# 6 Completions of locally constant functions

Definition 6.1
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Let $V$ be a semi-normed group, and $X$ a compact topological space. We denote by $V(X)$ the normed abelian group of locally constant functions $X \to V$ with respect to the sup norm. With $\widehat V(X)$ we denote the completion of $V(X)$.

These constructions are functorial in bounded group homomorphisms $V \to V'$ and contravariantly functorial in continuous maps $f \colon X \to X'$.

Note in particular that $V(f)$ and $\widehat V(f)$ are norm-nonincreasing morphisms of semi-normed groups.

Lemma 6.2
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Let $r \in \mathbb R_{{\gt} 0}$, and let $V$ be an $r$-normed $\mathbb Z[T^{\pm 1}]$-module. Let $X$ be a compact space. Then $\widehat V(X)$ is naturally an $r$-normed $\mathbb Z[T^{\pm 1}]$-module, with the action of $T$ given by post-composition.

Proof

Formalised, but omitted from this text.

We continue to use the notation of before: let $r' {\gt} 0, c \ge 0$ be real numbers, and let $M$ be a profinitely filtered pseudo-normed group with $r'$-action by $T^{-1}$ (see Section 3).

Lemma 6.3
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Let $f$ be a basic universal map from exponent $m$ to $n$. We get an induced homomorphism of profinitely filtered pseudo-normed groups $M^m \to M^n$ bounded by the maximum (over all $i$) of $\sum _j |f_{ij}|$, where the $f_{ij}$ are the coefficients of the $n \times m$-matrix representing $f$.

This construction is functorial in $f$.

Proof

Omitted.

Definition 6.4
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Let $f$ be a basic universal map from exponent $m$ to $n$, and let $(c_2, c_1)$ be $f$-suitable. We get an induced map

$V(f) \colon V(M_{\le c_1}^n) \to V(M_{\le c_2}^m)$

induced by the morphism of profinitely filtered pseudo-normed groups $M^m \to M^n$.

This construction is functorial in $f$.

Definition 6.5
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Let $f = \sum _g n_g g$ be a universal map from exponent $m$ to $n$, and let $(c_2, c_1)$ be $f$-suitable. We get an induced map

$V(f) \colon V(M_{\le c_1}^n) \to V(M_{\le c_2}^m)$

that is the sum $\sum n_g V(g)$.

This construction is functorial in $f$.

Definition 6.6
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Let $f$ be a universal map from exponent $m$ to $n$, and let $(c_2, c_1)$ be $f$-suitable. We get an induced map

$\widehat V(f) \colon \widehat V(M_{\le c_1}^n) \to \widehat V(M_{\le c_2}^m)$

that is the completion of $V(f)$.

This construction is functorial in $f$.

Let $r {\gt} 0$, and assume now that $V$ is an $r$-normed $\mathbb Z[T^{\pm 1}]$-module. Assume $r' \le 1$.

Definition 6.7
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There are two natural actions of $T^{-1}$ on $\widehat V(M_{\le c})$. The first comes from the $r'$-action of $T^{-1}$ on $M$ which gives a continuous map

$M_{\le cr'} \to M_{\le c}$

and thus a normed group morphism $V(M_{\le c}) \to V(M_{\le cr'})$ which can be extended by completion to

$(T^{-1})^* \colon \widehat V(M_{\le c}) \to \widehat V(M_{\le cr'}).$

The other comes from Lemma 6.2, using the $r$-normed $\mathbb Z[T^{\pm 1}]$-module $V$. Again by extension to completion, we get a map

$[T^{-1}] \colon \widehat V(M_{\le c}) \to \widehat V(M_{\le c}),$

that we can compose with the map $\widehat V(M_{\le c}) \to \widehat V(M_{\le cr'})$, obtained from the natural inclusion $M_{\le cr'} \to M_{\le c}$. We thus end up with two maps

$(T^{-1})^*, [T^{-1}] \colon \widehat V(M_{\le c}) \to \widehat V(M_{\le cr'}).$

and we define $\widehat V(M_{\le c})^{T^{-1}}$ to be the equalizer of $(T^{-1})^*$ and $[T^{-1}]$. In other words, the kernel of $(T^{-1})^* - [T^{-1}]$.

We will also need to understand the image of $(T^{-1})^* - [T^{-1}]$. The next lemma ensures it is surjective with controlled preimages, see Definition 5.1.

Lemma 6.8
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Let $M$ be a profinitely filtered pseudo-normed group with action of $T^{-1}$. For any $r ∈ (0, 1)$, any $r$-normed $\mathbb Z[T^{\pm 1}]$-module $V$, any $c{\gt}0$ and any $a$, the map

$\widehat{V}(M_{\leq c}^a)\xrightarrow {T^{-1}-[T^{-1}]^\ast } \widehat{V}(M_{\leq r'c}^a)$

has norm bounded by $r^{-1}+1$ and is $\frac{r}{1-r}(1+\epsilon )$-surjective.

Proof

The norm bound is clear because $[T^{-1}]^\ast$ is norm non-increasing and $T⁻¹$ scales norm by $r⁻¹$. Quantitative surjectivity will follow from Lemma 5.2 once we’ll have proven that $T^{-1}-[T^{-1}]^\ast : \widehat{V}(M_{\leq c}^a) → \widehat{V}(M_{\leq r'c}^a)$ is $r/(1-r)$-surjective onto $V(M_{\leq r'c}^a)$.

