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

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.


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 \(\operatorname{Hom}(\Lambda ,M)_{\leq c}\).


Omitted (but done in Lean).

Lemma 1.7.6

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

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


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

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


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.


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

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 \(1\).


For every constant \(c\), consider the pullback

      & 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}

We therefore get morphisms of cochain complexes

      \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

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 \(c \ge c_0'\).


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

      \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)

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 \(c_0\).


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

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


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

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.