# Blueprint for the Liquid Tensor Experiment

## 1.4 Some normed homological algebra

It will be convenient to use the following definition generalizing the notion of a bound on the norm of a right inverse of a normed group morphism (note that the morphisms we consider have no reason to have a right inverse, even when they are surjective).

Definition 1.4.1
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Let $$G$$ and $$H$$ be semi-normed groups, let $$K$$ be a subgroup of $$H$$ and $$C$$ be a positive real number. A morphism $$f : G → H$$ is $$C$$-surjective onto $$K$$ if, for all $$x$$ in $$K$$, there exists some $$g$$ in $$G$$ such that $$f(g) = x$$ and $$\| g\| ≤ C\| x\|$$. If $$K = H$$ we simply say $$f$$ is $$C$$-surjective.

The following controlled surjectivity lemma will be used to prove Lemma 1.4.3 and Lemma 1.5.8.

Lemma 1.4.2
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Let $$G$$ and $$H$$ be normed groups. Let $$K$$ be a subgroup of $$H$$ and $$f$$ a morphism from $$G$$ to $$H$$. Assume that $$G$$ is complete and $$f$$ is $$C$$-surjective onto $$K$$. Then $$f$$ is $$(C + ε)$$-surjective onto the topological closure of $$K$$ for every positive $$ε$$.

Proof

Let $$x$$ be any element of the closure of $$K$$. First note the conclusion is trivial when $$x = 0$$, so we can assume $$x ≠ 0$$. Then write $$x$$ as a sum $$\sum _{i \ge 0} x_i$$ with all $$x_i \in K$$, $$\| x - x_0\| ≤ ε_0$$ and $$‖x_i‖\leq \epsilon _i$$ for $$i{\gt}0$$ for some sequence of positive numbers $$\epsilon _i$$ to be chosen later. By assumption, we can then lift each $$x_i$$ to $$g_i$$ such that $$f(g_i) = x_i$$ and $$‖g_i‖\leq C‖x_i‖$$, and then set $$g = \sum g_i$$. Because $$G$$ is complete, this sum converges provided the $$ε_i$$ sequence converges fast enough to zero. We then have $$f(g) = x$$ and

$‖g‖ ≤ C\sum _{i \geq 0} ‖x_i‖ ≤ C(\| x\| + ε_0) + C\sum _{i{\gt}0} ε_i ≤ (C + ε)‖x‖$

where the last inequality holds provided the $$ε_i$$ sequence converges fast enough to zero. For instance $$ε_i = ε∥x∥/(2^{i+1}C)$$ satisfies all our constraints on the $$ε_i$$ sequence (in particular they are positive because $$x ≠ 0$$).

The first application of the above lemma is a completion result for a quantitative version of being a complex.

Lemma 1.4.3
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Let $$f : M_0 → M_1$$ and $$g : M_1 → M_2$$ be bounded maps between normed groups. Assume there are positive constants $$C$$ and $$D$$ such that:

• $$f$$ is $$C$$-surjective onto $$\ker g$$.

• $$g$$ is $$D$$-surjective onto its image.

Then for every positive $$ε$$, $$\widehat{f}$$ is $$(C + ε)$$-surjective onto $$\ker \widehat{g}$$.

Proof

Since $$f$$ is $$C$$-surjective onto $$\ker g$$, $$\widehat{f}$$ is $$C$$-surjective onto $$\ker g$$ seen as a subset of $$\widehat{M_1}$$. Hence this lemma will follow directly from Lemma 1.4.2 once we’ll have proven that $$\ker g$$ is dense in $$\ker \widehat{g}$$. Let $$\widehat y$$ be an element of $$\ker \widehat{g}$$. Pick any $$\delta {\gt} 0$$ and take $$y\in M_1$$ such that $$‖\widehat{y}-y‖\leq \delta$$. Let $$z=g(y)\in M_2$$, which has norm $$‖z‖=‖g(y)‖=‖g(y-\widehat{y})‖$$ bounded by $$C_{g}\delta$$, where $$C_{g}$$ is the norm of $$g$$. We can thus find some $$y'\in M_1$$ with $$‖y'‖\leq DC_{g}\delta$$ and $$g(y')=z$$. Replacing $$y$$ by $$y-y'$$, we can thus find $$y\in \ker (g: M_1\to M_2)$$ such that still $$‖\widehat{y}-y‖\leq (1+DC_{g})\delta$$; as $$\delta$$ was arbitrary, this gives the desired density.

