${\mathrm{Ext}_{\mathcal{C}}^{n+k}(A,\widetilde{B}_{k})}$${{[}\widetilde{A}_{n+k}{,}\,\widetilde{B}_{k}{]}_{\mathcal{C}}}$${\mathrm{Ext}_{\mathcal{C}}^{n+k+1}(A,\widetilde{B}_{k+1})}$${{[}\widetilde{A}_{n+k+1}{,}\,\widetilde{B}_{k+1}{]}_{\mathcal{C}}}$$\scriptstyle{\beta_{n+k}(\widetilde{B}_{k})}$$\scriptstyle{\delta^{n+k}}$$\scriptstyle{t_{\widetilde{A}_{n+k}{,}\,\widetilde{B}_{k}}}$$\scriptstyle{\beta_{n+k+1}(\widetilde{B}_{k+1})}$
(5.5)

commutes. The induced homomorphism

$\beta^{n}(B):=\varinjlim_{k\in\mathbb{N}_{0},n+k\geq
1}\beta_{n+k}(\widetilde{B}_{k}):\widehat{\mathrm{Ext}}_{\mathcal{C}}^{n}(A,B)\rightarrow
BC_{\mathcal{C}}^{n}(A,B)$

is an isomorphism.

######  Proof.

Note that we impose $n+k\geq 1$ in Diagram 5.5 and in the definition of
$\beta^{n}(B)$ so that the morphisms $\beta_{n+k}(B)$ are well defined. To
prove that the diagram is commutative, let
$f+\mathrm{Im}(\mathrm{Hom}_{\mathcal{C}}(\partial_{n+k},\widetilde{A}_{k})$
be an element in $\mathrm{Ext}_{\mathcal{C}}^{n+k}(A,\widetilde{B}_{k})$. It
suffices to show that the diagram

${A_{n+k+1}}$${\widetilde{A}_{n+k+1}}$${A_{n+k}}$${\widetilde{A}_{n+k}}$${\widetilde{B}_{k+1}}$${B_{k}}$${\widetilde{B}_{k}}$$\scriptstyle{\pi_{n+k+1}}$$\scriptstyle{F}$$\scriptstyle{\iota_{n+k+1}}$$\scriptstyle{\overline{\alpha_{n+k}(\widetilde{B}_{k})(f)}^{\ast}}$$\scriptstyle{\pi_{n+k}}$$\scriptstyle{f}$$\scriptstyle{\overline{\alpha_{n+k}(\widetilde{B}_{k})(f)}}$$\scriptstyle{\alpha_{n+k}(\widetilde{B}_{k})(f)}$$\scriptstyle{\iota_{k+1}}$$\scriptstyle{\pi_{k}}$
(5.6)

commutes where we explain in the following how to construct it. The right most
triangle corresponds to Diagram 5.4. We can lift
$\alpha_{n+k}(\widetilde{B}_{k})(f)$ to
$\overline{\alpha_{n+k}(\widetilde{B}_{k})(f)}$ and
$\overline{\alpha_{n+k}(\widetilde{B}_{k})(f)}^{\ast}$ as we have done in
Diagram 5.2. In particular, the morphism
$\overline{\alpha_{n+k}(\widetilde{B}_{k})(f)}^{\ast}$ is a representative of
$t_{\widetilde{A}_{n+k},\widetilde{B}_{k}}\circ\beta_{n+k}(\widetilde{B}_{k})(f+\mathrm{Im}(\mathrm{Hom}_{\mathcal{C}}(\partial_{n+k},\widetilde{B}_{k})))$.
Writing $F$ for
$\overline{\alpha_{n+k}(\widetilde{B}_{k})(f)}^{\ast}\circ\pi_{n+k+1}$, let us
deduce that

$\delta^{n+k}\big{(}f+\mathrm{Im}(\mathrm{Hom}_{\mathcal{C}}(\partial_{n+k},\widetilde{B}_{k}))\big{)}=F+\mathrm{Im}(\mathrm{Hom}_{\mathcal{C}}(\partial_{n+k+1},\widetilde{B}_{k+1}))$

in $\mathrm{Ext}_{\mathcal{C}}^{n+k+1}(A,\widetilde{B}_{k+1})$. Following the
construction of the connecting homomorphism $\delta^{n+k}$ by means of the
Snake Lemma as in [42, p. 11–12], we see that
$\overline{\alpha_{n+k}(\widetilde{B}_{k})(f)}$ is a lift of $f$ along
$\pi_{k}$. As the boundary map $\partial_{n+k+1}$ equals
$\iota_{n+k+1}\circ\pi_{n+k+1}:A_{n+k+1}\rightarrow A_{n+k}$ and
$\overline{\alpha_{n+k}(\widetilde{B}_{k})(f)}\circ\partial_{n+k+1}$ equals
$\iota_{k+1}\circ F$, $F$ is a representative of
$\delta^{n+k}(f+\mathrm{Im}(\mathrm{Hom}_{\mathcal{C}}(\partial_{n+k},\widetilde{B}_{k})))$.
Thus, $\overline{\alpha_{n+k}(\widetilde{B}_{k})(f)}^{\ast}$ is a
representative of
$\beta_{n+k+1}(\widetilde{B}_{k+1})\circ\delta^{n+k}(f+\mathrm{Im}(\mathrm{Hom}_{\mathcal{C}}(\partial_{n+k},\widetilde{B}_{k})))$
and Diagram 5.5 commutes.

To show that $\beta^{n}(B)$ is an isomorphism, let us first prove that it is
surjective. For this, let
$g+\mathcal{P}_{R}(\widetilde{A}_{n+k},\widetilde{B}_{k})\in[\widetilde{A}_{n+k},\widetilde{B}_{k}]_{\mathcal{C}}$.
Due to the diagram

${A_{n+k}}$${\widetilde{A}_{n+k}}$${\widetilde{B}_{k}}$$\scriptstyle{\pi_{n+k}}$$\scriptstyle{g\circ\pi_{n+k}}$$\scriptstyle{g}$

and Diagram 5.4, we see that
$\alpha_{n+k}(\widetilde{B}_{k})(g\circ\pi_{n+k})=g$ and thus

$\beta_{n+k}(\widetilde{B}_{k})(g\circ\pi_{n+k}+\mathrm{Im}(\mathrm{Hom}_{\mathcal{C}}(\partial_{n+k},\widetilde{B}_{k})))=g+\mathcal{P}_{R}(\widetilde{A}_{n+k},\widetilde{B}_{k})\,.$

Since $\beta_{n+k}(\widetilde{B}_{k})$ is surjective and direct limits in the
category of abelian groups are exact,
$\beta^{n}(B)=\varinjlim_{k\in\mathbb{N}_{0},n+k\geq
1}\beta_{n+k}(\widetilde{B}_{k})$ is surjective.

In order to demonstrate that $\beta^{n}(B)$ is injective, let
$\varphi\in\widehat{\mathrm{Ext}}_{\mathcal{C}}^{n}(A,B)$ be an element such
that $\beta^{n}(B)(\varphi)=0$. By the resolution construction,
$\widehat{\mathrm{Ext}}_{\mathcal{C}}^{n}(A,B)$ can be defined as the direct
limit
$\varinjlim_{k\in\mathbb{N}_{0}}\mathrm{Ext}_{\mathcal{C}}^{n+k}(A,\widetilde{B}_{k})$
in the category $\mathbf{Ab}$. By Proposition 5.6, there exists
$k\in\mathbb{N}_{0}$ and
$h\in\mathrm{Ker}\big{(}\mathrm{Hom}_{\mathcal{C}}(\partial_{n+k+1},\widetilde{B}_{k}))\big{)}$
such that the element
$h+\mathrm{Im}(\mathrm{Hom}_{\mathcal{C}}(\partial_{n+k},\widetilde{B}_{k}))$
of $\mathrm{Ext}_{\mathcal{C}}^{n+k}(A,\widetilde{B}_{k})$ is mapped to
$\varphi\in\widehat{\mathrm{Ext}}_{\mathcal{C}}^{n}(A,B)$ by the homomorphism
to the direct limit. Still using Proposition 5.6, we may assume
$\beta_{n+k}(\widetilde{B}_{k})\big{(}h+\mathrm{Im}(\mathrm{Hom}_{\mathcal{C}}(\partial_{n+k},\widetilde{B}_{k}))\big{)}=0$.
This implies that the morphism
$\alpha_{n+k}(\widetilde{B}_{k})(h):\widetilde{A}_{n+k}\rightarrow\widetilde{B}_{k}$
factors through a projective. Because the morphism $\pi_{k}$ at the bottom of
Diagram 5.6 is an epimorphism, it follows from the lifting property of the
previous projective object that there is a morphism $l$ making the diagram

