \(X \in \CHaus\) and \(M \in \Ab\text{,}\) there are natural (in \(X\)) isomorphisms

\begin{equation*} H^i_{\sheaf}(X, M) \cong H^i_{\cond}(X, M) \end{equation*}

where \(H^i_{\sheaf}(X, \bullet)\) denotes the usual right derived functors of the global sections functor.

Proof.

We loosely follow [6], Theorem 3.2. Using Remark 4.2.2, we may construct a simplicial hypercovering \(X_\bullet \to X\) by objects of \(\Prof\text{.}\) By Corollary 8.1.3, the terms of this hypercovering are acyclic and so we may compute \(R\Gamma_{\cond}(X, M)\) by the Čech complex

\begin{equation*} 0 \to \Gamma(X_0, M) \to \Gamma(X_1, M) \to \cdots. \end{equation*}

Since continuous functions from any \(X_n\) to the discrete group \(M\) are locally constant, the augmented version of this complex is itself a colimit of complexes associated to simplicial hypercoverings of finite sets, which are acyclic by Corollary 8.1.3 again.

Remark 8.1.5.

For \(X \in \CHaus\text{,}\) we can also interpret \(H^i_{\sheaf}(X,S)\) as Čech cohomology ([11], Théorème 5.10.1). In some cases we can also interpret it as singular cohomology, see for example Subsection 8.2.

We next consider coefficients in the topological (not discrete) group \(\RR\text{.}\)

Theorem 8.1.6.

For \(X \in \CHaus\text{,}\) \(H^i_{\cond}(X, \RR) = 0\) for all \(i \gt 0\text{.}\) (Note that \(H^0_{\cond}(X, \RR)\) is the group of continuous maps \(S \to \RR\text{.}\)) More precisely, for any simplicial hypercovering \(X_\bullet \to X\) by objects of \(\Prof\text{,}\) the augmented Čech complex

\begin{equation*} 0 \to \Cts(X, \RR) \to \Cts(X_0, \RR) \to \Cts(X_1, \RR) \to \cdots \end{equation*}

has the property that if we interpret each term as a Banach space for the supremum norm, then for every \(\epsilon \gt 0\text{,}\) every cochain \(f \in \Cts(X_n, \RR)\) has the form \(d(g)\) for some \(g \in \Cts(X_{n-1}, \RR)\) with \(\left| g \right|_{X_{n-1}} \leq (1+\epsilon) \left| f \right|_{X_n}\text{.}\)

Proof.

Suppose first that \(X\) and the \(X_n\) are all finite. Then the hypercover splits; this means that the augmented Čech complex admits a chain homotopy which is “combinatorial” in the sense that every map is given by pullback along a map of underlying sets. (This amounts to a very basic case of the Dold–Kan correspondence.) In this case the claim evidently holds even for \(\epsilon = 0\text{.}\)

By the same token, if \(X\) and the \(X_n\) are all profinite, then we can write the augmented Cech complex as a completed colimit of sequences of the previous form. By some elementary analysis (left to the reader), this implies the claim but now with \(\epsilon \gt 0\text{.}\)

In the general case, given \(f \in \Cts(X_n, \RR)\) nonzero with \(df = 0\text{,}\) for any \(x \in X\) we can apply the previous case to the restriction of \(f\) to \(X_n \times_X \{x\}\text{.}\) We may then deduce the general case by spreading the results out using Tietze’s extension theorem, invoking compactness to see that the spread out functions cover \(X\text{,}\) and constructing a suitable partition of unity argument to patch together the local choices. See [6], Theorem 3.3 for the detailed argument.

Subsection 8.2 Cohomology of CW complexes

Definition 8.2.1.

A CW complex is a topological space obtained as a colimit \(X_0 \subseteq X_1 \subseteq \cdots\) in which each \(X_k\) is obtained from \(X_{k-1}\) by glueing in some copies of the \(k\)-dimensional ball, with the boundaries contained in \(X_{k-1}\text{.}\)

Theorem 8.2.2.

For \(X\) a sequential CW complex, there is a canonical isomorphism \(H^i_{\cond}(X, M) \cong H^i_{\sing}(X, M)\text{.}\)

Proof.

For a CW complex, we have \(H^i_{\sing}(X, M) \cong H^i_{\sheaf}(X, M)\text{.}\) We can thus deduce this from Theorem 8.1.4.

Remark 8.2.3.

Theorem 8.2.2 can be thought of as a reinterpretation of the Dold–Thom theorem in algebraic topology.

