Blueprint for the Liquid Tensor Experiment: Key technical result

1.7 Key technical result

Now we state the following result, which is the key technical result on our to the main goal.

Theorem 1.7.1 ✓

Let $\mathsf{BD}= (n,f,h)$ be a Breen–Deligne package, and let $\kappa = (\kappa _0, \kappa _1, \kappa _2, \dots )$ be a sequence of constants in $\mathbb R_{\ge 0}$ that is $\mathsf{BD}$-suitable. Fix radii $1{\gt}r'{\gt}r{\gt}0$. For any $m$ there is some $k$ and $c_0$ such that for all profinite sets $S$ and all $r$-normed $\mathbb Z[T^{\pm 1}]$-modules $V$, the system of complexes

\[ C^{\mathsf{BD}}_{\kappa }(\overline{\mathcal L}_{r'}(S))_c^\bullet \colon \widehat{V}(\overline{\mathcal L}_{r'}(S)_{\leq c})^{T^{-1}} \longrightarrow \widehat{V}(\overline{\mathcal L}_{r'}(S)_{\leq \kappa _1c}^2)^{T^{-1}} \longrightarrow \ldots \]

is $\leq k$-exact in degrees $\leq m$ for $c\geq c_0$.

We will prove Theorem 1.7.1 by induction on $m$. Unfortunately, the induction requires us to prove a stronger statement.

Theorem 1.7.2 ✓

Fix radii $1{\gt}r'{\gt}r{\gt}0$. For any $m$ there is some $k$ such that for all polyhedral lattices $\Lambda $ there is a constant $c_0(\Lambda ){\gt}0$ such that for all profinite sets $S$ and all $r$-normed $\mathbb Z[T^{\pm 1}]$-modules $V$, the system of complexes

\[ C_{\Lambda ,c}^\bullet \colon \widehat{V}(\operatorname{Hom}(\Lambda ,\overline{\mathcal L}_{r'}(S))_{\leq c})^{T^{-1}} \longrightarrow \widehat{V}(\operatorname{Hom}(\Lambda ,\overline{\mathcal L}_{r'}(S))_{\leq \kappa _1c}^2)^{T^{-1}} \longrightarrow \ldots \]

is $\leq k$-exact in degrees $\leq m$ for $c\geq c_0(\Lambda )$.

Proof of Theorem 1.7.1 ▼

Use $\Lambda = \mathbb Z$, and the isomorphism $\operatorname{Hom}(\mathbb Z, A) \cong A$.

A word on universal constants: We fix once and for all, the constants $0 {\lt} r {\lt} r' \le 1$ a Breen–Deligne package $\mathsf{BD}$, and a sequence of positive constants $\kappa $ that is very suitable for $(\mathsf{BD}, r, r')$. Once the full proof is formalized, we can come back to this place and write a bit more about the other constants.

The global strategy of the proof is to construct a system of double complexes such that its first row is the system $C_{\Lambda , \bullet }^\bullet $ occuring in Theorem 1.7.2. We can then verify the conditions to Proposition 1.4.15 and conclude from there. For the time being, we will let $M$ denote an arbitrary profinitely filtered pseudo-normed group with action of $T^{-1}$, and whenever needed we can specialize to $M = \overline{\mathcal L}_{r'}(S)$.

Further choices of constants: We will argue by induction on $m$, so assume the result for $m-1$ (this is no assumption for $m=0$, so we do not need an induction start). This gives us some $k{\gt}1$ for which the statement of Theorem 1.7.2 holds true for $m-1$; if $m=0$, simply take any $k{\gt}1$. In the proof below, we will increase $k$ further in a way that depends only on $m$ and $r$. After this modified choice of $k$, we fix $\epsilon $ and $k_0$ as provided by Proposition 1.4.15. Fix a sequence $(\kappa '_i)_i$ of nonnegative reals that is adept to $(\mathsf{BD}, \kappa )$. (Such a sequence exists by Lemma 1.1.20.) Moreover, we let $k'$ be the supremum of $k_0$ and the $c_i'$ for $i=0,\ldots ,m+1$. Finally, choose a positive integer $b$ so that $2k'(\tfrac r{r’})^b\leq \epsilon $, and let $N$ be the minimal power of $2$ that satisfies

\[ k'/N\leq (r')^b. \]

Then in particular $r^bN\leq 2k'(\tfrac {r}{r’})^b\leq \epsilon $.

Definition 1.7.3 ✓

Let $\Lambda ^{(\bullet )}$ be the cosimplicial polyhedral lattice of Definition 1.6.11, and recall from 1.6.12 that $\operatorname{Hom}(\Lambda ^{(m)}, M)$ is a profinitely filtered pseudo-normed group with action of $T^{-1}$.