${\widetilde{A}_{n+k}}$${B_{k}}$${\widetilde{B}_{k}}$$\scriptstyle{\alpha_{n+k}(\widetilde{B}_{k})(h)}$$\scriptstyle{l}$$\scriptstyle{\pi_{k}}$

commute. From this diagram and Diagram 5.6 we infer that

$\pi_{k}\circ(l\circ\pi_{n+k})=\alpha_{n+k}(\widetilde{B}_{k})(h)\circ\pi_{n+k}=\pi_{k}\circ\overline{\alpha_{n+k}(\widetilde{B}_{k})(h)}\,.$

In particular,
$\pi_{k}\circ(\overline{\alpha_{n+k}(\widetilde{B}_{k})(h)}-l\circ\pi_{n+k})=0$.
Note that the kernel of $\pi_{k}$ is given by the morphism
$\iota_{k+1}:\widetilde{B}_{k+1}\rightarrow B_{k}$ whence there is a $L$ such
that

${A_{n+k}}$${\widetilde{B}_{k+1}}$${B_{k}}$$\scriptstyle{\overline{\alpha_{n+k}(\widetilde{B}_{k})(h)}-l\circ\pi_{n+k}}$$\scriptstyle{L}$$\scriptstyle{\iota_{k+1}}$

commutes. Using the morphism $\iota_{n+k+1}:\widetilde{A}_{n+k+1}\rightarrow
A_{n+k}$ from Diagram 5.6, we see that

$\iota_{k+1}\circ
L\circ\iota_{n+k+1}=(\overline{\alpha_{n+k}(\widetilde{B}_{k})(h)}-l\circ\pi_{n+k})\circ\iota_{n+k+1}\,.$

As $\pi_{n+k}\circ\iota_{n+k+1}=0$, we deduce from Diagram 5.6 that

$\iota_{k+1}\circ
L\circ\iota_{n+k+1}=\overline{\alpha_{n+k}(\widetilde{B}_{k})(h)}\circ\iota_{n+k+1}=\iota_{k+1}\circ\overline{\alpha_{n+k}(\widetilde{B}_{k})(h)}^{\ast}\,.$

Because $\iota_{n+k+1}$ is a monomorphism, we conclude that
$L\circ\iota_{n+k+1}=\overline{\alpha_{n+k}(\widetilde{B}_{k})(f)}^{\ast}$.
This conclusion together with the previous diagram and Diagram 5.6 can be
pictorially summarised as

${A_{n+k+1}}$${\widetilde{A}_{n+k+1}}$${\widetilde{A}_{n+k}}$${\widetilde{B}_{k+1}}$${B_{k}}$$\scriptstyle{\pi_{n+k+1}}$$\scriptstyle{H}$$\scriptstyle{\iota_{n+k+1}}$$\scriptstyle{\overline{\alpha_{n+k}(\widetilde{B}_{k})(h)}^{\ast}}$$\scriptstyle{L}$$\scriptstyle{\overline{\alpha_{n+k}(\widetilde{B}_{k})(h)}}$$\scriptstyle{\iota_{k+1}}$

Recalling that $\pi_{n+k+1}\circ\iota_{n+k+1}=\partial_{n+k+1}$, we have
$H=L\circ\partial_{n+k+1}\in\mathrm{Hom}_{\mathcal{C}}(\partial_{n+k+1},\widetilde{B}_{k+1})$.
Since this is a representative in $\mathrm{Im}(\delta^{n+k})$, we infer that
$\delta^{n+k}(h+\mathrm{Im}(\mathrm{Hom}_{\mathcal{C}}(\partial_{n+k},\widetilde{B}_{k})))=0$.
According to the resolution construction, the latter as well as
$h+\mathrm{Im}(\mathrm{Hom}_{\mathcal{C}}(\partial_{n+k},\widetilde{B}_{k}))$
are both mapped to $\varphi\in\widehat{\mathrm{Ext}}_{\mathcal{C}}^{n}(A,B)$.
Therefore $\varphi=0$, $\mathrm{Ker}(\beta^{n}(B))=0$ and $\beta^{n}(B)$ is
injective. $\square$

######  Proposition 5.8.

For every $n\in\mathbb{Z}$ the isomorphism $\beta^{n}(-)$ is natural. That is,
for every $f\in\mathrm{Hom}_{\mathcal{C}}(B,C)$ the square

${\widehat{\mathrm{Ext}}_{\mathcal{C}}^{n}(A{,}\,B)}$${BC_{\mathcal{C}}^{n}(A{,}\,B)}$${\widehat{\mathrm{Ext}}_{\mathcal{C}}^{n}(A{,}\,C)}$${BC_{\mathcal{C}}^{n}(A{,}\,C)}$$\scriptstyle{\beta^{n}(B)}$$\scriptstyle{\widehat{\mathrm{Ext}}_{\mathcal{C}}^{n}(A{,}\,f)}$$\scriptstyle{BC_{\mathcal{C}}^{n}(A{,}\,f)}$$\scriptstyle{\beta^{n}(C)}$

commutes.

######  Proof.

By construction of $\widehat{\mathrm{Ext}}_{\mathcal{C}}^{n}(A,f)$ through the
resolution construction from Proposition 4.2 and of
$BC_{\mathcal{C}}^{n}(A,f)$ from Definition 5.4, it suffices to consider the
cube

${\mathrm{Ext}_{\mathcal{C}}^{n+k}(A,\widetilde{B}_{k})}$${{[}\widetilde{A}_{n+k}{,}\,\widetilde{B}_{k}{]}_{\mathcal{C}}}$${\mathrm{Ext}_{\mathcal{C}}^{n+k}(A,\widetilde{C}_{k})}$${{[}\widetilde{A}_{n+k}{,}\,\widetilde{C}_{k}{]}_{\mathcal{C}}}$${\mathrm{Ext}_{\mathcal{C}}^{n+k+1}(A,\widetilde{B}_{k+1})}$${{[}\widetilde{A}_{n+k+1}{,}\,\widetilde{B}_{k+1}{]}_{\mathcal{C}}}$${\mathrm{Ext}_{\mathcal{C}}^{n+k+1}(A,\widetilde{C}_{k+1})}$${{[}\widetilde{A}_{n+k+1}{,}\,\widetilde{C}_{k+1}{]}_{\mathcal{C}}}$$\scriptstyle{\mathrm{Ext}_{\mathcal{C}}^{n+k}(A{,}\widetilde{f}_{k})}$$\scriptstyle{\beta_{n+k}(\widetilde{B}_{k})}$$\scriptstyle{\delta^{n+k}}$$\scriptstyle{{[}\widetilde{A}_{n+k}{,}\,\widetilde{f}_{k}{]}_{\mathcal{C}}}$$\scriptstyle{t_{\widetilde{A}_{n+k},\widetilde{B}_{k}}}$$\scriptstyle{\beta_{n+k}(\widetilde{C}_{k})}$$\scriptstyle{\delta^{n+k}}$$\scriptstyle{\mathrm{Ext}_{\mathcal{C}}^{n+k+1}(A{,}\widetilde{f}_{k+1})}$$\scriptstyle{\beta_{n+k+1}(\widetilde{B}_{k+1})}$$\scriptstyle{{[}\widetilde{A}_{n+k+1}{,}\,\widetilde{f}_{k+1}{]}_{\mathcal{C}}}$$\scriptstyle{\beta_{n+k+1}(\widetilde{C}_{k+1})}$$\scriptstyle{t_{\widetilde{A}_{n+k},\widetilde{C}_{k}}}$

for any $k\in\mathbb{N}_{0}$ with $n+k\geq 1$. The right hand side commutes
due to the choice of the sequence of morphisms
$(\widetilde{f}_{k}:\widetilde{B}_{k}\rightarrow\widetilde{C}_{k})_{k\in\mathbb{N}_{0}}$
and the left hand side since connecting homomorphisms are natural. The front
and back side correspond to Diagram 5.5. Regarding its top and bottom side,
let
$g\in\mathrm{Ker}(\mathrm{Hom}_{\mathcal{C}}(\partial_{n+k+1},\widetilde{B}_{k}))$.
Then

