# Blueprint for the Liquid Tensor Experiment

## 1.5 Completions of locally constant functions

Definition 1.5.1
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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 $$\hat V(X)$$ 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 $$\hat V(f)$$ are norm-nonincreasing morphisms of semi-normed groups.

Lemma 1.5.2
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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 $$\hat V(X)$$ 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 1.2).

Lemma 1.5.3
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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 1.5.4
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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

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

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

This construction is functorial in $$f$$.

Definition 1.5.5
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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

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

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

This construction is functorial in $$f$$.

Definition 1.5.6
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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

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

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 1.5.7
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There are two natural actions of $$T^{-1}$$ on $$\hat V(M_{\le c})$$. The first comes from the $$r'$$-action of $$T^{-1}$$ on $$M$$ which gives a continuous map

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

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

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

The other comes from Lemma 1.5.2, using the $$r$$-normed $$\mathbb Z[T^{\pm 1}]$$-module $$V$$. Again by extension to completion, we get a map

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

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

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

and we define $$\hat V(M_{\le c})^{T^{-1}}$$ 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 1.4.1.

Lemma 1.5.8
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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 1.4.2 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 1.5.9
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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 1.5.6 restricts to a map

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

Lemma 1.5.10
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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

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

is norm-nonincreasing.

Proof

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

• $$f$$ is bound by $$N$$ (see Definition 1.1.3)

• $$(c_2, c')$$ is $$f$$-suitable

• $$r ^b N ≤ 1$$

• $$c' ≤ (r') ^b c_1$$

Now, notice that the norm of $$\hat V(f)$$ is at most $$N$$, and $$\hat V(f)$$ 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 1.5.11
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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 1.5.9 induced by the universal maps $$f_i$$ from the Breen–Deligne $$\mathsf{BD}= (n,f)$$ assemble into a complex of normed abelian groups

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

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

By Lemma 1.5.10 the differentials are norm-nonincreasing. It is clear that the restriction maps are also norm-nonincreasing, and therefore the system is admissible.