Definition 1.4.4
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A system of complexes of normed abelian groups is for each $$c \in \mathbb R_{\ge 0}$$ a complex

$C_c^\bullet : C_c^0\longrightarrow C_c^1\longrightarrow \ldots$

of normed abelian groups together with maps of complexes $$\mathrm{res}_{c',c}: C_{c'}^\bullet \to C_c^\bullet$$, for $$c' ≥ c$$, satisfying $$\mathrm{res}_{c,c}=\mathrm{id}$$ and the obvious associativity condition. In other words, a functor from $$(\mathbb R_{\ge 0})^{\mathrm{op}}$$ to cochain complexes of semi-normed groups.

By convention, for every system of complexes $$C_\bullet ^\bullet$$, we will set $$C^{-1}_c = 0$$ for all $$c$$. This will come up each time we write $$C^{i-1}_c$$ and $$i$$ could be $$0$$.

In this subsection, given $$x ∈ C^•_{c'}$$ and $$c_0\leq c ≤ c'$$ we will use the notation $$x_{|c} := \mathrm{res}_{c', c}(x)$$.

Definition 1.4.5
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A system of complexes is admissible if all differentials and maps $$\mathrm{res}_{c',c}^i$$ are norm-nonincreasing.

Throughout the rest of this subsection, $$k$$ (and $$k'$$, $$k''$$) will denote reals at least 1, $$m$$ will be a non-negative integer, and $$K$$, $$K'$$, $$K''$$ will denote non-negative reals.

Definition 1.4.6
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A cochain complex $$C$$ of semi-normed groups is normed exact if for all $$i \ge 0$$, all $$\varepsilon {\gt} 0$$, and all $$x \in C^i$$ with $$d(x) = 0$$ there exists a $$y \in C^{i-1}$$ such that $$d(y) = x$$ and $$\| y\| \le (1 + \varepsilon )\| x\|$$.

Definition 1.4.7
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Let $$C_\bullet ^\bullet$$ be a system of complexes. For an integer $$m\geq 0$$ and reals $$k \ge 1$$, $$K \ge 0$$ and $$c_0 \ge 0$$, we say the datum $$C_\bullet ^\bullet$$ is $$\leq k$$-exact in degrees $$\leq m$$ and for $$c\geq c_0$$ with bound $$K$$ if the following condition is satisfied. For all $$c\geq c_0$$ and all $$x\in C_{kc}^i$$ with $$i\leq m$$ there is some $$y\in C_c^{i-1}$$ such that

$‖x_{|c} - dy‖ ≤ K ‖dx‖.$

We will also need a version where the inequality is relaxed by some arbitrary small additive constant.

Definition 1.4.8
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Let $$C_\bullet ^\bullet$$ be a system of complexes. For an integer $$m\geq 0$$ and reals $$k \ge 1$$, $$K \ge 0$$ and $$c_0 \ge 0$$, the datum $$(C_c^\bullet )_c$$ is weakly $$\leq k$$-exact in degrees $$\leq m$$ and for $$c\geq c_0$$ with bound $$K$$ if the following condition is satisfied. For all $$c\geq c_0$$, all $$x\in C_{kc}^i$$ with $$i\leq m$$ and any $$ε {\gt} 0$$ there is some $$y\in C_c^{i-1}$$ such that

$‖x_{|c} - dy‖ ≤ K ‖dx‖ + ε.$

We first note that the difference between those two definitions is only about cocyles if we are ready to lose a tiny something on the norm bound $$K$$.

Lemma 1.4.9
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Let $$C_\bullet ^\bullet$$ be a system of complexes. If $$C_\bullet ^\bullet$$ is weakly $$\leq k$$-exact in degrees $$\leq m$$ and for $$c\geq c_0$$ with bound $$K$$ and if, for all $$c\geq c_0$$ and all $$x\in C_{kc}^i$$ with $$i\leq m$$ such that $$dx = 0$$ there is some $$y\in C_c^{i-1}$$ such that $$x_{|c} = dy$$ then, for every positive $$δ$$, $$C_\bullet ^\bullet$$ is $$\leq k$$-exact in degrees $$\leq m$$ and for $$c\geq c_0$$ with bound $$K + δ$$.

