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Notes on condensed mathematics

Kiran S. Kedlaya

Section 6 Solid condensed abelian groups

In this section, we introduce the solid condition on condensed abelian groups, and the concept of solidification which is a sort of universal adic completion. We will later show that the solid condensed abelian groups form a category equivalent to the solid abelian groups (Theorem 7.4.1); this will allow us to use the latter to reason about the former.

Reference.

This section is based on [8], Lecture 5. See also [6], Lecture 5.

Subsection 6.1 The finite difference operator

We next introduce another important structure of the sequence space \(P\) which we use to define solid condensed abelian groups.

Definition 6.1.1.

Define the sequence space \(P := \ZZ[\underline{\NN_\infty}]/\ZZ[\underline{\infty}]\) as in Definition 5.3.1. Consider the map \(n \mapsto n+1\) on \(\NN_\infty\) (fixing \(\infty\)). It is continuous and therefore by adjunction induces a map \(\ZZ[\underline{\NN_\infty}]\text{;}\) the latter map preserves \(\ZZ[\underline{\infty}]\) and thus induces an endomorphism \(\sigma \colon P \to P\text{,}\) which we call the left shift (see below for why “left” instead of “right”).
Define the “finite difference” map \(\Delta \colon P \to P\) by \(\Delta := \id_P - \sigma\text{.}\) Via the ring homomorphism \(\ZZ[x] \to P\) taking \(x^n\) to \([n]\text{,}\) this corresponds to multiplication by \(1-x\text{.}\)
For any topological abelian group \(M\text{,}\) \(\Hom_{\CAb}(P, \underline{M})\) consists of null sequences (Remark 5.3.2) and \(\Delta^*\) acts on these via
\begin{equation*} (m_0, m_1, \dots) \mapsto (m_0-m_1, m_1-m_2, \dots)\text{.} \end{equation*}

Remark 6.1.2.

If \(M\) is an abelian group topologized by some nonarchimedean norm (so in particular the discrete topology is allowed), then every null sequence is summable and so the action of \(\Delta^*\) on \(\Hom_{\CAb}(P, \underline{M})\) is inverted by the “telescoping” map
\begin{equation*} (m_0, m_1, \dots) \mapsto \left(\sum_{n=0}^\infty m_n, \sum_{n=1}^\infty m_n, \dots\right)\text{.} \end{equation*}
In particular, \(\Delta^*\) acts isomorphically on \(\Hom_{\CAb}(P, \underline{M})\text{.}\) We will use this observation as the basis for defining the solid property for condensed abelian groups Definition 6.2.1. For the moment, we observe that the property that every null sequence is uniquely summable is preserved under arbitrary products of topological abelian groups, not just countable ones.

Remark 6.1.3.

The fact that \(\Delta\) is a homomorphism in \(\CAb\) reflects the fact that if two sequences are summable, their pointwise sum is summable and the sum is obtained by adding the sums of the original sequences. One can similarly translate other properties of summation of infinite sequences into algebraic statements about \(P\text{,}\) such as the following.
  • A sequence of the form \((m,0,0,\dots)\) is summable with sum \(m\text{.}\) This corresponds to the fact that the coefficient of \([0]\) in \(\Delta([0])\) is \(1\text{.}\)
  • The sequence \((m_0, m_1, \dots)\) is summable if and only if \((0, m_0, m_1, \dots)\) is summable, in which case the sums coincide. This corresponds to the fact that \(f_1 \circ \Delta = \Delta \circ f_2\) where
    \begin{gather*} f_1 \colon P \to P, \qquad [0] \mapsto [0], [n] \mapsto [n-1] \quad (n \gt 0)\\ f_2 \colon P \to P, \qquad [0] \mapsto [\infty], [n] \mapsto [n-1] \quad (n \gt 0) \end{gather*}
    and so \(f_1([0]) = [0]\) and
    \begin{align*} f_1^*(m_0,m_1,\dots) &= (m_0, m_0, m_1, \dots)\\ f_2^*(m_0,m_1,\dots) &= (0, m_0, m_1, \dots)\\ (f_1 \circ \Delta = \Delta \circ f_2)^*(m_0, m_1, \dots) & = (0, m_0-m_1, m_1-m_2, \dots)\text{.} \end{align*}
  • The sequence \((m_0, m_1, \dots)\) is summable if and only if \((m_0, 0, m_1, 0, \dots)\) is summable, in which case the sums coincide. This corresponds to the fact that \(f_1 \circ \Delta = \Delta \circ f_2\) where
    \begin{gather*} f_1 \colon P \to P, \qquad [n] \mapsto [\lceil n/2 \rceil] \quad (n \geq 0)\\ f_2 \colon P \to P, \qquad [2n] \mapsto [n], [2n+1] \mapsto [\infty] \quad (n \geq 0) \end{gather*}
    and so \(f_1([0]) = [0]\) and
    \begin{align*} f_1^*(m_0,m_1,\dots) &= (m_0, m_1, m_1, m_2, \dots)\\ f_2^*(m_0,m_1,\dots) &= (m_0, 0, m_1, 0, \dots)\\ (f_1 \circ \Delta = \Delta \circ f_2)^*(m_0, m_1, \dots) & = (m_0-m_1, 0, m_1-m_2, 0, \dots)\text{.} \end{align*}
We can thus use the object \(P\) to formulate and study an analogue of the property (of a topological abelian group) that every null sequence is uniquely summable. We pick up this thread in Section 7.