$\displaystyle\big{(}[\widetilde{A}_{n+k},\widetilde{f}_{k}]_{\mathcal{C}}\circ\beta_{n+k}(\widetilde{B}_{k})\big{)}$
$\displaystyle\big{(}g+\mathrm{Im}(\mathrm{Hom}_{\mathcal{C}}(\partial_{n+k},\widetilde{B}_{k}))\big{)}$
$\displaystyle=[\widetilde{A}_{n+k},\widetilde{f}_{k}]_{\mathcal{C}}\big{(}\alpha_{n+k}(g)+\mathcal{P}_{\mathcal{C}}(\widetilde{A}_{n+k},\widetilde{B}_{k})\big{)}$
$\displaystyle=\widetilde{f}_{k}\circ\alpha_{n+k}(g)+\mathcal{P}_{\mathcal{C}}(\widetilde{A}_{n+k},\widetilde{C}_{k})\,.$

Note that

$\widetilde{f}_{k}\circ\alpha_{n+k}(\widetilde{B}_{k})(g)\circ\pi_{n+k}=\widetilde{f}_{k}\circ
g=\alpha_{n+k}(\widetilde{C}_{k})(\widetilde{f}_{k}\circ g)\circ\pi_{n+k}\,.$

By definition of the morphism
$\alpha_{n+k}(\widetilde{C}_{k}):\mathrm{Ext}_{\mathcal{C}}^{n+k}(A,\widetilde{C}_{k})\rightarrow[\widetilde{A}_{n+k},\widetilde{C}_{k}]_{\mathcal{C}}$
pictorially given in Diagram 5.4, we deduce that
$\widetilde{f}_{k}\circ\alpha_{n+k}(\widetilde{B}_{k})(g)=\alpha_{n+k}(\widetilde{C}_{k})(\widetilde{f}_{k}\circ
g)$. In particular,

$\displaystyle\big{(}[\widetilde{A}_{n+k},\widetilde{f}_{k}]_{\mathcal{C}}\circ\beta_{n+k}$
$\displaystyle(\widetilde{B}_{k})\big{)}\big{(}g+\mathrm{Im}(\mathrm{Hom}_{\mathcal{C}}(\partial_{n+k},\widetilde{B}_{k}))\big{)}$
$\displaystyle=\alpha_{n+k}(\widetilde{C}_{k})(\widetilde{f}_{k}\circ
g)+\mathcal{P}_{\mathcal{C}}(\widetilde{A}_{n+k},\widetilde{C}_{k})$
$\displaystyle=\beta_{n+k}(\widetilde{C}_{k})\big{(}\widetilde{f}_{k}\circ
g+\mathrm{Im}(\mathrm{Hom}_{\mathcal{C}}(\partial_{n+k},\widetilde{C}_{k}))\big{)}$
$\displaystyle=\big{(}\beta_{n+k}(\widetilde{C}_{k})\circ\mathrm{Ext}_{\mathcal{C}}^{n+k}(A,\widetilde{f}_{k})\big{)}\big{(}g+\mathrm{Im}(\mathrm{Hom}_{\mathcal{C}}(\partial_{n+k},\widetilde{B}_{k}))\big{)}\,.$

Thus, the top and bottom side of the above cube also commute and form a direct
system of commuting squares. The statement of the proposition follows by
passing to the direct limit. $\square$

###  5.3 Connecting homomorphisms of the naïve construction

To render the naïve construction into a cohomological functor, a connecting
homomorphism is needed. It is formed similarly to the morphisms
$\varepsilon^{-m}:S^{-m}T\rightarrow S^{-m+1}T$ between satellite functors
that we have established at the start of Section 3. Namely, let $0\rightarrow
B\xrightarrow{f}C\xrightarrow{g}D\rightarrow 0$ be a short exact sequence in
$\mathcal{C}$. By Diagram 4.9, we can lift the morphisms $f$ and $g$ such that
the diagram

${0}$${\widetilde{B}_{k+1}}$${\widetilde{C}_{k+1}}$${\widetilde{D}_{k+1}}$${0}$${0}$${B_{k}}$${C_{k}}$${D_{k}}$${0}$${0}$${\widetilde{B}_{k}}$${\widetilde{C}_{k}}$${\widetilde{D}_{k}}$${0}$$\scriptstyle{\widetilde{f}_{k+1}}$$\scriptstyle{\widetilde{g}_{k+1}}$$\scriptstyle{f_{k}}$$\scriptstyle{g_{k}}$$\scriptstyle{\widetilde{f}_{k}}$$\scriptstyle{\widetilde{g}_{k}}$

commutes and has short exact sequences as rows for every $k\in\mathbb{N}_{0}$.
As the domain of the connecting homomorphism is supposed to be
$BC_{R}^{n}(A,D)=\varinjlim_{k\in\mathbb{N}_{0},n+k\geq
0}[\widetilde{A}_{n+k},\widetilde{D}_{k}]_{\mathcal{C}}$, let
$l\in\mathrm{Hom}_{\mathcal{C}}(\widetilde{A}_{n+k},\widetilde{D}_{k})$.
According to Diagram 5.2, we can lift $l$ such that the diagram

${0}$${\widetilde{A}_{n+k+1}}$${A_{n+k}}$${\widetilde{A}_{n+k}}$${0}$${0}$${\widetilde{D}_{k+1}}$${D_{k}}$${\widetilde{D}_{k}}$${0}$$\scriptstyle{\overline{l}^{\ast}}$$\scriptstyle{\overline{l}}$$\scriptstyle{l}$

commutes and
$t_{\widetilde{A}_{n+k},\widetilde{D}_{k}}(l+\mathcal{P}_{\mathcal{C}}(\widetilde{A}_{n+k},\widetilde{D}_{k}))=\overline{l}^{\ast}+\mathcal{P}_{\mathcal{C}}(\widetilde{A}_{n+k+1},\widetilde{D}_{k+1})$.
Using Diagram 3.2, we can extend it in a commutative manner to

${0}$${\widetilde{A}_{n+k+1}}$${A_{n+k}}$${\widetilde{A}_{n+k}}$${0}$${0}$${\widetilde{D}_{k+1}}$${D_{k}}$${\widetilde{D}_{k}}$${0}$${0}$${\widetilde{B}_{k}}$${\widetilde{C}_{k}}$${\widetilde{D}_{k}}$${0}$$\scriptstyle{\overline{l}^{\ast}}$$\scriptstyle{\overline{l}}$$\scriptstyle{l}$$\scriptstyle{h^{\ast}}$$\scriptstyle{h}$$\scriptstyle{\mathrm{id}_{\widetilde{D}_{k}}}$$\scriptstyle{\widetilde{f}_{k}}$$\scriptstyle{\widetilde{g}_{k}}$

We define the map

$\displaystyle\tau_{n+k,k}:=[\widetilde{A}_{n+k+1},h^{\ast}]_{\mathcal{C}}\circ
t_{\widetilde{A}_{n+k},\widetilde{D}_{k}}:\;\;[\widetilde{A}_{n+k},\widetilde{D}_{k}]_{\mathcal{C}}$
$\displaystyle\rightarrow[\widetilde{A}_{n+k+1},\widetilde{B}_{k}]_{\mathcal{C}}$
$\displaystyle
l+\mathcal{P}_{\mathcal{C}}(\widetilde{A}_{n+k},\widetilde{D}_{k})$
$\displaystyle\mapsto
h^{\ast}\circ\overline{l}^{\ast}+\mathcal{P}_{\mathcal{C}}(\widetilde{A}_{n+k+1},\widetilde{B}_{k})$

Following the proof of Proposition 5.2, it is well defined in the sense that
it does not depend on the particular lift
$h^{\ast}:\widetilde{D}_{k+1}\rightarrow\widetilde{B}_{k}$ of
$\mathrm{id}_{\widetilde{D}_{k}}$. By Proposition 5.1 and 5.2, $\tau_{n+k,k}$
is a homomorphism.

