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Notes on condensed mathematics

Kiran S. Kedlaya

Section 7 Explicit solidifications

In this section, we show that the functor defined in Proposition 6.4.2 from solid abelian groups as defined in Section 1. to solid condensed abelian groups is an equivalence of categories. This will allow us to employ the language of solid abelian groups while still drawing conclusions about condensed abelian groups.
In the process, we will also identify the solidifications of the sequence space \(P\) and of the groups \(\ZZ[\underline{S}]\) for \(S \in \Prof\text{.}\) Even if one is content to work solely with solid objects in the language of condensed abelian groups, these identifications are crucial to have any control over the solid objects.

Reference.

This section is loosely based on [8], Lecture 5, but with some changes to accommodate the discussion of solid modules in Section 1. See also [6], Lecture 5.

Subsection 7.1 Setting up a comparison of solidifications

Our first goal is to show that the coordinate map \(c \colon P \to \prod_\NN \underline{\ZZ}\) induces an isomorphism \(P_\solid \cong \prod_\NN \underline{\ZZ}\) (Proposition 7.2.10). Since both \(P\) and \(\prod_\NN \underline{\ZZ}\) correspond to sequential topological abelian groups (see Definition 5.3.1), we can interpret \(c\) as a map in that category. However, to overcome a subtle topological issue (Remark 7.2.7) we need to refactor this map as follows.

Definition 7.1.1.

Note that \(\prod_\NN \underline{\ZZ}\) admits the subobject
\begin{equation*} \prod_\NN^{\bd} \underline{\ZZ} := \colim_{n \in \NN} \prod_\NN \underline{\{-n,\dots,n\}}\text{.} \end{equation*}
In terms of evaluations at \(S \in \Prof\text{,}\) \(\left(\prod_\NN \underline{\ZZ}\right)(S)\) consists of continuous maps \(S \to \prod_\NN \ZZ\text{,}\) or equivalently families of continuous maps \(S \to \ZZ\) indexed by \(\NN\text{.}\) Each of these maps has bounded image, but \(\left( \prod_\NN^{\bd} \underline{\ZZ} \right)(S)\) picks out the families with uniformly bounded image.
From the explicit description of \(\ZZ[\NN_\infty]\) (Example 5.1.7), we can factor \(c\colon P \to \prod_\NN \underline{\ZZ}\) through \(\prod_\NN^{\bd} \underline{\ZZ}\) to yield a sequence of inclusions
\begin{equation} P \stackrel{c}{\to} \prod_\NN^{\bd} \underline{\ZZ} \to \prod_\NN \underline{\ZZ}\text{.}\tag{7.1} \end{equation}
Namely, \(\Hom_{\CAb}(P, \prod_\NN \underline{\ZZ}) = \prod_\NN \Hom_{\CAb}(P, \underline{\ZZ})\) consists of infinite families of null sequences in \(\ZZ\text{,}\) while \(\Hom_{\CAb}(P, \prod_\NN^{\bd} \underline{\ZZ})\) picks out those families whose images are uniformly bounded. The sequences corresponding to the factors of the map \(c\) only take values in \(\{0,1\}\text{,}\) so they form a family of the latter form.
Fortunately, it is easy to compare the solidifications of \(\prod_\NN^{\bd} \underline{\ZZ}\) and \(\prod_\NN \underline{\ZZ}\text{.}\)

Proof.

By analogy with the construction of \(\prod_\NN^{\bd} \underline{\ZZ}\text{,}\) we define
\begin{equation*} \prod_\NN^{\bd} \underline{\RR} := \colim_{n \in \NN} \prod_\NN \underline{[-n,n]}\text{;} \end{equation*}
we will construct an isomorphism (at the level of topological abelian groups)
\begin{equation*} \left( \prod_\NN \underline{\ZZ}\right) / \left( \prod_\NN^{\bd} \underline{\ZZ} \right) \cong \left( \prod_\NN \underline{\RR} \right) / \left( \prod_\NN^{\bd} \underline{\RR} \right)\text{.} \end{equation*}
Since the right-hand side carries an evident \(\underline{\RR}\)-module structure, so does the left-hand side. By Corollary 6.3.3, this will yield the claim.
The proposed isomorphism is formally equivalent to the claim that
\begin{equation*} \left( \prod_\NN^{\bd} \underline{\RR} \right) / \left( \prod_\NN^{\bd} \underline{\ZZ} \right) \cong \left( \prod_\NN \underline{\RR} \right) / \left( \prod_\NN \underline{\ZZ} \right)\text{;} \end{equation*}
we get this isomorphism of abelian groups because \(\RR/\ZZ\) is compact.