Proof

Let $$δ$$ be some positive real number. Let $$x$$ be an element of $$C_{kc}^i$$ for some $$c ≥ c_0$$ and $$i ≤ m$$. If $$dx = 0$$ then the assumption we made about exact elements is exactly what we want.

Assume now that $$dx ≠ 0$$. The weak exactness assumption applied to $$ε = δ‖dx‖$$ gives some $$y\in C_c^{i-1}$$ such that

\begin{align*} ‖x_{|c} - dy‖ & ≤ K‖dx‖ + δ‖dx‖ \\ & = (K + δ)‖dx‖ \end{align*}

Lemma 1.4.10
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Let $$k \ge 1$$, $$c_0 \ge 0$$ be real numbers, and $$m \in \mathbb N$$. Let $$C_\bullet ^\bullet$$ be a system of complexes, and for each $$c \ge 0$$ let $$D_c$$ be a cochain complex of semi-normed groups. Let $$f_c \colon C^\bullet _{kc} \to D^\bullet _c$$ and $$g_c \colon D^\bullet _c \to C^\bullet _c$$ be norm-nonincreasing morphisms of cochain complexes of semi-normed groups such that $$g_c \circ f_c$$ is the restriction map $$C^\bullet _{kc} \to C^\bullet _c$$. Assume that for all $$c \ge c_0$$ the cochain complex $$D_c$$ is normed exact. Then $$C_\bullet ^\bullet$$ is weakly $$\le k$$-exact in degrees $$\le m$$ and for $$c \ge c_0$$ with bound $$1$$.

Proof

Fix $$c \ge c_0$$, $$i \le m$$, $$x \in C_{kc}^i$$, and $$\varepsilon {\gt} 0$$. Denote by $$\delta$$ the positive real number $$\frac{\varepsilon }{\| x\| + 1}$$.

Clearly $$f(d(x))$$ is killed by $$d$$, so by normed exactness of $$D_c$$ we find $$x' \in D_c^i$$ such that $$d(x') = f(d(x))$$ and $$\| x'\| \le (1 + \delta )\| f(d(x))\|$$. Similarly $$d(f(x) - x') = 0$$, so by exactness of $$D_c$$ we find $$y \in D_c^{i-1}$$ such that $$d(y) = f(x) - x'$$.

We are done if we show that $$\| x_{|c} - d(g(y))\| \le \| d(x)\| + \varepsilon$$. Observe that $$x_{|c} - d(g(y)) = g(f(x)) - g(d(y)) = g(x')$$, and therefore we shall show $$\| g(x')\| \le \| d(x)\| + \varepsilon$$.

Now we use that $$f$$ and $$g$$ are norm-nonincreasing to calculate

$\| g(x')\| \le \| x'\| \le (1 + \delta ) \| f(d(x))\| \le (1 + \delta ) \| d(x)\| .$

Finally, we have $$(1 + \delta ) \| d(x)\| \le \| d(x)\| + \varepsilon$$ by our choice of $$\delta$$.

Lemma 1.4.11
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Let $$M^\bullet _\bullet$$ be an admissible collection of complexes of complete normed abelian groups.

Assume that $$M^\bullet _c$$ is weakly $$\leq k$$-exact in degrees $$\leq m$$ for $$c\geq c_0$$ with bound $$K$$. Then $$M^\bullet _c$$ is $$\leq k^2$$-exact in degrees $$\leq m$$ for $$c\geq c_0$$ with bound $$K+δ$$, for every $$δ {\gt} 0$$.

Proof

Lemma 1.4.9 ensures we only need to care about cocycles of $$M$$. More precisely, let $$x$$ be a cocycle in $$M^i_{k^2c}$$ for some $$i ≤ m$$ and $$c ≥ c_0$$. We need to find $$y \in M^{i-1}_c$$ such that $$dy = x_{|c}$$.

