Blueprint for the Liquid Tensor Experiment: Some normed homological algebra

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 ✓

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 ✓

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 ✓

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 ✓

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 ✓

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 ✓

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 ✓

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 ✓

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 ✓

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 ✓

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 ✓

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 ✓

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 ✓

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 $M^{p,q}_c$, $p,q\geq 0$, $c\geq c_0$,

$\begin{tikzcd} M^{0,0}_c \ar[r]{d'^{0,0}_c}\ar[d]{d^{0,0}_c} & M^{0,1}_c\ar[r]{d'^{0,1}_c}\ar[d]{d^{0,1}_c} & M^{0,2}_c\ar[r]{d'^{0,2}_c}\ar[d]{d^{0,2}_c} & \ldots\\ M^{1,0}_c\ar[r]{d'^{1,0}_c}\ar[d]{d^{1,0}_c} & M^{1,1}_c\ar[r]{d'^{1,1}_c}\ar[d]{d^{1,1}_c} & M^{1,2}_c\ar[r]{d'^{1,2}_c}\ar[d]{d^{1,2}_c} & \ldots\\ M^{2,0}_c\ar[r]{d'^{2,0}_c}\ar[d]{d^{2,0}_c} & M^{2,1}_c\ar[r]{d'^{2,1}_c}\ar[d]{d^{2,1}_c} & \ddots\\ \vdots & \vdots \end{tikzcd}$

of complete normed abelian groups.

Definition 1.4.14 ✓

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

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

1.4.3

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

Proposition 1.4.15 ✓

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

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 $K$;
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 $K$;
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 $c\geq c_0$.

Proof ▼

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

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 $a$.