Subsection 7.2 A comparison of solidifications

We now finish comparing the effect of solidification across the terms of (7.1). The strategy is to build a diagram that allows us to exploit the fact that \(\Delta\colon P \to P\) becomes an isomorphism after solidification.

Definition 7.2.1.

Let \(\Delta\colon \ZZ[\NN] \to \ZZ[\NN]\) denote the map \([n] \mapsto [n]-[n+1]\text{.}\) We then obtain a commutative diagram
Figure 7.2.2.
of discrete abelian groups in which the top row maps \([m] \otimes [n]\) to \([n]\) if \(m = n\) and 0 otherwise, and the bottom row maps \([m] \otimes [n]\) to \([n]\) if \(m \leq n\) and 0 otherwise. Note that the two rows are equalized by \(\ZZ[\NN] \to \ZZ[\NN] \otimes \ZZ[\NN]\) via \([n] \mapsto [n] \otimes [n]\text{.}\)
By encoding the various maps in terms of maps of profinite sets via the isomorphism \(\ZZ[\underline{\NN_\infty \times \NN_\infty}] \cong \ZZ[\underline{\NN_\infty}] \otimes \ZZ[\underline{\NN_\infty}]\text{,}\) we promote the previous diagram to a new diagram:
Figure 7.2.3.
where again the rows are equalized by a map \(P \to P \otimes P\) acting via \([n] \mapsto [n] \otimes [n]\text{.}\)

Proof.

By imitating the construction of the diagram from Definition 7.2.1, we obtain a diagram of the form
Figure 7.2.6.
where now the top row is \([m] \otimes (c_0,c_1,\dots) \mapsto (0,\dots,0,c_m,0,\dots)\text{,}\) the bottom row is \([m] \otimes (c_0,c_1,\dots) \mapsto (0,\dots,0,c_m,c_{m+1},\dots)\text{,}\) and the two rows are equalized by a map \(P \to P \otimes \prod_\NN^{\bd} \underline{\ZZ}\) acting via \([n] \mapsto [n] \otimes c([n])\text{.}\) The bottom row is split by a map of the form \(s \mapsto [0] \otimes s\text{.}\)
To get what we need, we must verify that the top row
\begin{equation} P \otimes \prod_\NN^{\bd} \underline{\ZZ} \to \prod_\NN^{\bd} \underline{\ZZ}\tag{7.2} \end{equation}
factors through \(c\text{.}\) The splitting of the resulting top row will be guaranteed by the fact that since \(c\) is injective, the composition \(P \to P \otimes \prod_\NN^{\bd} \underline{\ZZ} \to P\) will again be the identity.
More precisely, we will show that that the formula
\begin{equation} \left( \sum_n c_n [n], (d_n)_n \right) \mapsto \sum_n c_n d_n [n]\tag{7.3} \end{equation}
defines a continuous bilinear map \(P \times\prod_\NN^{\bd} \underline{\ZZ} \to P \) at the level of topological abelian groups. Composing (7.3) with \(c\) will then yield the map (7.2)
To check continuity of (7.3), it will suffice to check that every null sequence is mapped to a null sequence. For this, note first that a sequence of sums \(\sum_n a_n [n] \in P\) is a null sequence iff \(\sum_n |a_n|\) is uniformly bounded (say by \(n_1\)) and for each fixed \(n\text{,}\) \(a_n\) is eventually zero. Meanwhile, a sequence of sequences \((b_n)_n \in\prod_\NN^{\bd} \underline{\ZZ}\) is a null sequence if and only if \(\sup_n \{|b_n|\}\) is uniformly bounded (say by \(n_2\)) and for each fixed \(n\text{,}\) \(b_n\) is eventually zero. From this, we see that \(\sum_n |a_nb_n|\) is uniformly bounded by \(n_1n_2\) and for each fixed \(n\text{,}\) \(a_nb_n\) is eventually zero; that is, \(\sum_n a_n b_n[n]\) is a null sequence in \(P\text{,}\) proving the claim.

Remark 7.2.7.

The continuous bilinear map (7.3) does not extend continuously to \(P \times \prod_\NN \underline{\ZZ}\text{:}\) the sequence \([n] \otimes nc([n])\) is a null sequence but its image in \(P\) is the sequence \(n[n]\) which is not a null sequence. This explains why the insertion of \(\prod_\NN^{\bd} \underline{\ZZ}\) in Definition 7.1.1 was necessary.

Proof.