Hence $\operatorname{Hom}(\Lambda ^{(\bullet )}, M)$ is a simplicial profinitely filtered pseudo-normed group with action of $T^{-1}$.

Now apply the construction of the system of complexes from Definition 1.5.11 to obtain a cosimplicial system of complexes

\[ C^{\mathsf{BD}}_{\kappa }(\operatorname{Hom}(\Lambda ^{(\bullet )}, M))_\bullet ^\bullet . \]

Now take the alternating face map cochain complex to obtain a system of double complexes, whose objects are

\[ \widehat{V}(\operatorname{Hom}(\Lambda ^{(m)},M)_{\leq \kappa _ic}^{n_i})^{T^{-1}}. \]

As final step, rescale the norm on the object in row $m$ by $m!$, so that all columns become admissible: the vertical differential from row $m$ to row $m+1$ is an alternating sum of $m+1$ maps that are all norm-nonincreasing.

Lemma 1.7.4 ✓

In particular, for any $c{\gt}0$, we have

\[ \operatorname{Hom}(\Lambda ',M)_{\leq c} = \operatorname{Hom}(\Lambda ,M)_{\leq c/N}^N, \]

with the map to $\operatorname{Hom}(\Lambda ,M)_{\leq c}$ given by the sum map.

Proof ▼

Omitted (but done in Lean).

Lemma 1.7.5 ✓

Similarly, for any $c{\gt}0$, we have

\[ \operatorname{Hom}(\Lambda '^{(m)},M)_{\leq c} = \operatorname{Hom}(\Lambda ',M)_{\leq c}^{m/\operatorname{Hom}(\Lambda ,M)_{\leq c}}, \]

the $m$-fold fibre product of $\operatorname{Hom}(\Lambda ',M)_{\leq c}$ over $\operatorname{Hom}(\Lambda ,M)_{\leq c}$.

Proof ▼

Omitted (but done in Lean).

Lemma 1.7.6 ✓

There is a canonical isomorphism between the first row of the double complex

\[ C^{\mathsf{BD}}_{\kappa }(\operatorname{Hom}(\Lambda ^{(1)}, M))^\bullet \]

and

\[ C^{N \otimes \mathsf{BD}}_{\kappa /N}(\operatorname{Hom}(\Lambda , M))^\bullet \]

which identifies the map induced by the diagonal embedding $\Lambda \to \Lambda ' = \Lambda ^{(1)}$ with the map induced by $\sigma ^N \colon N \otimes \mathsf{BD}\to \mathsf{BD}$.

Proof ▼

Omitted (but done in Lean).

Proposition 1.7.7 ✓

Let $\pi : X \to B$ be a surjective morphism of profinite sets, and let $S_\bullet \to S_{-1}$, $S_{-1} := B$, be its augmented Čech nerve. Let $V$ be a semi-normed group. Then the complex

\[ 0\longrightarrow \widehat{V}(S_{-1})\longrightarrow \widehat{V}(S_0)\longrightarrow \widehat{V}(S_1)\longrightarrow \cdots \]

is exact. Furthermore, for all $\epsilon {\gt} 0$ and $f \in \ker (\widehat{V}(S_{m}) \to \widehat{V}(S_{m+1}))$, there exists some $g\in \widehat{V}(S_{m-1})$ such that $d(g) = f$ and $‖g‖\leq (1+\epsilon ) \cdot ‖f‖$. In other words, the complex is normed exact in the sense of Definition 1.4.6.

Proof ▼

We argue similarly to [ Sch19 , Theorem 3.3 ] , as follows. By applying Lemma 1.4.3, we first reduce to a statement which does not involve $\epsilon $ or completions. Explicitly, we must show that

\[ 0 \longrightarrow V(S_{1}) \longrightarrow V(S_{0}) \longrightarrow V(S_{1}) \longrightarrow \cdots \]

is exact, and that whenever $f \in \ker (V(S_{m}) \to V(S_{m+1}))$, there exists $g \in V(S_{m-1})$ such that $‖g‖\leq ‖f‖$ and $d(g) = f$. The map $V(S_{-1}) \to V(S_{0})$ is the one induced by $S_0 \to S_{-1}$ which agrees with $X \to B$. Since $X \to B$ is surjective, we easily see that $V(S_{-1}) \to V(S_{0})$ is injective.