Remark 6.1.4.

By analogy with the split exact sequence
\begin{equation*} 0 \to \ZZ[x] \stackrel{\times 1-x}{\to} \ZZ[x] \stackrel{Q(x) \mapsto Q(1)}{\longrightarrow} \ZZ \to 0\text{,} \end{equation*}
one might expect that \(\coker(\Delta) \cong \underline{\ZZ}\) via the map which sends \([n]\) to \(1\) for all \(n \in \NN\text{.}\) But this map is not continuous for the discrete topology on \(\ZZ\text{:}\) the sequence \(\{[n]\}_n\) is a null sequence in \(P\text{,}\) so it cannot map to a nonzero constant sequence.
In fact \(\coker(\Delta)\) is a sheaf that assigns \(S \in \Prof\) to a certain subset of the functions \(S \to \ZZ\text{:}\) these are functions with bounded image with locally closed level sets. For example, for any closed subset \(U \subseteq S\) we get the function mapping \(U\) to 1 and \(S \setminus U\) to 0. This does not correspond to any topology on \(\ZZ\text{.}\)
We end up with a nonsplit exact sequence in \(\CAb\text{:}\)
\begin{equation} 0 \to P \stackrel{\Delta}{\to} P \to \coker(\Delta) \to 0\text{.}\tag{6.1} \end{equation}
This in turn induces an exact sequence
\begin{equation} 0 \to P \stackrel{\Delta}{\to} P/\ZZ \to \coker(\Delta)/\ZZ \to 0\tag{6.2} \end{equation}
which (because \(P/\ZZ \cong P\)) is an internally projective resolution of \(\coker(\Delta)/\ZZ\) of length \(1\text{.}\) See Remark 10.3.5 for an application of this construction.

Subsection 6.2 Solid condensed abelian groups

Definition 6.2.1.

An object \(M \in \CAb\) is solid if
\begin{equation} \Delta^*\colon \iHom_{\CAb}(P, M) \to \iHom_{\CAb}(P, M)\tag{6.3} \end{equation}
is an isomorphism in \(\CAb\text{.}\) Let \(\CAb_\solid\) denote the full subcategory of \(\CAb\) consisting of solid objects.
We may also equate (6.3) with the condition that
\begin{equation} \Delta^*\colon R\iHom_{\CAb}(P, M) \to R\iHom_{\CAb}(P, M)\tag{6.4} \end{equation}
is an isomorphism, as by Proposition 5.4.8 all of the higher Ext objects are automatically zero.
Recall that we already introduced another notion of solid \(\ZZ\)-modules in Section 1 (which we started calling solid abelian groups in Section 2). We will reconcile terminology later by exhibiting an equivalence between \(\CAb_\solid\) and the category \(\Ab_\solid\) of solid abelian groups (Theorem 7.4.1).
Our first order of business is to check that Definition 6.2.1 is nonempty. This amounts to a mild upgrade of Remark 6.1.2 in the simplest case: we must replace the external Hom with an internal Hom.

