## 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).

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.

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 \(ε\).

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

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.

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 \(\ker \widehat{g}\).

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.

A *system of complexes* of normed abelian groups is for each \(c \in \mathbb R_{\ge 0}\) a complex

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 \(x_{|c} := \mathrm{res}_{c', c}(x)\).

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.

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 \(\| y\| \le (1 + \varepsilon )\| x\| \).

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

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

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

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 \(K\).

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 \(K + δ\).

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

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 \(1\).

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

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

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 \(δ {\gt} 0\).

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

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

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

We have

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 \(x_{|c} = dy\).

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

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 \(K'(KK'' + 1)\).

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

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

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

Using the norm bound on \(f\), we get

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

which combines with the previous estimate to give:

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

In particular,

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

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

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?)

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 \(K + r_1r_2KK'\).

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

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

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

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

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

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

In particular we get

Now let

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

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.

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

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

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

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

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

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 \(\epsilon \).

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

On the other hand, by (3),

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

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

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 \(\max (k^4,k^3+k+1)\).

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

On the other hand, by (3),

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

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

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

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