By weak $$\leq k$$-exactness applied to $$x$$ and a sequence $$ε_j$$ to be chosen later, we can find a sequence $$w^j \in M^{i-1}_{kc}$$ such that

$‖x_{kc} - dw^j‖ ≤ ε_j.$

Then, by weak $$\leq k$$-exactness applied to each $$w^{j + 1} - w^j$$ and a sequence $$δ_j$$ to be chosen later, we can find a sequence $$z^j \in M^{i-2}_{c}$$ such that

$‖(w^{j+1} - w^j)_{|c} - dz^j‖ ≤ K‖dw^{j+1} - dw^j‖ + δ_j.$

We set $$y^j := w^j_{|c} - \sum _{l=0}^{j-1} dz^l ∈ M^{i-1}_c$$.

We have

\begin{align*} ‖y^{j + 1} - y^j‖ & = \left\| (w^{j + 1} - w^j)_{|c} - dz^j\right\| \\ & ≤ K‖dw^{j+1} - dw^j‖ + δ_j \\ & ≤ 2Kε_j + δ_j. \end{align*}

So $$y^j$$ is a Cauchy sequence as long as we make sure $$2Kε_j + δ_j ≤ 2^{-j}$$ for instance. Since $$M^{i-1}_c$$ is complete, this sequence converges to some $$y$$. Because $$dy^j = dw^j_{|c}$$, we get that $$‖x_{|c} - dy^j‖ ≤ ε_j$$ and in the limit $$x_{|c} = dy$$.

Proposition 1.4.12
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Let $$M^\bullet _\bullet$$ and $$M'^\bullet _\bullet$$ be two admissible collections of complexes of complete normed abelian groups. For each $$c\geq c_0$$ let $$f^\bullet _c: M^\bullet _c\to M'^\bullet _c$$ be a collection of maps between these collections of complexes that are norm-nonincreasing and which all commute with all restriction maps, and assume that there exists these maps satisfy

$‖x_{|c}‖ ≤ K''‖f(x)‖$

for all $$i ≤ m+1$$ and all $$x\in M^i_{k''c}$$. Let $$N^\bullet _c=M'^\bullet _c/M^\bullet _c$$ be the collection of quotient complexes, with the quotient norm; this is again an admissible collection of complexes.

Assume that $$M^\bullet _c$$ (resp. $$M'^\bullet _c$$) is weakly $$\leq k$$-exact (resp. $$≤ k'$$-exact) in degrees $$\leq m$$ for $$c\geq c_0$$ with bound $$K$$ (resp. $$K'$$). Then $$N^\bullet _c$$ is weakly $$\leq kk'k''$$-exact in degrees $$\leq m-1$$ for $$c\geq c_0$$ with bound $$K'(KK'' + 1)$$.

Proof

Let $$n \in N^i_{kk'k''c}$$ for $$i\leq m-1$$. We fix $$ε {\gt} 0$$. We need to find an element $$y \in N^{i-1}_c$$ such that

$‖n_{|c} - dy‖ \leq K'(KK'' + 1)‖dn‖ + \epsilon .$

Pick any preimage $$m' \in M'^i_{kk'k''c}$$ of $$n$$. In particular $$dm'$$ is a preimage of $$dn$$. By definition of the quotient norm, we can find $$m_1 ∈ M^{i+1}_{kk'k''c}$$ and $$m_1'' ∈ (M')^{i+1}_{kk'k''c}$$ such that

$dm' = f(m_1) + m_1''$

with $$‖m_1''‖ \leq ‖dn‖ + ε_1$$, for some positive $$ε_1$$ to be chosen later.

Applying the differential to the last displayed equation, and using that this kills the image of $$d$$, and that $$f$$ is a map of complexes, we see that

$f(dm_1) = -dm_1''.$

Using the norm bound on $$f$$, we get

\begin{aligned} ‖dm_{1|kk'c}‖ & ≤ K”‖f(dm_1)‖ = K”‖dm_1”‖\\ & ≤ K”‖m_1”‖ ≤ K”‖dn‖ + K”ε_1. \end{aligned}

On the other hand, weak exactness of $$M$$ applied to $$m_{1|kk'c}$$ gives $$m_0 ∈ M^i_{k'c}$$ such that

$‖m_{1|kk'c|k'c} - dm_0‖ \leq K‖dm_{1|kk'c}‖ + ε_1$

which combines with the previous estimate to give:

$‖m_{1|k'c} - dm_0‖ \leq K K'' \left\| d n\right\| + (KK'' + 1)ε_1.$

Now let $$m'_{\mathrm{new}} = m'_{|k'c} - f(m_0) \in M'^i_{k'c}$$; this is a lift of $$n_{|k'c}$$. Then

$dm'_{\mathrm{new}} = dm'_{|k'c} - f(m_{1|k'c}) + f(m_{1|k'c} - dm_0) = m''_{1|k'c} + f(m_{1|k'c} - dm_0).$

In particular,

$‖dm'_{\mathrm{new}}‖ ≤ (KK'' + 1)\left\| dn\right\| + (KK'' + 2) ε_1.$

Now weak exactness of $$M'$$ gives $$x \in M'^{i-1}_c$$ such that

$‖m'_{\mathrm{new}|c} - dx‖ ≤ K'‖dm'_{\mathrm{new}}‖ + ε_1 \leq K'((K K'' + 1) \left\| dn\right\| + (KK'' + 2) ε_1) + ε_1.$

In particular, letting $$y \in N^{i-1}_c$$ be the image of $$x$$, we get

$‖n_{|c} - dy‖ ≤ K'(K K'' + 1)\left\| dn\right\| + (K'(K K'' + 2) + 1) ε_1,$

which is exactly what we wanted if we choose $$ε_1 = ε/(K'(K K'' + 2) + 1)$$.

We also need the ‘dual’ version of 1.4.12, for kernels instead of cokernels. Note that this is actually a bit easier to prove. (Should we put it first in the presentation?)

Proposition 1.4.13
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Let $$M^\bullet _\bullet$$ and $$M'^\bullet _\bullet$$ be two admissible collections of complexes of complete normed abelian groups. For each $$c\geq c_0$$ let $$f^\bullet _c: M^\bullet _c\to M'^\bullet _c$$ be a collection of maps between these collections of complexes which all commute with all restriction maps. Assume moreover that we are given two constants $$r_1, r_2 \geq 0$$ such that:

• for all $$i, c\geq c_0$$ and all $$x\in M^i_c$$

$‖f(x)‖ ≤ r_1‖x‖;$
• for all $$i ≤ m+1, c \geq c_0$$ and all $$y\in M'^i_c$$, there exists $$x\in M^i_c$$ such that

$f(x) = y \mbox{ and } ‖x‖ ≤ r_2‖y‖.$

Let $$N^\bullet _c$$ be the collection of kernel complexes, with the induced norm; this is again an admissible collection of complexes.

Assume that $$M^\bullet _c$$ (resp. $$M'^\bullet _c$$) is weakly $$\leq k$$-exact (resp. $$≤ k'$$-exact) in degrees $$\leq m$$ for $$c\geq c_0$$ with bound $$K$$ (resp. $$K'$$). Then $$N^\bullet _c$$ is weakly $$\leq kk'$$-exact in degrees $$\leq m-1$$ for $$c\geq c_0$$ with bound $$K + r_1r_2KK'$$.

Proof

Let $$n \in N^i_{kk'c} \subseteq M^i_{kk'c}$$ for $$i\leq m-1$$ and let $$ε {\gt} 0$$. We need to find an element $$y \in N^{i-1}_c$$ such that

$‖n_{|c} - dy‖ \leq K + r_1r_2KK'‖dn‖ + \epsilon .$

By weak exactness of $$M^\bullet _\bullet$$, we can find $$m \in M^{i-i}_{k'c}$$ such that

$‖n_{|k'c} - dm‖ \leq K‖dn‖ + \epsilon _1,$

where $$\epsilon _1 {\gt} 0$$ to be chosen later. By weak exactness of $$M'^\bullet _\bullet$$, we can find $$m' \in M'^{i-2}_c$$ such that

$‖f(m)_{|c} - dm'‖ \leq K'‖df(m)‖ + \epsilon _2,$

where $$\epsilon _2 {\gt} 0$$ to be chosen later. Let $$m_1 \in M^{i-2}_c$$ be a lift of $$m'$$ and let $$m_2 \in M^{i-1}_c$$ be such that