We solidify the diagram in Lemma 7.2.4 to obtain
Figure 7.2.9.
where again rows are split surjections.
We check that the left vertical arrow in the diagram is an isomorphism. By Yoneda’s lemma, it suffices to check that this arrow induces an isomorphism after taking internal Homs into any \(N \in \CAb_\solid\text{.}\) To see this, we apply Hom-tensor adjunction to write
\begin{equation*} \iHom_{\CAb}\left( P \otimes \prod_\NN^{\bd} \underline{\ZZ}, N\right) = \iHom_{\CAb}\left( P, \iHom_{\CAb}\left(\prod_\NN^{\bd} \underline{\ZZ}, N\right)\right) \end{equation*}
and then invoke Proposition 6.2.4 to see that \(\iHom_{\CAb}(\prod_\NN^{\bd} \underline{\ZZ}, N)\) is solid.
At this point we conclude that \(c_\solid\) is a split surjection. Since the top row is also surjective, tracing clockwise from \(\left( \prod_\NN^{\bd} \underline{\ZZ} \right)_\solid\) back to \(P_\solid\) yields a surjection whose postcomposition with \(c_\solid\) is the identity map. That surjection must therefore be an isomorphism, which proves that \(c_\solid\) is also injective.

Proof.

Proof.

This is straightforward if \(I\) is finite, and follows from Proposition 5.4.8 with Proposition 7.2.10 (specifically the isomorphism at the level of derived solidifications) if \(I\) is countable.

Proof.

This is a consequence of Corollary 7.2.11: we have an exact functor \(\CAb_\solid \to \Ab_\solid\) taking \(X\) to the functor
\begin{equation*} \prod_I \ZZ_\solid \mapsto \Hom_{\CAb_\solid}\left(\prod_I \underline{\ZZ}, X\right) \end{equation*}
and the composition \(\Ab_\solid \to \CAb_\solid \to \Ab_\solid\) is an equivalence.

Remark 7.2.13.

In principle one could use the explicit description of \(\ZZ[\underline{\NN_\infty}]\) from Example 5.1.7 to recover the isomorphism \(P_\solid \cong \prod_{\NN} \underline{\ZZ}\text{.}\) We will do something similar later; see Proposition 10.6.6.

Subsection 7.3 More comparisons of solidifications

It still remains to confirm that every solid condensed abelian group arises from a solid abelian group. For this, we calculate some more solidifications using a process similar to the computation of \(P_\solid\text{;}\) however, now we may take a shortcut and compare things to \(P_\solid\) rather than directly to \(\prod_\NN \underline{\ZZ}\text{.}\)

Definition 7.3.1.

For \(S \in \Prof\) infinite, we construct a morphism \(g\colon P \to \ZZ[\underline{S}]\) as follows. Per Remark 3.3.1, we may write \(S\) as a sequential inverse limit of finite sets with surjective transition maps: \(S = \varprojlim_{n \in \NN} S_n\text{.}\) Let \(\pi_n\colon S \to S_n\) denote the projection map.
We first inductively choose subsets \(T_n \subset S_n\) and maps \(i_n\colon T_n \to S\) subject to the following conditions:
  • Each composition \(\pi_n \circ i_n \colon T_n \to S \to S_n\) coincides with the inclusion \(T_n \subseteq S_n\text{.}\)
  • Each \(T_n\) is the complement of the images of the maps \(\pi_n \circ i_m\colon T_m \to S \to S_n\) for all \(m \lt n\text{.}\) In particular, \(T_1 = S_1\text{.}\)
Let \(j_n\colon S_n \to S\) be the map obtained by collating the maps \(i_m\) for \(m \leq n\text{.}\) That is, each element of \(S_{n}\) projects to \(T_m\) for a unique choice of \(m\text{,}\) and then we apply \(i_m\) for that choice of \(m\text{.}\)
Define \(T = T_1 \sqcup T_2 \sqcup \cdots\) and fix a bijection \(\NN \cong T\text{.}\) We then define the map
\begin{equation*} g\colon P \cong \ZZ[\underline{T_\infty}]/\ZZ[\underline{\infty}] \to \ZZ[\underline{S}] \end{equation*}
to act on \(T_0\) via \(i_0\text{,}\) and on \(T_n\) for \(n \gt 0\) via the difference between \(T_n \stackrel{i_n}{\to} S\) and \(T_n \to S_{n-1} \stackrel{j_{n-1}}{\to} S\text{,}\) or equivalently via \((j_n \circ \pi_n - j_{n-1} \circ \pi_{n-1}) \circ i_n\text{.}\)

Remark 7.3.2.

If we run the construction in Definition 7.3.1 with \(S = \NN_\infty\text{,}\) then all of the choices are forced: we must take \(T_n = \{n\}\) and \(i_n\) to be the map \(n \mapsto n\text{.}\) In this case \(j_{n-1} \circ \pi_{n-1} \circ i_n = 0\text{,}\) so we recover precisely the map \(P \to \ZZ[\underline{\NN_\infty}]\) splitting the projection.

Remark 7.3.3.