######  Lemma 5.9.

Both the squares

${{\scriptstyle\mathrm{Ext}_{\mathcal{C}}^{n+k}(A{,}\widetilde{D}_{k})}}$${{\scriptstyle{[}\widetilde{A}_{n+k}{,}\,\widetilde{D}_{k}{]}_{\mathcal{C}}}}$${{\scriptstyle{[}\widetilde{A}_{n+k}{,}\,\widetilde{D}_{k}{]}_{\mathcal{C}}}}$${{\scriptstyle{[}\widetilde{A}_{n+k+1}{,}\,\widetilde{B}_{k}{]}_{\mathcal{C}}}}$${{\scriptstyle\mathrm{Ext}_{\mathcal{C}}^{n+k+1}(A{,}\widetilde{B}_{k})}}$${{\scriptstyle{[}\widetilde{A}_{n+k+1}{,}\,\widetilde{B}_{k}{]}_{\mathcal{C}}}}$${{\scriptstyle{[}\widetilde{A}_{n+k+1}{,}\,\widetilde{D}_{k+1}{]}_{\mathcal{C}}}}$${{\scriptstyle{[}\widetilde{A}_{n+k+2}{,}\,\widetilde{B}_{k+1}{]}_{\mathcal{C}}}}$$\scriptstyle{\scriptscriptstyle\beta_{n+k}(\widetilde{D}_{k})}$$\scriptstyle{\scriptscriptstyle\delta^{n+k}}$$\scriptstyle{\scriptscriptstyle\tau_{n+k,k}}$and$\scriptstyle{\scriptscriptstyle(-1)^{k}\tau_{n+k,k}}$$\scriptstyle{\scriptscriptstyle
t_{\widetilde{A}_{n+k},\widetilde{D}_{k}}}$$\scriptstyle{\scriptscriptstyle
t_{\widetilde{A}_{n+k+1},\widetilde{B}_{k}}}$$\scriptstyle{\scriptscriptstyle\beta_{n+k+1}(\widetilde{B}_{k})}$$\scriptstyle{\scriptscriptstyle(-1)^{k+1}\tau_{n+k+1,k+1}}$

commute for every $k\in\mathbb{N}_{0}$ with $n+k\geq 1$. In particular, the
induced homomorphism

$\tau^{n}:=\varinjlim_{k\in\mathbb{N}_{0},n+k\geq
1}(-1)^{k}\tau_{n+k,k}:BC_{\mathcal{C}}^{n}(A,D)\rightarrow
BC_{\mathcal{C}}^{n+1}(A,B)$

fits into the commuting square

${\widehat{\mathrm{Ext}}_{\mathcal{C}}^{n}(A{,}\,D)}$${BC_{\mathcal{C}}^{n}(A{,}\,D)}$${\widehat{\mathrm{Ext}}_{\mathcal{C}}^{n+1}(A{,}\,B)}$${BC_{\mathcal{C}}^{n+1}(A{,}\,B)}$$\scriptstyle{\beta^{n}(D)}$$\scriptstyle{\widehat{\delta}^{n}}$$\scriptstyle{\tau^{n}}$$\scriptstyle{\beta^{n+1}(B)}$

Therefore, $\tau^{n}:BC_{\mathcal{C}}^{n}(A,-)\rightarrow
BC_{\mathcal{C}}^{n+1}(A,-)$ represents the (proposed) connecting homomorphism
of $BC_{\mathcal{C}}^{\bullet}(A,-)$.

######  Proof.

Taking a morphism $h^{\ast}:\widetilde{D}_{k+1}\rightarrow\widetilde{B}_{k}$
lifting
$\mathrm{id}_{\widetilde{D}_{k}}:\widetilde{D}_{k}\rightarrow\widetilde{D}_{k}$
as before, we apply the Ext-functor
$\mathrm{Ext}_{\mathcal{C}}^{\bullet}(A,-)$ to obtain the commutative square

${\mathrm{Ext}_{\mathcal{C}}^{n+k}(A{,}\,\widetilde{D}_{k})}$${\mathrm{Ext}_{\mathcal{C}}^{n+k+1}(A{,}\,\widetilde{D}_{k+1})}$${\mathrm{Ext}_{\mathcal{C}}^{n+k}(A{,}\,\widetilde{D}_{k})}$${\mathrm{Ext}_{\mathcal{C}}^{n}(A{,}\,\widetilde{B}_{k})}$$\scriptstyle{\delta^{n+k}}$$\scriptstyle{\mathrm{id}}$$\scriptstyle{\mathrm{Ext}_{\mathcal{C}}^{n+k+1}(A{,}\,h^{\ast})}$$\scriptstyle{\delta^{n+k}}$

Using this and the definition of the morphism $\tau_{n+k,k}$, we can factorise
the top left square in the statement of the Lemma as

${\mathrm{Ext}_{\mathcal{C}}^{n+k}(A{,}\,\widetilde{D}_{k})}$${{[}\widetilde{A}_{n+k}{,}\,\widetilde{D}_{k}{]}_{\mathcal{C}}}$${\mathrm{Ext}_{\mathcal{C}}^{n+k+1}(A{,}\,\widetilde{D}_{k+1})}$${{[}\widetilde{A}_{n+k+1}{,}\,\widetilde{D}_{k+1}{]}_{\mathcal{C}}}$${\mathrm{Ext}_{\mathcal{C}}^{n+k+1}(A{,}\,\widetilde{B}_{k})}$${{[}\widetilde{A}_{n+k+1}{,}\,\widetilde{B}_{k}{]}_{\mathcal{C}}}$$\scriptstyle{\beta_{n+k}(\widetilde{D}_{k})}$$\scriptstyle{\delta^{n+k}}$$\scriptstyle{t_{\widetilde{A}_{n+k},\widetilde{D}_{k}}}$$\scriptstyle{\beta_{n+k+1}(\widetilde{D}_{k+1})}$$\scriptstyle{\mathrm{Ext}_{\mathcal{C}}^{n+k+1}(A{,}\,h^{\ast})}$$\scriptstyle{{[}\widetilde{A}_{n+k+1}{,}\,h^{\ast}{]}_{\mathcal{C}}}$$\scriptstyle{\beta_{n+k+1}(\widetilde{B}_{k})}$

It commutes by Lemma 5.7 and Proposition 5.8. As for the top right square in
the statement of the lemma, we can fit it as the right hand side of the cube

${\mathrm{Ext}_{\mathcal{C}}^{n+k}(A{,}\,\widetilde{D}_{k})}$${{[}\widetilde{A}_{n+k}{,}\,\widetilde{D}_{k}{]}_{\mathcal{C}}}$${\mathrm{Ext}_{\mathcal{C}}^{n+k+1}(A{,}\,\widetilde{B}_{k})}$${{[}\widetilde{A}_{n+k+1}{,}\,\widetilde{B}_{k}{]}_{\mathcal{C}}}$${\mathrm{Ext}_{\mathcal{C}}^{n+k+1}(A{,}\,\widetilde{D}_{k+1})}$${{[}\widetilde{A}_{n+k+1}{,}\,\widetilde{D}_{k+1}{]}_{\mathcal{C}}}$${\mathrm{Ext}_{\mathcal{C}}^{n+k+2}(A{,}\,\widetilde{B}_{k+1})}$${{[}\widetilde{A}_{n+k+1}{,}\,\widetilde{B}_{k+1}{]}_{\mathcal{C}}}$$\scriptstyle{(-1)^{k}\delta^{n+k}}$$\scriptstyle{\beta_{n+k}(\widetilde{D}_{k})}$$\scriptstyle{\delta^{n+k}}$$\scriptstyle{(-1)^{k}\tau_{n+k,k}}$$\scriptstyle{t_{\widetilde{A}_{n+k},\widetilde{D}_{k}}}$$\scriptstyle{\beta_{n+k+1}(\widetilde{B}_{k})}$$\scriptstyle{\delta^{n+k+1}}$$\scriptstyle{(-1)^{k+1}\delta^{n+k+1}}$$\scriptstyle{\beta_{n+k+1}(\widetilde{D}_{k+1})}$$\scriptstyle{(-1)^{k+1}\tau_{n+k+1,k+1}}$$\scriptstyle{\beta_{n+k+2}(\widetilde{B}_{k+1})}$$\scriptstyle{t_{\widetilde{A}_{n+k+1},\widetilde{B}_{k}}}$

We have just seen that the top and bottom side commutes. The left hand side
corresponds to Diagram 4.10 while the front and back side commute by Lemma
5.7. The right hand commutes as desired since
$\beta_{n+k+1}(\widetilde{B}_{k})$ is an epimorphism by the proof of Lemma
5.7. Lastly, the top and bottom sides of these cubes form a direct system of
commuting squares in whose direct limit we retrieve the bottom square from the
statement of the lemma. $\square$

######  Theorem 5.10.

The naïve construction $(BC_{\mathcal{C}}^{\bullet}(A,-),\tau^{\bullet})$
forms a cohomological functor and
$\beta^{\bullet}:(\widehat{\mathrm{Ext}}_{\mathcal{C}}^{\bullet}(A,-),\widehat{\delta}^{\bullet})\rightarrow(BC_{\mathcal{C}}^{\bullet}(A,-),\tau^{\bullet})$
is an isomorphism of cohomological functors.