Subsection 8.3 Locally compact abelian groups

See [15] for a much more detailed development of this topic.

Theorem 8.3.1.

Let \(\LCAb\) denote the category of sequential locally compact abelian groups.

Every \(A \in \LCAb\) has the form \(\RR^n \times A'\) for some nonnegative integer \(n\) and some \(A' \in \LCAb\) admitting a compact open subgroup. In particular, \(A'\) is an extension of a discrete abelian group by a compact group.
Let \(\TT \cong \RR/\ZZ \in \Top\) be the circle group, so that we have an exact sequence
\begin{equation} 0 \to \ZZ \to \RR \to \TT \to 0\text{.}\tag{8.1} \end{equation}
The Pontryagin duality functor
\begin{equation*} A \mapsto D(A) := \Hom_{\LCAb}(A, \TT) \end{equation*}
takes values in \(\LCAb\) and induces a contravariant autoduality on this category. In particular, there is a functorial isomorphism \(A \to D(D(A))\text{.}\)
The functor \(D\) restricts to a duality between discrete abelian groups and compact abelian groups.

Proof.

See [14], specifically Theorem 24.30 for the first point, Theorem 24.8 for the second, and Theorem 23.17 for the third.

Lemma 8.3.2.

For any discrete abelian group \(M\) and any at most countable index set \(I\text{,}\) we have

\begin{equation*} R\Hom_{\CAb}\left( \prod_I \TT, \underline{M} \right) = \bigoplus_I M[-1]. \end{equation*}

More precisely, the map from right to left is the product over \(i \in I\) of the maps

\begin{equation*} M[-1] = R\Hom_{\CAb}(\underline{\ZZ}[1], M) \cong R\Hom_{\CAb}(\underline{\TT}, M) \to R\Hom_{\CAb}(\prod_I \underline{\TT}, M) \end{equation*}

where the passage from \(\ZZ\) to \(\TT\) is the connecting homomorphism from (8.1) and the last arrow is induced by the \(i\)-th projection \(\prod_I \underline{\TT} \to \underline{\TT}\text{.}\)

Proof.

Since \(\TT\) is a compact CW complex, this can be deduced from Theorem 8.1.4 and Theorem 8.2.2; but it can also be seen more directly as follows. Since \(\underline{M}\) is solid by Theorem 7.4.1, we have \(R\Hom_{\CAb}(\prod_I \underline{\RR}, \underline{M})\) by Corollary 6.3.3. Meanwhile, via Theorem 7.4.1 we have

\begin{equation*} R\Hom_{\CAb}(\prod_I \underline{\ZZ}, \underline{M}) = R\Hom_{\ZZ_\solid}(\prod_I \ZZ_\solid, M_\solid) = \bigoplus_I M[0] \end{equation*}

because \(\prod_I \underline{\ZZ}\) is projective in \(\Mod_{\ZZ_\solid}\text{.}\) This yields the claim.

Theorem 8.3.3.

The assignment \(A \mapsto \underline{A}\) defines a fully faithful functor from the bounded derived category of \(\LCAb\) to the derived category of \(\CAb\text{.}\)

Proof.

In light of point 1 of Theorem 8.3.1, it suffices to check that \(R\Hom_{\LCAb}(A,B) \cong R\Hom_{\CAb}(\underline{A}, \underline{B})\) when each of \(A,B\) is either compact, discrete, or isomorphic to \(\RR\text{.}\) We break down the argument as follows.

Suppose first that \(A\) is discrete. Using resolutions and filtered colimits we may reduce to the case \(A = \ZZ\text{.}\) In this case, \(A\) and \(\underline{A}\) are projective in their respective categories and the result is clear.

Suppose next that \(A\) is compact. By taking a resolution of the dual group, we may reduce to the case where \(A = \prod_I \TT\) with \(I\) at most countable. At this point, when \(B\) is discrete we may deduce the desired isomorphism by comparing Lemma 8.3.2 with [15], Example 4.11; when \(B = \RR\) we may instead compare Theorem 8.1.6 with [15], Proposition 4.15 for the same effect. Combining these and using the exact sequence (8.1), we deduce the claim also when \(B = \TT\text{,}\) and consequently when \(B = \prod_I \TT\text{;}\) again we may formally promote this to the general case of \(B\) compact.

Finally, to resolve the case \(A = \RR\) we may use (8.1) to reduce to the previously treated cases \(A = \ZZ\) (discrete) and \(A = \TT\) (compact).

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