$f(m_2) = f(m_{|c} - dm_1) \mbox{ and } ‖m_2‖ \leq r_2 ‖f(m_{|c} - dm_1)‖.$

Set $$y = m_{|c} - dm_1 - m_2 \in M^{i-1}_c$$. By construction $$f(y) = 0$$, so $$y \in N^{i-1}_c$$. We compute

\begin{gather*} ‖n_{|c} - dy‖ = ‖n_{|c} - dm_{|c} + d^2m_1 - dm_2‖ = ‖n_{|c} - dm_{|c} - dm_2‖ \leq \\ ‖n_{|c} - dm_{|c} ‖ + ‖dm_2‖ = ‖(n_{|k'c} - dm)_{|c}‖ + ‖dm_2‖ \leq ‖(n_{|k'c} - dm)‖ + ‖dm_2‖ \leq \\ K‖dn‖ + \epsilon _1 + ‖dm_2‖. \end{gather*}

Where we have used the defining property of $$m$$ and admissibility of $$M^\bullet _\bullet$$. By the same assumption and since $$f(n_{|k'c}) = f(n)_{|k'c} = 0$$, we have

\begin{gather*} ‖dm_2‖ \leq ‖m_2‖ \leq r_2 ‖f(m_{|c} - dm_1)‖ = r_2 ‖f(m)_{|c} - df(m_1)‖ = r_2 ‖f(m)_{|c} - dm’‖ \leq \\ r_2(K’‖df(m)‖ + \epsilon _2) = r_2(K’‖f(dm)‖ + \epsilon _2) = r_2(K’‖f(n_{|k'c}) - f(dm)‖ + \epsilon _2) = \\ r_2(K’‖f(n_{|k'c} - dm)‖ + \epsilon _2) \leq r_2(K’r_1‖n_{|k'c} - dm‖ + \epsilon _2) \leq r_2(K’r_1(K‖dn‖ + \epsilon _1) + \epsilon _2) \end{gather*}

In particular we get

\begin{gather*} ‖n_{|c} - dy‖ \leq K‖dn‖ + \epsilon _1 + r_2(K’r_1(K‖dn‖ + \epsilon _1) + \epsilon _2) = \\ (K + r_1r_2KK’)‖dn‖ + \epsilon _1(1+r_1r_2K’) + r_2\epsilon _2. \end{gather*}

Now let

$\epsilon _1 = \frac{\epsilon }{2(1+r_1r_2K')} \mbox{ and } \epsilon _2 = \begin{cases} \frac{\epsilon }{2r_2} \mbox{ if } r_2 \neq 0 \\ 1 \mbox{ if } r_2 = 0 \end{cases}$

In any case $$r_2 \epsilon _2 \leq \frac{\epsilon }{2}$$ and so

$‖n_{|c} - dy‖ \leq (K + r_1r_2KK')‖dn‖ + \epsilon$

as required.

If $$i=0$$, then all $$m$$, $$m'$$, $$m_1$$ and $$m_2$$ are $$0$$, so $$y=0$$ as required.

Consider a system of double complexes $$M^{p,q}_c$$, $$p,q\geq 0$$, $$c\geq c_0$$,

of complete normed abelian groups.

Definition 1.4.14
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We say that the system of double complexes $$M^{p,q}_c$$ satisfies the normed spectral homotopy condition for $$m \in \mathbb {N}$$ and $$H, c_0, \epsilon \in \mathbb {R}_{\ge 0}$$ if the following condition is satisfied:

For $$q=0,\ldots ,m$$ and $$c\geq c_0$$, there is a map $$h^q_{k'c} \colon M^{0,q+1}_{k'c}\to M^{1,q}_c$$ with

$\| h^q_{k'c}(x)\| _{M^{1,q}_c} \leq H\| x\| _{M^{0,q+1}_{k'c}}$

for all $$x\in M^{0,q+1}_{k'c}$$, and such that for all $$c\geq c_0$$ and $$q=0,\ldots ,m$$ the “homotopic” map

$\mathrm{res}_{k'^2c,k'c}^{1,q}\circ d^{0,q} + h^q_{k'^2c}\circ d'^{0,q}_{k'^2c} + d'^{1,q-1}_{k'c}\circ h^{q-1}_{k'^2c} \colon M^{0,q}_{k'^2c}\to M^{1,q}_{k'c}$

factors as a composite of the restriction $$\mathrm{res}_{k'^2c,c}^{0,q}$$ and a map

$\delta ^{0,q}_c \colon M^{0,q}_c\to M^{1,q}_{k'c}$

that is a map of complexes (in degrees $$\leq m$$), and satisfies the estimate

$$\label{eq:homotopicmapsmall} \| \delta ^{0,q}_c(x)\| _{M^{1,q}_{k'c}}\leq \epsilon \| x\| _{M^{0,q}_c}$$
1.4.3

for all $$x\in M^{0,q}_c$$.