######  Proof.

We already know that
$(\widehat{\mathrm{Ext}}_{\mathcal{C}}^{\bullet}(A,-),\widehat{\delta}^{\bullet})$
is a cohomological functor by Lemma 3.2, that $\beta^{\bullet}$ is a natural
transformation by Proposition 5.8 and that every $\beta^{n}$ is an isomorphism
by Lemma 5.7. Hence, we can reiterate the proof of Theorem 4.5 where we invoke
Lemma 5.9 instead of Lemma 4.4 whenever required. $\square$

######  Remark 5.11.

On page 299 of [28], G. Mislin constructs for every $n\in\mathbb{Z}$ an
isomorphism
$\varinjlim_{k\in\mathbb{N}_{0}}S^{-k}\mathrm{Ext}^{n+k}(A,-)\rightarrow
BC^{n}(A,-)$ in the case of modules over a ring. Then he asserts that one can
turn $BC^{\bullet}(A,-)$ into a Mislin completion by transferring the
structure of $\widehat{\mathrm{Ext}}^{\bullet}(A,-)$ and claims that
homomorphisms of the form
$[\widetilde{A}_{n+k},\widetilde{D}_{k}]\rightarrow[\widetilde{A}_{n+k+1},\widetilde{B}_{k}]$
induce the connecting homomorphism $\tau^{n}:BC^{n}(A,D)\rightarrow
BC^{n+1}(A,B)$. However, he does not explain how to construct the latter
homomorphisms and in particular, does not indicate that there alternating
signs involved in the direct system giving rise to $\tau^{n}$ as can be seen
in Lemma 5.9.

##  6 The hypercohomology and the resolution construction

We demonstrate in this section that the hypercohomology construction gives
rise to the Mislin completion of unenriched Ext-functors in the same way as
the naïve construction. For this, we relate in Subsection 6.1 the
hypercohomology complex with chain maps and the Vogel complex with so-called
almost chain maps. It is known that unenriched Ext-groups can be retrieved as
cohomology groups of the former complex. In Subsection 6.2 we establish an
explicit isomorphism between these two forms of Ext-groups and use it to
construct connecting homomorphisms through (almost) chain maps. This sets the
base for Subsection 6.3 where we construct a natural isomorphism from the
cohomology groups of the resolution construction to the ones of the
hypercohomolgy construction. In Subsection 6.4, we extend it to an isomorphism
of cohomological functors. Moreover, we provide an explicit isomorphism of
cohomological functors from the Mislin completion resulting from the
hypercohomology construction to the one resulting from the naïve construction.
Lastly, although S. Guo and L. Liang have turned the cohomology groups of
the Vogel complex into a cohomological functor in [19, Proposition 4.8], we do
not know whether their construction forms a Mislin completion (Question 6.19).
Similarly, J. Hu et al. introduce the Vogel complex in [22, p. 7] in a far
more general setting than we or S. Guo and L. Liang consider, but we also do
not know whether it yields a Mislin completion.

###  6.1 Defining Ext-functors through chain maps

There is a reformulation of the hypercohomology construction occurring in D.
J. Benson and J. F. Carlson’s paper [5, p. 109] in terms of almost chain
maps. For this, we introduce the following notation. If
$(M_{k},\mu_{k})_{k\in\mathbb{Z}}$ is a chain complex in $\mathcal{C}$ with
boundary maps $\mu_{k}:M_{k}\rightarrow M_{k-1}$ and $n\in\mathbb{Z}$, then we
define $(M[n]_{k},\mu[n]_{k})_{k\in\mathbb{Z}}$ to be $M[n]_{k}:=M_{k+n}$ and
$\mu[n]_{k}:=(-1)^{n}\mu_{k+n}$ [15, p. 154], [42, p. 9–10]. Recall that we
have constructed the hypercohomoology complex
$\mathrm{Hyp}_{\mathcal{C}}(M_{\bullet},N_{\bullet})_{\bullet}$ for any other
chain complexes $(N_{k},\nu_{k})_{k\in\mathbb{Z}}$ before Definition 2.11.
According to [42, p. 62–63], an $n$-cocycle of
$\mathrm{Hyp}_{\mathcal{C}}(M_{\bullet},N_{\bullet})_{\bullet}$ is exactly a
chain map of the form $M[n]_{\bullet}\rightarrow N_{\bullet}$ where
$n$-coboundaries are nullhomotopic chain maps of this form. In the same way,
we observe that an $n$-cocycle of the Vogel complex
$\mathrm{Vog}_{\mathcal{C}}(M_{\bullet},N_{\bullet})_{\bullet}$ is a
collection of morphisms $(f_{k+n}:M[n]_{k}\rightarrow N_{k})_{k\in\mathbb{Z}}$
such that all but finitely many of the squares

${M_{k+n}}$${N_{k}}$${M_{k-1+n}}$${N_{k-1}}$$\scriptstyle{f_{k+n}}$$\scriptstyle{(-1)^{n}\mu_{n+k}}$$\scriptstyle{\nu_{k}}$$\scriptstyle{f_{k-1+n}}$

commute. We follow the convention in [5, p. 109] by calling this an almost
chain map of degree $n$. We call $f_{\bullet+n}:M[n]_{\bullet}\rightarrow
N_{\bullet}$ nullhomotopic if there is a collection of morphisms
$(e_{k+n}:M[n]_{k}\rightarrow N_{k+1})_{k\in\mathbb{Z}}$ such that
$f_{k+n}=e_{k-1+n}\circ\mu[n]_{k}+\mu[n]_{k+1}\circ e_{k+n}$ for all but
finitely many $k\in\mathbb{Z}$. Then we observe that an $n$-cboundary of
$\mathrm{Vog}_{\mathcal{C}}(M_{\bullet},N_{\bullet})_{\bullet}$ is exactly
such a nullhomotopic almost chain map. Denote by
$\mathrm{Hom}_{\mathrm{Ch}(\mathcal{C})}(M[n]_{\bullet},N_{\bullet})$ the set
of chain maps $M[n]_{\bullet}\rightarrow N_{\bullet}$ and by
$\mathrm{Null}_{\mathrm{Ch}(\mathcal{C})}(M[n]_{\bullet},N_{\bullet})$ the
subset of nullhomotopic maps. Analogously, write
$\widehat{\mathrm{Hom}}_{\mathrm{Ch}(\mathcal{C})}(M[n]_{\bullet},N_{\bullet})$
for the set of almost chain maps $M[n]_{\bullet}\rightarrow N_{\bullet}$ and
by
$\widehat{\mathrm{Null}}_{\mathrm{Ch}(\mathcal{C})}(M[n]_{\bullet},N_{\bullet})$
the subset of nullhomotopic almost chain maps. Above, we have just seen that

$\displaystyle
H^{n}(\mathrm{Hyp}_{\mathcal{C}}(M_{\bullet},N_{\bullet})_{\bullet})$
$\displaystyle=\mathrm{Hom}_{\mathrm{Ch}(\mathcal{C})}(M[n]_{\bullet},N_{\bullet})/\mathrm{Null}_{\mathrm{Ch}(\mathcal{C})}(M[n]_{\bullet},N_{\bullet})\text{
and}$ $\displaystyle
H^{n}(\mathrm{Vog}_{\mathcal{C}}(M_{\bullet},N_{\bullet})_{\bullet})$
$\displaystyle=\widehat{\mathrm{Hom}}_{\mathrm{Ch}(\mathcal{C})}(M[n]_{\bullet},N_{\bullet})/\widehat{\mathrm{Null}}_{\mathrm{Ch}(\mathcal{C})}(M[n]_{\bullet},N_{\bullet})\,.$