Proof.

We must check that \(\Delta^*\) is an isomorphism on \(\iHom_{\CAb}(P, \underline{\ZZ})\text{.}\) By (5.2) and Hom-tensor adjunction, this sheaf on \(\Prof\) can be written as
\begin{align*} S & \mapsto \Hom_{\CAb}(P \otimes \ZZ[\underline{S}], \underline{\ZZ}) \\ & = \Hom_{\CAb}(\ZZ[\underline{\NN_\infty \times S}]/\ZZ[\underline{\{\infty\} \times S}], \underline{\ZZ})\text{.} \end{align*}
The latter group consists of the continuous maps \(f\colon \NN_\infty \times S \to \ZZ\) which vanish on \(\{\infty\} \times S\text{.}\) The action of \(\Delta^*\) carries \(f\) to the map
\begin{equation*} (n,s) \mapsto f(n,s) - f(n+1,s); \end{equation*}
it is inverted by the map taking \(f\) to
\begin{equation*} (n,s) \mapsto f(n,s) + f(n+1,s) + \cdots \end{equation*}
where again continuity of the map ensures that the sum is finite.
Alternatively, (5.3), \(\iHom_{\CAb}(P, \underline{\ZZ})\) is represented by the subgroup of the topological group \(\Cts(\NN_\infty,\ZZ)\) consisting of continuous functions mapping \(\infty\) to \(0\text{.}\) Then the argument in Remark 6.1.2 can be upgraded to show that \(\Delta^*\) induces a homeomorphism of this topological group with itself.

Remark 6.2.3.

By (5.2) if \(M \in \CAb\) is solid, then the map \(\Delta^*\colon \Hom_{\CAb}(P, M) \to \Hom_{\CAb}(P, M)\) is an isomorphism. The proof of Lemma 6.2.2 shows that the converse holds when \(M \in \TopAb\text{,}\) but it can fail in other cases, e.g., for Example 5.1.4.

Proof.

Closure under limits and colimits (including kernels and cokernels) follows from the fact that by Proposition 5.4.8, the functor \(\iHom_{\CAb}(P, \bullet)\) commutes with all limits and colimits. Closure under extensions is immediate from (6.4). Closure under \(N \mapsto R\iHom_{\CAb}(\bullet,N)\) follows from Hom-tensor adjunction (and Proposition 5.4.8 to conflate \(\iHom_{\CAb}(P, N)\) with \(R\iHom_{\CAb}(P, N)\)):
\begin{align*} \iHom_{\CAb}(P, R\iHom_{\CAb}(\bullet,N)) &\cong R\iHom_{\CAb}(P \otimes \bullet,N)\\ & \cong R\iHom_{\CAb}(\bullet \otimes P,N)\\ & \cong \iHom_{\CAb}(\bullet, R\iHom_{\CAb}(P,N))\text{.} \end{align*}

Definition 6.2.5.

By the adjoint functor theorem, the embedding of \(\CAb_\solid \to \CAb\) admits a left adjoint. We denote this by \(M \mapsto M_\solid\) and refer to it as solidification.
Since Proposition 6.2.4 does not include compatibility with tensor products, we instead define the solid tensor product on \(\CAb_\solid\) by the formula
\begin{equation*} M \otimes_\solid N := (M \otimes N)_\solid. \end{equation*}
For example, for \(M = N = \prod_\NN \underline{\ZZ}\text{,}\) \(M \otimes_\solid N = \prod_{\NN \times \NN} \underline{\ZZ}\) but this is not equal to \(M \otimes N\text{.}\)
As usual for a functor defined as an adjoint, the definition does not give much direct control over the behavior of solidification on explicit objects. Instead, we will have to take more indirect steps to evaluate some solidifications.

Remark 6.2.6.