Proposition 1.4.15
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Fix an integer $$m\geq 0$$ and constants $$k$$, $$K$$. Then there exists an $$\epsilon {\gt}0$$ and constants $$k_0$$, $$K_0$$, depending (only) on $$k$$, $$K$$ and $$m$$, with the following property.

Let $$M^{p,q}_c$$ be a system of double complexes as above, and assume that it is admissible. Assume further that there is some $$k'\geq k_0$$ and some $$H{\gt}0$$, such that

1. for $$i=0,\ldots ,m+1$$, the rows $$M^{i,q}_c$$ are weakly $$\leq k$$-exact in degrees $$\leq m-1$$ for $$c\geq c_0$$ with bound $$K$$;

2. for $$j=0,\ldots ,m$$, the columns $$M^{p,j}_c$$ are weakly $$\leq k$$-exact in degrees $$\leq m$$ for $$c\geq c_0$$ with bound $$K$$;

3. it satisfies the normed spectral homotopy condition for $$m$$, $$H$$, $$c_0$$, and $$\epsilon$$.

Then the first row is weakly $$\leq k'^2$$ exact in degrees $$\leq m$$ for $$c\geq c_0$$ with bound $$2K_0H$$.

We note that the homotopy is of a peculiar nature, namely that the homotopic map factors via a deep restriction of $$x$$, as well as its norm being bound by $$\epsilon$$.

Proof of Proposition 1.4.15

First, we treat the case $$m=0$$. If $$m=0$$, we claim that one can take $$\epsilon =\tfrac 1{2k}$$ and $$k_0=k$$. We have to prove exactness at the first step. Let $$x_{k'^2c}\in M^{0,0}_{k'^2c}$$ and denote $$x_{k'c}=\mathrm{res}_{k'^2c,k'c}^{0,0}(x)$$ and $$x_c=\mathrm{res}_{k'^2c,c}^{0,0}(x)$$. Then by assumption (2) (and $$k'\geq k$$), we have

$‖x_c‖_{M^{0,0}_c}\leq k‖d^{0,0}_{k'c}(x_{k'c})‖_{M^{1,0}_{k'c}}.$

On the other hand, by (3),

$‖\mathrm{res}_{k'^2c,k'c}^{1,0}(d^{0,0}_{k'^2c}(x))\pm h^0_{k'^2c}(d'^{0,0}_{k'^2c}(x))‖_{M^{1,0}_{k'c}}\leq \epsilon ‖x_c‖_{M^{0,0}_c}.$

In particular, noting that $$\mathrm{res}_{k'^2c,k'c}^{1,0}(d^{0,0}_{k'^2c}(x)) = d^{0,0}_{k'c}(x_{k'c})$$, we get

$‖x_c‖_{M^{0,0}_c}\leq k‖d^{0,0}_{k'c}(x_{k'c})‖_{M^{1,0}_{k'c}}\leq k\epsilon ‖x_c‖_{M^{0,0}_c} + kH ‖d'^{0,0}_{k'^2c}(x)‖_{M^{0,1}_{k'^2c}}.$

Thus, taking $$\epsilon =\tfrac 1{2k}$$ as promised, this implies

$‖x_c‖_{M^{0,0}_c}\leq 2kH ‖d'^{0,0}_{k'^2c}(x)‖_{M^{0,1}_{k'^2c}}.$

This gives the desired $$\leq \max (k'^2,2k_0H)$$-exactness in degrees $$\leq m$$ for $$c\geq c_0$$.