If $M_{\bullet}$ is a projective resolution of an object
$M\in\mathrm{obj}(\mathcal{C})$ and $N_{\bullet}$ a projective resolution of
$N\in\mathrm{obj}(\mathcal{C})$, then the former equals the unenriched Ext-
group $\mathrm{Ext}_{\mathcal{C}}^{n}(M,N)$ according to [15, p. 166] and [42,
Section 10.7]. The proper setting to illustrate this are derived categories.
The idea and basic properties of the derived category
$\mathcal{D}(\mathcal{C})$ of $\mathcal{C}$ are summarised on pages 143–144 in
S. I. Gelfand and Yu. I. Manin’s book on homological algebra [15].
Morally, objects of $\mathcal{D}(\mathcal{C})$ are chain complexes
$(M_{k})_{n\in\mathbb{Z}}$ in $\mathcal{C}$ and morphisms are chain maps where
any chain map $(f_{k}:M_{k}\rightarrow N_{k})_{k\in\mathbb{Z}}$ is taken to be
an isomorphism if for every $n\in\mathbb{Z}$ the induced morphism in homology
$H_{n}(f_{\bullet}):H_{n}(M_{\bullet})\rightarrow H_{n}(N_{\bullet})$ is an
isomorphism. Every $A\in\mathrm{obj}(\mathcal{C})$ can be turned into an
object $\iota(A)_{\bullet}\in\mathrm{obj}(\mathcal{D}(\mathcal{C}))$ where
$\iota(A)_{0}=A$, $\iota(A)_{k}=0$ for $k\neq 0$ and all boundary maps are
zero. If $(A_{k})_{k\in\mathbb{N}_{0}}$ is a projective resolution of $A$,
then we can extend it to a chain complex $(A_{k})_{k\in\mathbb{Z}}$ by setting
$A_{k}=0$ for $k<0$ and all other boundary maps to be zero. By means of the
augmentation map $A_{0}\rightarrow A$ we see that $\iota(A)_{\bullet}$ is
isomorphic to $A_{\bullet}$ in $\mathcal{D}(\mathcal{C})$. As we index our
projective resolutions to have positive degrees, our notation in the following
agrees with the one found in [42] and disagrees with the one in [15].
According to [15, p. 166] and [42, p. 399], one can define unenriched Ext-
groups as
$\mathrm{Ext}_{\mathcal{C}}^{n}(A,B):=\mathrm{Hom}_{\mathcal{D}(\mathcal{C})}(\iota(A)[n]_{\bullet},\iota(B)_{\bullet})$.
Note that
$\mathrm{Ext}_{\mathcal{C}}^{n}(A,B)\cong\mathrm{Hom}_{\mathcal{D}(\mathcal{C})}(A[n]_{\bullet},\iota(B)_{\bullet})$.
Since homotopic chain maps are identified in $\mathcal{D}(\mathcal{C})$ [15,
p. 159], this agrees with the definition of
$\mathrm{Ext}_{\mathcal{C}}^{n}(-,B)$ as the $n^{\text{th}}$ right derived
functors of the unenriched Hom-functor $\mathrm{Hom}_{\mathcal{C}}(-,B)$. We
refer the reader to Sections 2.4 and 2.5 of [42] for a thorough account on
derived categories. In order to distinguish the two emerging notions of Ext-
functors, we introduce the following notation.

######  Notation 6.1.

From this point on, if $(A_{k})_{k\in\mathbb{Z}}$ is a chain complex in
$\mathcal{C}$, then it is assumed that $(A_{k})_{k\in\mathbb{N}_{0}}$ is a
projective resolution of an object $A$ in $\mathcal{C}$ and $A_{k}=0$ for
$k<0$. We write

$\mathcal{E}xt_{\mathcal{C}}^{n}(A,B):=\mathrm{Hom}_{\mathrm{Ch}(\mathcal{C})}(A[n]_{\bullet},B_{\bullet})/\mathrm{Null}_{\mathrm{Ch}(\mathcal{C})}(A[n]_{\bullet},B_{\bullet})$

whenever we consider the $n^{\text{th}}$ Ext-group as arising from chain maps
modulo chain homotopy. Analogously, we write

$\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A,B):=\widehat{\mathrm{Hom}}_{\mathrm{Ch}(\mathcal{C})}(A[n]_{\bullet},B_{\bullet})/\widehat{\mathrm{Null}}_{\mathrm{Ch}(\mathcal{C})}(A[n]_{\bullet},B_{\bullet})\,.$

These agree with the $n^{\text{th}}$ cohomology group of the hypercohomology
complex and of the Vogel complex respectively. If $a_{k}:A_{k}\rightarrow
A_{k-1}$ denote the boundary maps of $A_{\bullet}$, then we continue to write

$\mathrm{Ext}_{\mathcal{C}}^{n}(A,B)=\mathrm{Ker}(\mathrm{Hom}_{\mathcal{C}}(a_{n+1},B))/\mathrm{Im}(\mathrm{Hom}_{\mathcal{C}}(a_{n},B))$

for the $n^{\text{th}}$ Ext-group defined as the $n^{\text{th}}$ derived
functor of $\mathrm{Hom}_{\mathcal{C}}(-,B)$.

For morphisms $f:B\rightarrow C$ and $g:C\rightarrow A$ in $\mathcal{C}$, let
us construct $\mathcal{E}xt_{\mathcal{C}}^{n}(A,f)$ and
$\mathcal{E}xt_{\mathcal{C}}^{n}(g,B)$ as well as
$\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A,f)$ and
$\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(g,B)$. Consider projective
resolutions $A_{\bullet}$, $B_{\bullet}$, $C_{\bullet}$ of $A$, $B$, $C$. By
the Comparison Theorem [42, Theorem 2.2.6] there is a chain map
$f_{\bullet}:B_{\bullet}\rightarrow C_{\bullet}$ that is unique up to chain
homotopy such that the diagram

${{}\dots}$${B_{2}}$${B_{1}}$${B_{0}}$${B}$${{}\dots}$${C_{2}}$${C_{1}}$${C_{0}}$${C}$$\scriptstyle{f_{2}}$$\scriptstyle{f_{1}}$$\scriptstyle{f_{0}}$$\scriptstyle{f}$

commutes. There is an analogous chain map $g_{\bullet}:C_{\bullet}\rightarrow
A_{\bullet}$ that is unique up to chain homotopy. Define
$g[n]_{\bullet}:C[n]_{\bullet}\rightarrow A[n]_{\bullet}$ by
$g[n]_{k}:=g_{k+n}$ for every $k\in\mathbb{Z}$ [42, p. 10]. Since the category
of chain complexes $\mathrm{Ch}(\mathcal{C})$ with chain maps as morphisms
forms an abelian category [42, Theorem 1.2.3], there are homomorphisms of
abelian groups

$\displaystyle\mathrm{Hom}_{\mathrm{Ch}(\mathcal{C})}(A[n]_{\bullet},f_{\bullet}):\mathrm{Hom}_{\mathcal{C}}(A[n]_{\bullet},B_{\bullet})\rightarrow\mathrm{Hom}_{\mathcal{C}}(A[n]_{\bullet},C_{\bullet}),\;\varphi_{\bullet}\mapsto
f_{\bullet}\circ\varphi_{\bullet}\text{ and}$
$\displaystyle\mathrm{Hom}_{\mathrm{Ch}(\mathcal{C})}(g[n]_{\bullet},B_{\bullet}):\mathrm{Hom}_{\mathcal{C}}(A[n]_{\bullet},B_{\bullet})\rightarrow\mathrm{Hom}_{\mathcal{C}}(C[n]_{\bullet},B_{\bullet}),\;\varphi_{\bullet}\mapsto\varphi_{\bullet}\circ
g[n]_{\bullet}$

It follows by definition and [42, p. 5] that any
$\widehat{\mathrm{Hom}}_{\mathrm{Ch}(\mathcal{C})}(A[n]_{\bullet},B_{\bullet})$
is an abelian group and that the maps

$\displaystyle\widehat{\mathrm{Hom}}_{\mathrm{Ch}(\mathcal{C})}(A[n]_{\bullet},f_{\bullet}):\widehat{\mathrm{Hom}}_{\mathcal{C}}(A[n]_{\bullet},B_{\bullet})\rightarrow\widehat{\mathrm{Hom}}_{\mathcal{C}}(A[n]_{\bullet},C_{\bullet}),\;\varphi_{\bullet}\mapsto
f_{\bullet}\circ\varphi_{\bullet}\text{ and}$
$\displaystyle\widehat{\mathrm{Hom}}_{\mathrm{Ch}(\mathcal{C})}(g[n]_{\bullet},B_{\bullet}):\widehat{\mathrm{Hom}}_{\mathcal{C}}(A[n]_{\bullet},B_{\bullet})\rightarrow\widehat{\mathrm{Hom}}_{\mathcal{C}}(C[n]_{\bullet},B_{\bullet}),\;\varphi_{\bullet}\mapsto\varphi_{\bullet}\circ
g[n]_{\bullet}$

are also homomorphisms of abelian groups.

######  Definition 6.2.

The above descend to homomorphisms

$\displaystyle\mathcal{E}xt_{\mathcal{C}}^{n}(A,f):\;\mathcal{E}xt_{\mathcal{C}}^{n}(A,B)$
$\displaystyle\rightarrow\mathcal{E}xt_{\mathcal{C}}^{n}(A,C)$
$\displaystyle\varphi_{\bullet}+\mathrm{Null}_{\mathrm{Ch}(\mathcal{C})}(A[n]_{\bullet},B_{\bullet})$
$\displaystyle\mapsto
f_{\bullet}\circ\varphi_{\bullet}+\mathrm{Null}_{\mathrm{Ch}(\mathcal{C})}(A[n]_{\bullet},C_{\bullet})$

and

$\displaystyle\mathcal{E}xt_{\mathcal{C}}^{n}(g,B):\;\mathcal{E}xt_{\mathcal{C}}^{n}(A,B)$
$\displaystyle\rightarrow\mathcal{E}xt_{\mathcal{C}}^{n}(C,B)$
$\displaystyle\varphi_{\bullet}+\mathrm{Null}_{\mathrm{Ch}(\mathcal{C})}(A[n]_{\bullet},B_{\bullet})$
$\displaystyle\mapsto\varphi_{\bullet}\circ
g[n]_{\bullet}+\mathrm{Null}_{\mathrm{Ch}(\mathcal{C})}(C[n]_{\bullet},B_{\bullet})\,.$

The homomorphisms

$\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A,f):\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A,B)\rightarrow\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A,C)\text{
and
}\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(g,B):\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A,B)\rightarrow\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(C,B)$

are defined analogously.

