Blueprint for the Liquid Tensor Experiment

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

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).

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

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

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

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.

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

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

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

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

\begin{tikzcd} 
      \hat V(M_{\le c_1}^n)^{T^{-1}}
      \rar{\mathrm{res}} &
      \hat V(M_{\le c' }^n)^{T^{-1}}
      \rar{\hat V(f)} &
      \hat V(M_{\le c_2}^m)^{T^{-1}}
    \end{tikzcd}

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.

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.