We first note that any locally constant function $φ ∈ V(M_{\leq r'c}^a)$ can be extended to a locally constant function $\barφ ∈ V(M_{\leq c}^a)$ with the same norm (recall $f$ takes finitely many values and its norm is the maximum of norms of these values).

Let $f$ be any element of $V(M_{\leq r'c}^a)$. We inductively define a sequence of locally constant functions $h_n ∈ V(M_{\leq c}^a)$ with $h_0 = T ∘ \bar f$ and $h_{n+1} = T ∘ \overline{[T^{-1}]^\ast h_n}$. Here we use the composition symbol to emphasize this is indeed the naive post-composition with $T$, there is no extra precomposition with a the inclusion map $ι : M_{\leq r'c}^a ↪ M_{\leq c}^a$ as in the definition of $T⁻¹$ seen as a map from $V(M_{\leq c}^a)$ to $V(M_{\leq r'c}^a)$.

Since $[T^{-1}]^\ast$ is norm non-increasing, extension is norm preserving and $T$ scales norm by $r$, we get that $\| h_n\| ≤ r^{n+1}\| f\|$. We then set $g_n = \sum _{i = 0}^n h_i$. The norm estimate on $h_n$ ensures $g$ is a Cauchy sequence in $V(M_{\leq c}^a)$ hence it converges to some $g$ in $\widehat{V}(M_{\leq c}^a)$. We compute:

\begin{align*} (T^{-1}-[T^{-1}]^\ast )g_n & = ∑_{k=0}^n \Big(T⁻¹h_k - [T^{-1}]^\ast h_k\Big) \\ & = T⁻¹h_0 + ∑_{k=0}^{n-1} \Big(T⁻¹h_{k+1} - [T^{-1}]^\ast h_k\Big) - [T^{-1}]^\ast h_n\\ & = \bar f ∘ ι + ∑_{k=0}^{n-1} \Big( T⁻¹∘ T ∘ \overline{[T^{-1}]^\ast h_k} ∘ ι - [T^{-1}]^\ast h_k\Big) - [T^{-1}]^\ast h_n\\ & = f - [T^{-1}]^\ast h_n \end{align*}

which converges to $f$ hence $(T^{-1}-[T^{-1}]^\ast )g = f$. In addition $\| g\| ≤ \sum _n r^{n+1}\| f\| = r/(1-r)\| f\|$.

Definition 6.9
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Let $f$ be a universal map from exponent $m$ to $n$, and let $(c_2, c_1)$ be $f$-suitable.

The natural map from Definition 6.6 restricts to a map

$\widehat V(f)^{T^{-1}} \colon \widehat V(M_{\le c_1}^n)^{T^{-1}} \to \widehat V(M_{\le c_2}^m)^{T^{-1}}$

Lemma 6.10
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Let $0 {\lt} r$ and $0 {\lt} r' \le 1$ be real numbers. Let $f$ be a universal map from exponent $m$ to $n$, and let $(c_2, c_1)$ be very suitable for $(f, r, r')$. Then

$\widehat V(f)^{T^{-1}} \colon \widehat V(M_{\le c_1}^n)^{T^{-1}} \to \widehat V(M_{\le c_2}^m)^{T^{-1}}$

is norm-nonincreasing.

Proof

Use the assumption that $(c_2, c_1)$ is very suitable for $(f, r, r')$ in order to find $N, b \in \mathbf{N}$ and $c' \in \mathbf{R}_{\ge 0}$ such that:

• $f$ is bound by $N$ (see Definition 2.4)

• $(c_2, c')$ is $f$-suitable

• $r ^b N ≤ 1$

• $c' ≤ (r') ^b c_1$

Now, notice that the norm of $\widehat V(f)$ is at most $N$, and $\widehat V(f)$ can be factored as

Now use the defining property of the equalizer to conclude that the restriction map has norm less than $1/N$, and therefore the composition is norm-nonincreasing.

Definition 6.11
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Let $0 {\lt} r$ and $0 {\lt} r' \le 1$ be real numbers, and let $V$ be an $r$-normed $\mathbb Z[T^{\pm 1}]$-module. Let $\mathsf{BD}= (n,f)$ be Breen–Deligne data, and let $\kappa = (\kappa _0, \kappa _1, \kappa _2, \dots )$ be a sequence of constants in $\mathbb R_{\ge 0}$ that is very suitable for $(\mathsf{BD}, r, r')$. Let $M$ be a profinitely filtered pseudo-normed group with $r'$-action of $T^{-1}$.

For every $c \in \mathbb R_{\ge 0}$, the maps from Definition 6.9 induced by the universal maps $f_i$ from the Breen–Deligne $\mathsf{BD}= (n,f)$ assemble into a complex of normed abelian groups

$C^{\text{BD}}_{\kappa }(M)_c^\bullet \colon 0 \to \dots \to \widehat V(M_{\le \kappa _i}^{n_i})^{T^{-1}} \to \widehat V(M_{\le \kappa _{i+1}}^{n_{i+1}})^{T^{-1}} \to \dots .$

Together, these complexes fit into a system of complexes with the natural restriction maps.

By Lemma 6.10 the differentials are norm-nonincreasing. It is clear that the restriction maps are also norm-nonincreasing, and therefore the system is admissible.