Although F. Goichot only works with modules over a ring, we generalise his
remark from [17, p. 41] that $\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A,B)$
does not depend on choices of projective resolutions of $A$ and $B$.

######  Proposition 6.3.

For any $n\in\mathbb{Z}$, both

$\mathcal{E}xt_{\mathcal{C}}^{n}(-,-):\mathcal{C}^{\mathrm{op}}\times\mathcal{C}\rightarrow\mathbf{Ab}\text{
and
}\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(-,-):\mathcal{C}^{\mathrm{op}}\times\mathcal{C}\rightarrow\mathbf{Ab}$

form well defined functors that are additive in both variables with respect to
the induced morphisms from Definition 6.2.

######  Proof.

Let $B_{\bullet}$, $B_{\bullet}^{\prime}$ be two projective resolutions of
$B$. According to [42, Theorem 2.2.6] there are chain maps
$\iota_{\bullet}:B_{\bullet}\rightarrow B_{\bullet}^{\prime}$ and
$\iota_{\bullet}^{\prime}:B_{\bullet}^{\prime}\rightarrow B_{\bullet}$ lifting
$\mathrm{id}_{B}:B\rightarrow B$ that are unique up to chain homotopy. In
particular, we see that

$\displaystyle\iota_{\bullet}^{\prime}\circ\iota_{\bullet}+\mathrm{Null}_{\mathrm{Ch}(\mathcal{C})}(B_{\bullet},B_{\bullet})$
$\displaystyle=\mathrm{id}+\mathrm{Null}_{\mathrm{Ch}(\mathcal{C})}(B_{\bullet},B_{\bullet})\text{
and}$
$\displaystyle\iota_{\bullet}\circ\iota_{\bullet}^{\prime}+\mathrm{Null}_{\mathrm{Ch}(\mathcal{C})}(B_{\bullet}^{\prime},B_{\bullet}^{\prime})$
$\displaystyle=\mathrm{id}+\mathrm{Null}_{\mathrm{Ch}(\mathcal{C})}(B_{\bullet}^{\prime},B_{\bullet}^{\prime})\,.$

This constitutes a direct proof that the abelian groups
$\mathcal{E}xt_{\mathcal{C}}(A,B)$ and
$\widehat{\mathcal{E}xt}_{\mathcal{C}}(A,B)$ do not depend on a choice of
projective resolution of $B$. We may conclude analogously that they do not
depend on the choice of projective resolution of $A$. If $f:A\rightarrow B$
and $g:C\rightarrow A$ are morphisms in $\mathcal{C}$, then this also proves
that $\mathcal{E}xt_{\mathcal{C}}(A,f)$ and
$\widehat{\mathcal{E}xt}_{\mathcal{C}}(A,f)$ as well as
$\mathcal{E}xt_{\mathcal{C}}(g,B)$ and
$\widehat{\mathcal{E}xt}_{\mathcal{C}}(g,B)$ are well defined homomorphisms.
We conclude from [42, p. 5] that $\mathcal{E}xt_{\mathcal{C}}^{n}(-,-)$ and
$\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(-,-)$ are additive in both
variables. $\square$

###  6.2 Connecting homomorphisms through chain maps

We construct an explicit isomorphism from the Ext-group
$\mathrm{Ext}_{\mathcal{C}}^{n}(A,B)$ to
$\mathcal{E}xt_{\mathcal{C}}^{n}(A,B)$ which allows us to determine connecting
homomorphisms for $\mathcal{E}xt_{\mathcal{C}}^{\bullet}(A,-)$ and
$\widehat{\mathcal{E}xt}_{\mathcal{C}}^{\bullet}(A,-)$ in terms of (almost)
chain maps. The construction of this isomorphism is of similar nature to pages
63–64 of C. A. Weibel’s book [42] where he compares the chain complex
$\mathrm{Hom}_{\mathcal{C}}(A_{\bullet},B)$ to
$\mathrm{Hyp}_{\mathcal{C}}(A_{\bullet},B_{\bullet})_{\bullet,\bullet}$ seen
as a double complex. However, as our techniques differ and we have not found
this isomorphism in the literature otherwise, we provide full details.

Given that Ext-groups vanish in negative degrees, let $n\in\mathbb{N}_{0}$.
Let $(A_{\bullet},a_{\bullet})$ be a projective resolution of $A$,
$(B_{\bullet},b_{\bullet})$ is a projective resolution of $B$ and
$\varphi\in\mathrm{Ker}(\mathrm{Hom}_{\mathcal{C}}(a_{k+1},B))$. By Diagram
5.4 and the Comparison Theorem [42, Theorem 2.2.6], there exists a chain map
$(\varphi[n]_{k}:A[n]_{k}\rightarrow B_{k})_{k\in\mathbb{N}_{0}}$ that is
unique up to chain homotopy making the diagram

${{}\dots}$${A_{n+2}}$${A_{n+1}}$${A_{n}}$${\widetilde{A}_{n}}$${{}\dots}$${B_{2}}$${B_{1}}$${B_{0}}$${B}$$\scriptstyle{(-1)^{n}a_{n+2}}$$\scriptstyle{\varphi{[}n{]}_{2}}$$\scriptstyle{(-1)^{n}a_{n+1}}$$\scriptstyle{\varphi{[}n{]}_{1}}$$\scriptstyle{\pi_{n}}$$\scriptstyle{\varphi}$$\scriptstyle{\varphi{[}n{]}_{0}}$$\scriptstyle{\alpha_{n}(B)(\varphi)}$$\scriptstyle{b_{2}}$$\scriptstyle{b_{1}}$$\scriptstyle{b}$
(6.1)

commute. Since $B_{k}=0$ for $k<0$, there is an extension
$(\varphi[n]_{k}:A[n]_{k}\rightarrow B_{k})_{k\in\mathbb{Z}}$. Hence the map

$\displaystyle\zeta_{n}^{\prime}:\mathrm{Ker}(\mathrm{Hom}_{\mathcal{C}}(a_{n+1},B))$
$\displaystyle\rightarrow\mathcal{E}xt^{n}_{\mathcal{C}}(A,B)$
$\displaystyle\varphi$
$\displaystyle\mapsto\varphi[n]_{\bullet}+\mathrm{Null}_{\mathrm{Ch}(\mathcal{C})}(A[n]_{\bullet},B_{\bullet})$

is a well defined homomorphism of abelian groups.

Assume that $n\geq 1$ and
$\psi\in\mathrm{Im}(\mathrm{Hom}_{\mathcal{C}}(a_{n},B))$. Then there is a
morphism $\psi:A_{n-1}\rightarrow B$ such that $\varphi=\psi\circ a_{n}$.
Because $A_{n-1}$ is projective and $b:B_{0}\rightarrow B$ an epimorphism,
there $e_{n-1}:A_{n-1}\rightarrow B_{0}$ such that $\psi=b\circ e_{n-1}$.
Knowing that $a_{n}\circ a_{n+1}=0$, we see that the diagram

${A_{n+1}}$${A_{n}}$${\widetilde{A}_{n}}$${B_{1}}$${B_{0}}$${B}$$\scriptstyle{(-1)^{n}a_{n+1}}$$\scriptstyle{0}$$\scriptstyle{\pi_{n}}$$\scriptstyle{\varphi}$$\scriptstyle{e_{n-1}\circ
a_{n}}$$\scriptstyle{\alpha_{n}(B)(\varphi)}$$\scriptstyle{b_{1}}$$\scriptstyle{b}$

is commutative. Set
$\varphi[n]_{\bullet}\in\mathrm{Hom}_{\mathrm{Ch}(\mathcal{C})}(A[n]_{\bullet},B_{\bullet})$
to be zero in every degree except for $k=0$ where $\varphi[n]_{0}=e_{n-1}\circ
a_{n}$. Similarly, set $e[n]_{\bullet}:A[n]_{\bullet}\rightarrow
B_{\bullet+1}$ to be also zero in every degree except for $k=-1$ where
$e[n]_{-1}=(-1)^{n}e_{n-1}:A_{n-1}\rightarrow B_{0}$. Then $e[n]_{\bullet}$ is
a chain homotopy between $\varphi[n]_{\bullet}$ and the zero chain map.
Therefore, $\zeta_{n}^{\prime}$ descends to a homomorphism

$\displaystyle\zeta_{n}:\;\mathrm{Ext}_{\mathcal{C}}^{n}(A,B)$
$\displaystyle\rightarrow\mathcal{E}xt_{\mathcal{C}}^{n}(A,B)$
$\displaystyle\varphi+\mathrm{Im}(\mathrm{Hom}_{\mathcal{C}}(a_{n},B))$
$\displaystyle\mapsto\varphi[n]_{\bullet}+\mathrm{Null}_{\mathrm{Ch}(\mathcal{C})}(A[n]_{\bullet},B_{\bullet})\,.$

As we have assumed $n\geq 1$ for this, we define
$\zeta_{0}:=\zeta_{0}^{\prime}$ where we exploit the fact that
$\mathrm{Ker}(\mathrm{Hom}_{\mathcal{C}}(a_{1},B))=\mathrm{Ext}_{\mathcal{C}}^{0}(A,B)$.
Given that unenriched Ext-functors of negative degree vanish, we set
$\zeta_{n}:=0$ for $n<0$.