If $X$ and $B$ are finite, then the remaining assertions follow from the existence of a splitting $\sigma : B \to X$ of $\pi : X \to B$, as follows. The map $\sigma $ provides maps $S_{m} \to S_{m+1}$ for all $m \geq -1$, defined explicitly as

\[ (a_{0},\ldots ,a_{m}) \mapsto (\sigma (\pi (a_{0})), a_{0},\ldots ,a_{m}) \]

if $m \geq 0$ and simply as $\sigma $ if $m = -1$. Here, for $m \geq 0$, we have identified $S_{m}$ with the $m+1$-fold fibered product $X \times _{B} \cdots \times _{B} X$. Applying $V(-)$, these maps induce $h_{m} : V(S_{m+1}) \to V(S_{m})$, which form a contracting homotopy for the complex in question, and which are norm nonincreasing by the definition of $V(-)$. If $f \in \ker (V_{m} \to V_{m+1})$ is as above, then $g := h_{m}(f)$ satisfies $d(g) = f$ and $‖g‖\leq ‖f‖$, as required.

In the general case, write $X = \varprojlim _{i} X_{i}$ where $X_{i}$ vary over the discrete (hence finite) quotients of $X$. Since $X \to B$ is surjective, for each $i$ there exists a unique maximal discrete quotient $B_{i}$ of $B$ such that $X \to B$ descends to $X_{i} \to B_{i}$. The maps $X_{i} \to B_{i}$ are again surjective, and one has

\[ (X \longrightarrow B) = \varprojlim _{i} (X_{i} \longrightarrow B_{i}). \]

Let $S_{i,\bullet } \to S_{i,-1}$, $S_{i,-1} := B_{i}$, denote the augmented Čech nerve of $X_{i} \to B_{i}$.

The terms in the Čech nerve are themselves limits, hence we have $S_{m} = \varprojlim _{i} S_{i,m}$, with each $S_{i,m}$ finite. The functor $V(-)$, when considered as taking values in abelian groups, sends cofiltered limits to filtered colimits. Also, if $f \in V(S_{m})$ is the pullback of $f_{i} \in V(S_{i,m})$, then for a sufficiently small index $j \leq i$, the image of $f : S_{m} \to V$ agrees with the image of $f_{j} : S_{j,m} \to V$, where $f_{j}$ is the image of $f_{i}$ under the map $V(S_{i,m}) \to V(S_{j,m})$ induced by the transition map $S_{j,m} \to S_{i,m}$.

Now suppose that $f \in \ker (V(S_{m}) \to V(S_{m+1}))$ is given. By the discussion above, there exists some $i$ and some $f_{i} \in V(S_{i,m})$ such that $f$ is the pullback of $f_{i}$ with respect to the morphism $S_{m} \to S_{i,m}$ and such that the following additional conditions hold:

One has $‖f_{i}‖ = ‖f‖$.
One has $f_{i} \in \ker (V(S_{i,m}) \to V(S_{i,m+1}))$.

Let $h_{m} : V(S_{i,m}) \to V(S_{i,m-1})$ be the map constructed in the argument for the finite case $X_{i} \to B_{i}$. Put $g_{i} := h_{m} (f_{i})$ and $g$ the image of $g_{i}$ in $V(S_{m-1})$. Since the maps $V(S_{i,\bullet }) \to V(S_{\bullet })$ commute with the differentials, we have $d(g) = f$. Finally, the map $V(S_{i,m-1}) \to V(S_{m-1})$ is norm nonincreasing as it is induced from $S_{m-1} \to S_{i,m-1}$, so that

\[ ‖g‖\leq ‖g_{i}‖ \le ‖f_{i}‖ = ‖f‖, \]

as contended.

Lemma 1.7.8 ✓

Let $M$ be a profinitely filtered pseudo-normed group with $T^{-1}$-action that is $N$-splittable with error term $d \ge 0$. Let $k \ge 1$ be a real number, and let $c_0 {\gt} 0$ satisfy $d \le \frac{(k - 1) c_0}{N}$. For every $c$, consider the Cech nerve of the summation map $M^N_{c/N} \to M_c$. By applying the functor $\hat V(\_ )$ and taking the alternating face map complex, we obtain a system of complexes

\[ \hat V(M_{\le c}) \longrightarrow \hat V(M^N_{\le c/N}) \longrightarrow \dots \]

This system of complexes is weakly $\le k$-exact in degrees $\le m$ and for $c \ge c_0$ with bound $1$.