Using (6.1), we can make a number of formal observations about Definition 6.2.1 and Definition 6.2.5. To begin with, we may rewrite (6.4) as the statement that
\begin{equation*} R\iHom_{\CAb}(\coker(\Delta), M) = 0. \end{equation*}
By (6.1) and Proposition 5.4.8, this in turn is equivalent to
\begin{equation*} \iHom_{\CAb}(\coker(\Delta), M) = 0, \qquad \iExt^1_{\CAb}(\coker(\Delta), M) = 0. \end{equation*}
Turning to solidification, note that \(\Delta_\solid\) induces an isomorphism on \(\Hom_{\CAb_\solid}(P_\solid, M) = \Hom_{\CAb}(P,M)\) for any \(M \in \CAb_\solid\text{,}\) so by Yoneda’s lemma it is itself an isomorphism. Consequently,
\begin{equation*} \coker(\Delta)_\solid = \coker(\Delta_\solid) = 0\text{;} \end{equation*}
since \(\Delta_\solid\) also induces an isomorphism on \(\Ext^i_{\CAb_\solid}(P_\solid, M) = \Ext^i_{\CAb}(P,M)\) for any \(i \gt 0\text{,}\) the derived solidification of \(\coker(\Delta)\) also vanishes.
Using the exact sequence
\begin{equation*} 0 \to \underline{\ZZ} \to \coker(\Delta) \to \coker(\Delta)/\underline{\ZZ} \to 0 \end{equation*}
we obtain the example \(\coker(\Delta)/\underline{\ZZ}\) whose solidification is zero, but whose derived solidification consists of \(\ZZ\) in degree \(-1\text{.}\) For \(M \in \CAb\) general, we obtain a canonical morphism
\begin{equation*} M = \iHom_{\CAb}(\underline{\ZZ}, M) \to \iExt^1_{\CAb}(\coker(\Delta)/\underline{\ZZ}, M) \end{equation*}
which is an isomorphism if and only if \(M \in \CAb_\solid\text{.}\)

Remark 6.2.7.

We say that an object \(C\) in the derived category \(D(\CAb)\) is solid if its cohomology groups belong to \(\CAb_\solid\text{.}\) It is equivalent (thanks to Proposition 5.4.8) to require
\begin{equation*} \Delta^*\colon R\iHom_{\CAb}(P, C) \to R\iHom_{\CAb}(P,C) \end{equation*}
to be an isomorphism. As in Remark 6.2.6, we have a canonical morphism
\begin{equation*} C \to R\iHom_{\CAb}(\coker(\Delta)/\underline{\ZZ}, C)[-1] \end{equation*}
which is an isomorphism if and only if \(C\) is solid. The forgetful functor from the full subcategory of solid objects of \(D(\CAb)\) back to \(D(\CAb)\) again has a left adjoint, called derived solidification.
Even when we start with a map \(M \to N\) in \(\CAb\text{,}\) there is a difference between saying that this map induces an isomorphism of solidifications versus an isomorphism of derived solidifications; the first is equivalent to having the natural isomorphism \(\iHom_{\CAb}(N, \bullet) \cong \iHom^i_{\CAb}(M, \bullet)\) of functors on \(\CAb_\solid\text{,}\) whereas the latter also includes the natural isomorphisms \(\iExt^i_{\CAb}(N, \bullet) \cong \iExt^i_{\CAb}(M, \bullet)\) for all \(i \gt 0\text{.}\)

Subsection 6.3 Solidification and the real numbers

As the definition of a solid condensed abelian group is meant to simulate the condition that null sequences are summable, it is perhaps not surprising that it raises havoc in an archimedean context. We record some results both to illustrate the incompatibility between solidification and the topology of the reals, but also to provide a technique that will be used in the course of comparing solid abelian groups with their condensed counterparts (see Lemma 7.1.2).

Proof.

