# Blueprint for the Liquid Tensor Experiment

## 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
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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
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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
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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
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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
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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
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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
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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:

1. One has $$‖f_{i}‖ = ‖f‖$$.

2. 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
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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

We therefore get morphisms of cochain complexes

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
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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

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
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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.

1. A back of the envelope calculation seems to suggest that $$k$$ is roughly doubly exponential in $$m$$, and that $$N$$ has to be taken of roughly the same magnitude.