Proof ▼

For every constant $c$, consider the pullback

$\begin{tikzcd} & M_c \rar & M_{kc} \\ & X_c \uar \rar & M^N_{kc/N} \uar \\ M^N_{c/N} \ar{uur} \ar{rru} \ar[dotted]{ur} \end{tikzcd}$

We therefore get morphisms of cochain complexes

$\begin{tikzcd} \hat V(M_{kc}) \rar\dar & \hat V(M_c) \rar\dar & \hat V(M_c) \dar \\ \hat V(M^N_{kc/N}) \rar\dar & \hat V(X_c) \rar\dar & \hat V(M^N_{c/N}) \dar \\ \vdots \rar & \vdots \rar & \vdots \end{tikzcd}$

where all the columns are of the form “alternating face map complex of $\hat V(\_ )$ applied to a Cech nerve”. Note that the horizontal maps are norm-nonincreasing and their compositions are restriction maps.

Claim: for $c \ge c_0$, the map $X_c \to M_c$ is surjective.

Indeed, by assumption every $x \in M_c$ can be decomposed into a sum $x = x_1 + \dots + x_N$ with $x_i \in M_{c/N+d} \subset M_{kc/N}$, since $c \ge c_0$ and $d \le \frac{(k-1)c_0}{N}$.

By Proposition 1.7.7, the middle column is normed exact (in the sence of Definition 1.4.6). The result follows from Lemma 1.4.10.

Proposition 1.7.9 ✓

Let $d$ be the constant from Proposition 1.6.3. Let $k {\gt} 1$ and $c_0 {\gt} 0$ be real numbers such that

\[ (k - 1) * c_0 / N \ge d. \]

Let $m$ be any natural number, and put

\[ K = (m + 2) + \frac{r + 1}{r(1 - r)} (m + 2)^2 \]

Finally, let $c_0'$ be $\frac{c_0}{r' \cdot n_i}$, where $n_i$ is the $i$-th index in our fixed Breen–Deligne data.

Then $i$-th column in the double complex is $(k^2, K)$-weak bounded exact in degrees $\le m$ for $c \ge c_0'$.

Proof ▼

Let $M^{(m)}$ denote $\operatorname{Hom}(\Lambda ^{(m)},\overline{\mathcal L}_{r'}(S))^{n_i}$. We also write $M$ for $M^{(0)} = \operatorname{Hom}(\Lambda ,\overline{\mathcal L}_{r'}(S))^{n_i}$ and $M'$ for $M^{(1)}$. By Proposition 1.6.4, $M$ is $N$-splittable with error term $d$.

Consider the diagram of morphisms of systems of complexes

$\begin{tikzcd} \hat V(M_c)^{T^{-1}} \rar\dar & \hat V(M_c) \rar["{T^{-1} - [T^{-1}]^*}"]\dar & \hat V(M_c) \dar \\ \hat V(M'_c)^{T^{-1}} \rar\dar & \hat V(M'_c) \rar["{T^{-1} - [T^{-1}]^*}"]\dar & \hat V(M'_c) \dar \\ \vdots \dar & \vdots \dar & \vdots \dar \\ \hat V(M^{(m)}_c)^{T^{-1}} \rar & \hat V(M^{(m)}_c) \rar["{T^{-1} - [T^{-1}]^*}"] & \hat V(M^{(m)}_c) \end{tikzcd}$

By Lemmas 1.7.8 and 1.7.5, we know that the two columns on the right are weakly $\le k$-exact in degrees $\le m$ and for $c \ge c_0$ with bound $1$.

The result now follows from Lemma 1.5.8, and Proposition 1.4.13.

Proposition 1.7.10 ✓

Let $h$ be the homotopy packaged with $\mathsf{BD}$, and let $h^N$ denote the $n$-th iterated composition of $h$ (see Def 1.1.11) which is a homotopy between $\pi ^N$ and $\sigma ^N \colon N \otimes \mathsf{BD}\to \mathsf{BD}$.

Let $H \in \mathbb {R}_{\ge 0}$ be such that for $i = 0, \dots , m$ the universal map $h^N_i$ is bound by $H$ (see Def 1.1.3).

Then the double complex satisfies the normed homotopy homotopy condition (Def 1.4.14) for $m$, $H$, and $c_0$.

Proof ▼

By Lemma 1.7.6 we may replace the first row by

\[ C^{N \otimes \mathsf{BD}}_{\kappa /N}(\operatorname{Hom}(\Lambda , M))^\bullet . \]

Now it is important to recall that we have chosen $k' \ge \kappa '_i$ for all $i = 0, \dots , m+1$.