The null sequence \(\{2^{-\lfloor \log_2 n \rfloor}\}_n\) in \(\RR\text{,}\) i.e.,
\begin{equation} 1, \tfrac{1}{2}, \tfrac{1}{2}, \tfrac{1}{4}, \tfrac{1}{4}, \tfrac{1}{4}, \tfrac{1}{4}, \dots\tag{6.5} \end{equation}
defines an element \(x \in \Hom_{\CAb}(P, \underline{\RR})\text{;}\) we may thus apply the inverse of \(\Delta^*\) to get an element \(y \in \Hom_{\CAb}(P, \underline{\RR}_\solid)\text{.}\) Taking the first coordinate (i.e., restricting along \(\ZZ \to \ZZ \cdot [1] \subset P\)) yields an element \(z \in \underline{\RR}_\solid\text{.}\)
By formally rearranging (6.5) as
\begin{equation*} 1 + \left( \tfrac{1}{2} + \tfrac{1}{2} \right) + \left( \tfrac{1}{4} + \tfrac{1}{4} \right) + \left( \tfrac{1}{4} + \tfrac{1}{4} \right) + \cdots \end{equation*}
and invoking Remark 6.1.3, we deduce that \(z = 1 + z\text{.}\) We deduce that \(0 = 1\) in \(\underline{\RR}_\solid\text{;}\) since \(\underline{\RR}_\solid\) is a ring object in \(\CAb\) (because \(\underline{\RR}\) is), it follows that \(\underline{\RR}_\solid = 0\text{.}\)

Proof.

The \(\underline{\RR}\)-module structure on \(M\) formally promotes to an \(\underline{\RR}_\solid\)-module structure on \(M_\solid\text{,}\) but then Proposition 6.3.1 implies that \(M_\solid\) is a module over the zero ring.

Proof.

For each \(i \gt 0\text{,}\) for any \(Q \in \CAb_\solid\text{,}\) \(\iExt^i_{\CAb}(N/M, Q)\) is solid by Proposition 6.2.4 and inherits an \(\underline{\RR}\)-module structure from \(N/M\text{,}\) hence is zero. This implies the second assertion, including the case \(i = 0\) from which we recover the first assertion using Yoneda’s lemma.

Remark 6.3.4.

By analogy with Remark 6.2.6, we can apply Proposition 6.3.1 to see that the derived solidification of \(\underline{\RR}/\underline{\ZZ}\) consists of \(\underline{\ZZ}\) placed in degree \(-1\text{.}\)

Remark 6.3.5.

Proposition 6.3.1 and its corollaries also apply with \(\underline{\RR}\) replaced by \(\underline{\QQ}\) equipped with the archimedean topology. By contrast, for the \(p\)-adic topology for some prime \(p\text{,}\) the solidification of \(\underline{\QQ}\) will end up being precisely \(\underline{\QQ_p}\text{.}\)

Remark 6.3.6.

Proposition 6.3.1 and its corollaries make clear that if we want any hope of doing analytic geometry over \(\RR\) or \(\CC\text{,}\) we cannot stick to solid abelian groups and must instead find a more flexible framework. We will eventually introduce such a framework in the form of analytic rings, where we define a class of “complete modules” with properties analogous to those of solid abelian groups (Section 10).

Subsection 6.4 Solid abelian groups as condensed abelian groups

At this point, we can exhibit a functor from solid \(\ZZ\)-modules to solid objects in \(\CAb\text{.}\) We will prove later that this yields an equivalence (Theorem 7.4.1).

Proof.

Since both objects are represented by sequential topological abelian groups, we may make the computation there, but we have already done this in Remark 2.1.1.

Proof.

By Lemma 6.2.2 and Proposition 6.2.4, we have \(\prod_I \underline{\ZZ} \in \CAb_\solid\) whenever \(I\) is at most countable. By Lemma 6.4.1, we get a fully faithful functor \(\ZZ_\solid \to \CAb_\solid\) carrying \(\prod_I \ZZ_\solid\) to \(\prod_I \underline{\ZZ}\text{.}\) Since \(\prod_\NN \ZZ_\solid\) is a compact generator of \(\Ab_\solid\text{,}\) we get the functor we want by forming a left Kan extension.

Remark 6.4.3.

We note in passing that Proposition 6.4.2 admits the following compatibility with solidification: applying the functor \(\Ab \to \Ab_\solid\) and then passing to \(\CAb\) gives the same result as passing from \(\Ab\) to \(\CAb\) (i.e., passing to discrete topological groups). In both cases the essential image is contained in \(\CAb_\solid\text{.}\)
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