From the construction in Definition 7.3.1, we see that \(g\) induces a bijection between \(\ZZ\)-valued measures on \(S\) and functions \(\NN \to \ZZ\text{:}\) the value of the resulting function at \(x \in T_n\) computes the measure of the preimage of \(x\) via \(S \to S_n\text{.}\) In this context, we will not need an analogue of replacing \(\prod_\NN \underline{\ZZ}\) with \(\prod_\NN^{\bd} \underline{\ZZ}\text{,}\) ultimately because \(\ZZ[\underline{S}]\) is itself locally compact (Example 5.1.7).
We next follow the strategy of Lemma 7.2.4.

Proof.

We first construct a diagram
Figure 7.3.6.
in which the top row maps \([n] \otimes [s]\) to \((j_0 \circ \pi_0)([s])\) if \(n=0\) and \((j_n \circ \pi_n - j_{n-1} \circ \pi_{n-1})([s])\) otherwise, while the bottom row maps \([n] \otimes [s]\) to \([s]\) if \(n=0\) and \((\id - j_{n-1} \circ \pi_{n-1})([s])\) otherwise. The bottom row is split by the map \(x \mapsto [0] \otimes x\text{.}\)
We next observe that the top row factors through \(g\) via the map \(P \otimes \ZZ[\underline{S}] \to P\) carrying \([n] \otimes [s]\) to \(\pi_n([s])\) if \(s\) projects to \(T_n\) and \(0\) otherwise. This will then be split by the map \(P \cong \ZZ[\underline{T_\infty}]/\ZZ[\underline{\infty}] \to P \otimes \ZZ[\underline{S}]\) carrying \([t]\) to \([n] \otimes [t]\) for \(t \in T_n\text{.}\)

Proof.

Remark 7.3.8.

Lemma 7.3.7 has the consequence that for any exact sequence
\begin{equation*} 0 \to M_1 \to M \to M_2 \to 0 \end{equation*}
in \(\CAb\) with \(M_1\) solid and any \(S \in \Prof\text{,}\) we have an exact sequence
\begin{equation*} 0 \to M_1(S) \to M(S) \to M_2(S) \to \Ext^1_{\CAb}(\ZZ[\underline{S}], M_1) = 0\text{.} \end{equation*}
This can be used to explain the empirical observation that in many natural examples, formation of cokernels of morphisms in \(\CAb\text{,}\) which nominally happens in the category of sheaves of abelian groups on \(\Prof\text{,}\) can actually be computed in the category of presheaves on \(\Prof\text{.}\) For instance, in the sequence
\begin{equation*} 0 \to \RR_{\disc} \to \RR \to \RR/\RR_{\disc} \to 0 \end{equation*}
the term \(\RR_{\disc}\) is solid, so \(\RR(S) \to (\RR/\RR_{\disc})(S)\) is surjective for \(S \in \Prof\text{;}\) we previously established this by a direct argument (Example 4.1.9).

Subsection 7.4 A characterization of solid condensed abelian groups

We conclude by asserting that every solid condensed abelian group “is” (better, is represented by) a solid abelian group.

Proof.

The category \(\CAb\) admits the objects \(\ZZ[\underline{S}]\) for \(S \in \Prof\) as a family of compact generators. By Lemma 7.3.7, \(\CAb_\solid\) admits \(\prod_\NN \underline{\ZZ}\) as a single compact generator, so we can identify any object as a colimit of a diagram arising from \(\Ab_{\solid}\text{;}\) this yield essential surjectivity. Everything else follows from Proposition 6.4.2 and Corollary 7.2.12.

Remark 7.4.2.

Theorem 7.4.1 has the effect that we can replace the study of \(\CAb_\solid\) with the equivalent but conceptually simpler category \(\Ab_{\solid}\text{,}\) whose definition contains no topology. One can also go in the opposite direction, as in the following.

Remark 7.4.3.

We give an alternate proof of Lemma 2.3.6 in the setting of condensed abelian groups. In this language, the statement is that for \(Q\) a torsion abelian group, \(\iHom_{\CAb}(\underline{Q}, \underline{\ZZ}) = 0\text{.}\) To prove this, evaluate at \(S \in \Prof\text{:}\)
\begin{align*} \iHom_{\CAb}(\underline{Q}, \underline{\ZZ})(S) &= \Hom_{\CAb}(\underline{Q} \otimes \ZZ[\underline{S}], \underline{\ZZ})\\ &= \Hom_{\CAb}(\underline{Q}, \iHom_{\CAb}(\ZZ[\underline{S}], \underline{\ZZ}))\\ & = \Hom_{\TopAb}(Q, \Cts(S, \ZZ)) \end{align*}
using (5.3) at the last step. But already \(\Hom_{\Ab}(Q, \Cts(S, \ZZ)) = 0\) because \(Q\) cannot admit a nonzero map to any torsion-free abelian group.
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