Our goal is to find, in degrees $\leq m$, a homotopy between the two maps from the first row

\[ \widehat{V}(\operatorname{Hom}(\Lambda ,M)_{\leq c})^{T^{-1}}\longrightarrow \widehat{V}(\operatorname{Hom}(\Lambda ,M)_{\leq \kappa _1c}^2)^{T^{-1}}\longrightarrow \ldots \]

to the second row

\[ \widehat{V}(\operatorname{Hom}(\Lambda ,M)_{\leq c/N}^N)^{T^{-1}}\longrightarrow \widehat{V}(\operatorname{Hom}(\Lambda ,M)_{\leq \kappa _1c/N}^{2N})^{T^{-1}}\longrightarrow \ldots \]

respectively induced by $\sigma ^N$ and $\pi ^N$ (which are maps $N \otimes \mathsf{BD}$

By Definition 1.1.11 and Lemma 1.1.18 we can find this homotopy between the complex for $k'c$ and the complex for $c$. (Here we use $k'\geq c_i'$ for $i=0,\ldots ,m$.) By assumption, the norm of these maps is bounded by $H$.

Finally, it remains to establish the estimate (eq. 1.4.3) on the homotopic map. We note that this takes $x\in \widehat{V}(\operatorname{Hom}(\Lambda ,M)_{\leq k'^2\kappa _ic}^{a_i})^{T^{-1}}$ (with $i=q$ in the notation of (eq. 1.4.3)) to the element

\[ y\in \widehat{V}(\operatorname{Hom}(\Lambda ,M)_{\leq k'\kappa _ic/N}^{Na_i})^{T^{-1}} \]

that is the sum of the $N$ pullbacks along the $N$ projection maps $\operatorname{Hom}(\Lambda ,M)_{\leq k'\kappa _ic/N}^{Na_i}\to \operatorname{Hom}(\Lambda ,M)_{\leq k'^2\kappa _ic}^{a_i}$. We note that these actually take image in $\operatorname{Hom}(\Lambda ,M)_{\leq \kappa _ic}^{a_i}$ as $N\geq k'$, so this actually gives a well-defined map

\[ \widehat{V}(\operatorname{Hom}(\Lambda ,M)_{\leq \kappa _ic}^{a_i})^{T^{-1}} \longrightarrow \widehat{V}(\operatorname{Hom}(\Lambda ,M)_{\leq k'\kappa _ic/N}^{Na_i})^{T^{-1}}. \]

We need to see that this map is of norm $\leq \epsilon $. Now note that by our choice of $N$, we actually have $k'\kappa _ic/N\leq (r')^b \kappa _ic$, so this can be written as the composite of the restriction map

\[ \widehat{V}(\operatorname{Hom}(\Lambda ,M)_{\leq \kappa _ic}^{a_i})^{T^{-1}} \longrightarrow \widehat{V}(\operatorname{Hom}(\Lambda ,M)_{\leq (r')^b \kappa _ic}^{a_i})^{T^{-1}} \]

and

\[ \widehat{V}(\operatorname{Hom}(\Lambda ,M)_{\leq (r')^b \kappa _ic}^{a_i})^{T^{-1}} \longrightarrow \widehat{V}(\operatorname{Hom}(\Lambda ,M)_{\leq k'\kappa _ic/N}^{Na_i})^{T^{-1}}. \]

The first map has norm exactly $r^b$, by $T^{-1}$-invariance, and as multiplication by $T$ scales the norm with a factor of $r$ on $\widehat{V}$. (Here is where we use $r'{\gt}r$, ensuring different scaling behaviour of the norm on source and target.) The second map has norm at most $N$ (as it is a sum of $N$ maps of norm $\leq 1$). Thus, the total map has norm $\leq r^bN$. But by our choice of $N$, we have $r^bN\leq \epsilon $, giving the result.

Proof of Theorem 1.7.2 ▼

By induction, the first condition of Proposition 1.4.15 is satisfied for all $c\geq c_0$ with $c_0$ large enough (depending on $\Lambda $ but not $V$ or $S$).

The second condition is Proposition 1.7.9, and the third condition has been checked in Proposition 1.7.10.

Thus, we can apply Proposition 1.4.15, and get the desired $\leq \max (k'^2,2k_0H)$-exactness in degrees $\leq m$ for $c\geq c_0$, where $k'$, $k_0$ and $H$ were defined only in terms of $k$, $m$, $r'$ and $r$, while $c_0$ depends on $\Lambda $ (but not on $V$ or $S$). This proves the inductive step.

Question 1.7.11

Can one make the constants explicit, and how large are they? ¹ Modulo the Breen-Deligne resolution, all the arguments give in principle explicit constants; and actually the proof of the existence of the Breen-Deligne resolution should be explicit enough to ensure the existence of bounds on the $c_i$ and $c_i'$.

Answer: yes, we can do this. And we should write up a clean account asap, when we have cleaned up the proof in Lean.