Now we argue by induction on $$m$$. Consider the complex $$N^{p,q}$$ given by $$M^{p,q+1}$$ for $$q\geq 1$$ and $$N^{p,0} = M^{p,1}/\overline{M^{p,0}}$$ (the quotient by the closure of the image, which is also the completion of $$M^{p,1}/M^{p,0}$$), equipped with the quotient norm. Using the normed version of the snake lemma, Proposition 1.4.12, one checks that this satisfies the assumptions for $$m-1$$, with $$k$$ replaced by $$\max (k^4,k^3+k+1)$$.

Proof

First, we treat the case $$m=0$$. If $$m=0$$, we claim that one can take $$\epsilon =\tfrac 1{2k}$$ and $$k_0=k$$. We have to prove exactness at the first step. Let $$x_{k'^2c}\in M^{0,0}_{k'^2c}$$ and denote $$x_{k'c}=\mathrm{res}_{k'^2c,k'c}^{0,0}(x)$$ and $$x_c=\mathrm{res}_{k'^2c,c}^{0,0}(x)$$. Then by assumption (2) (and $$k'\geq k$$), we have

$‖x_c‖_{M^{0,0}_c}\leq k‖d^{0,0}_{k'c}(x_{k'c})‖_{M^{1,0}_{k'c}}.$

On the other hand, by (3),

$‖\mathrm{res}_{k'^2c,k'c}^{1,0}(d^{0,0}_{k'^2c}(x))+ h^0_{k'^2c}(d'^{0,0}_{k'^2c}(x))‖_{M^{1,0}_{k'c}}\leq \epsilon ‖x_c‖_{M^{0,0}_c},$

noting that the left-hand side agrees with $$\delta ^{0,0}_c(x_c)$$ by assumption. In particular, noting that $$\mathrm{res}_{k'^2c,k'c}^{1,0}(d^{0,0}_{k'^2c}(x)) = d^{0,0}_{k'c}(x_{k'c})$$, we get

$‖x_c‖_{M^{0,0}_c}\leq k‖d^{0,0}_{k'c}(x_{k'c})‖_{M^{1,0}_{k'c}}\leq k\epsilon ‖x_c‖_{M^{0,0}_c} + kH ‖d'^{0,0}_{k'^2c}(x)‖_{M^{0,1}_{k'^2c}}.$

Thus, taking $$\epsilon =\tfrac 1{2k}$$ as promised, and bringing $$\tfrac 12‖x_c‖_{M^{0,0}_c}$$ to the left-hand side, this implies

$‖x_c‖_{M^{0,0}_c}\leq 2kH ‖d'^{0,0}_{k'^2c}(x)‖_{M^{0,1}_{k'^2c}}.$

This gives the desired $$\leq \max (k'^2,2k_0H)$$-exactness in degrees $$\leq m$$ for $$c\geq c_0$$.

Now we argue by induction on $$m$$. Consider the complex $$N^{p,q}$$ given by $$M^{p,q+1}$$ for $$q\geq 1$$ and $$N^{p,0} = M^{p,1}/\overline{M^{p,0}}$$ (the quotient by the closure of the image, which is also the completion of $$M^{p,1}/M^{p,0}$$), equipped with the quotient norm. Using the normed version of the snake lemma, Proposition 1.4.12, one checks that this satisfies the assumptions for $$m-1$$, with $$k$$ replaced by $$\max (k^4,k^3+k+1)$$. To verify condition (3), note that the maps $$\delta ^{0,q}_c$$ induce similar maps after passing to this quotient complex. To verify the estimate 1.4.3, note that it is nontrivial only for $$N^{0,0} = M^{0,1}/\overline{M^{0,0}}$$. In that case, for any given $$a{\gt}0$$ one can lift $$x\in N^{0,0}_c$$ to $$\tilde{x}\in M^{0,1}_c$$ with $$‖\tilde{x}‖_{N^{0,0}_c}\leq ‖x‖_{M^{0,1}_c}+a$$. This implies

$‖\delta ^{0,q}_c(x)‖_{N^{1,0}_{k'c}}\leq ‖\delta ^{0,q}_c(\tilde{x})‖_{M^{1,1}_{k'c}}\leq \epsilon ‖\tilde{x}‖_{M^{0,1}_c}\leq \epsilon ‖x‖_{M^{0,1}_c} + \epsilon a$

for all $$a{\gt}0$$, and hence the desired inequality by taking the infimum over all $$a$$.