######  Proposition 6.4.

For every $n\in\mathbb{Z}$,
$\zeta_{n}:\mathrm{Ext}^{n}(A,B)\rightarrow\mathcal{E}xt_{\mathcal{C}}^{n}(A,B)$
is a isomorphism that is natural in both variables $A$ and $B$.

######  Proof.

Since $\zeta_{n}$ is a natural isomorphism for $n<0$, assume that
$n\in\mathbb{N}_{0}$ for the rest of the proof. To prove that $\zeta_{n}$ is
surjective, let
$\psi_{\bullet+n}\in\mathrm{Hom}_{\mathcal{C}}(A[n]_{\bullet},B_{\bullet})$.
As the diagram

${A_{n+1}}$${A_{n}}$${B_{1}}$${B_{0}}$${B}$$\scriptstyle{(-1)^{n}a_{n+1}}$$\scriptstyle{\psi_{n+1}}$$\scriptstyle{b\circ\psi_{n}}$$\scriptstyle{\psi_{n}}$$\scriptstyle{b_{1}}$$\scriptstyle{b}$

commutes, $B_{k}=$ for $k<0$ and $b_{1}\circ b=0$, we infer that
$b\circ\psi_{n}\in\mathrm{Ker}(\mathrm{Hom}_{\mathcal{C}}(a_{n+1},B))$ and
that

$\zeta_{n}(b\circ\psi_{0}+\mathrm{Im}(\mathrm{Hom}_{\mathcal{C}}(a_{n},B)))=\psi_{\bullet+n}+\mathrm{Null}_{\mathrm{Ch}(\mathcal{C})}(A[n]_{\bullet},B_{\bullet})\,.$

Let us show that $\zeta_{n}$ is injective. Assume first that $n\geq 1$. If
$\varphi\in\mathrm{Ker}(\mathrm{Hom}_{\mathcal{C}}(a_{n+1},B))$ is such that
$\zeta_{n}(\varphi+\mathrm{Im}(\mathrm{Hom}_{\mathcal{C}}(a_{n},B)))=0$, then
there are morphisms $\varphi_{n}:A_{n}\rightarrow B_{0}$,
$e_{n}:A_{n}\rightarrow B_{1}$ and $e_{n-1}:A_{n-1}\rightarrow B_{0}$ such
that

$\varphi=b\circ\varphi_{n}=b\circ\big{(}e_{n-1}\circ(-1)^{n}a_{n}+b_{1}\circ
e_{n}\big{)}=(-1)^{n}(b\circ e_{n})\circ a_{n}\,.$

In particular, $\varphi\in\mathrm{Im}(\mathrm{Hom}_{\mathcal{C}}(a_{n},B))$
and $\zeta_{n}$ is injective. Suppose that $n=0$ and that
$\varphi\in\mathrm{Ker}(\mathrm{Hom}_{\mathcal{C}}(a_{1},B))$ is such that
$\zeta_{0}(\varphi)=0$. Because $A_{-1}=0$ and $a_{0}=0$, we conclude that
$\varphi=0$ and that $\zeta_{0}$ is also injective.

Let us demonstrate that $\zeta_{n}$ is natural in both variables. Let
$g:C\rightarrow A$ and $f:B\rightarrow D$ be a morphism in $\mathcal{C}$ and
$\varphi\in\mathrm{Ker}(\mathrm{Hom}_{\mathcal{C}}(a_{n+1},B))$. Taking lifts
$f_{\bullet}:B_{\bullet}\rightarrow D_{\bullet}$ and
$g[n]_{\bullet}:C[n]_{\bullet}\rightarrow A[n]_{\bullet}$ as needed in
Definition 6.2, we see that the diagram

${{}\dots}$${C_{n+2}}$${C_{n+1}}$${C_{n}}$${{}\dots}$${A_{n+2}}$${A_{n+1}}$${A_{n}}$${{}\dots}$${B_{2}}$${B_{1}}$${B_{0}}$${B}$${{}\dots}$${D_{2}}$${D_{1}}$${D_{0}}$${C}$$\scriptstyle{(-1)^{n}c_{n+2}}$$\scriptstyle{g{[}n{]}_{2}}$$\scriptstyle{(-1)^{n}c_{n+1}}$$\scriptstyle{g{[}n{]}_{1}}$$\scriptstyle{g{[}n{]}_{0}}$$\scriptstyle{(-1)^{n}a_{n+2}}$$\scriptstyle{\varphi{[}n{]}_{2}}$$\scriptstyle{(-1)^{n}a_{n+1}}$$\scriptstyle{\varphi{[}n{]}_{1}}$$\scriptstyle{\varphi}$$\scriptstyle{\varphi{[}n{]}_{0}}$$\scriptstyle{b_{2}}$$\scriptstyle{f_{2}}$$\scriptstyle{b_{1}}$$\scriptstyle{f_{1}}$$\scriptstyle{b}$$\scriptstyle{f_{0}}$$\scriptstyle{f}$$\scriptstyle{d_{2}}$$\scriptstyle{d_{1}}$$\scriptstyle{d}$

commutes. This shows that

$\zeta_{n}\Big{(}\mathrm{Ext}_{\mathcal{C}}^{n}(A,f)\big{(}\varphi+\mathrm{Im}(\mathrm{Hom}_{\mathcal{C}}(a_{n+1},B))\big{)}\Big{)}=\mathcal{E}xt_{\mathcal{C}}^{n}(A,f)\Big{(}\zeta_{n}\big{(}\varphi+\mathrm{Im}(\mathrm{Hom}_{\mathcal{C}}(a_{n+1},B))\big{)}\Big{)}$

and that

$\zeta_{n}\Big{(}\mathrm{Ext}_{\mathcal{C}}^{n}(g,B)\big{(}\varphi+\mathrm{Im}(\mathrm{Hom}_{\mathcal{C}}(a_{n+1},B))\big{)}\Big{)}=\mathcal{E}xt_{\mathcal{C}}^{n}(g,B)\Big{(}\zeta_{n}\big{(}\varphi+\mathrm{Im}(\mathrm{Hom}_{\mathcal{C}}(a_{n+1},B))\big{)}\Big{)}\,.$

In short,
$\zeta_{n}\circ\mathrm{Ext}_{\mathcal{C}}^{n}(A,-)=\mathcal{E}xt_{\mathcal{C}}^{n}(A,-)\circ\zeta_{n}$
and
$\zeta_{n}\circ\mathrm{Ext}_{\mathcal{C}}^{n}(-,B)=\mathcal{E}xt_{\mathcal{C}}^{n}(-,B)\circ\zeta_{n}$.
$\square$

Our construction of the desired connecting homomorphisms for
$\mathcal{E}xt_{\mathcal{C}}^{\bullet}(A,-)$ and
$\widehat{\mathcal{E}xt}_{\mathcal{C}}^{\bullet}(A,-)$ is similar to the
definitions found in [13, p. 2202]. We consider a projective resolution
$(D_{\bullet},d_{\bullet})$ of $D$. For any $k\in\mathbb{N}_{0}$ define the
chain complex $(D_{\bullet}^{+1},d_{\bullet}^{+1})$ by setting
$(D_{m}^{+1},d_{m}^{+1})_{m\in\mathbb{N}}:=(D_{m+1},d_{m+1})_{m\in\mathbb{N}}$,
$D_{0}^{+1}:=D_{1}$ and $D_{m}^{+1}:=0$ for $m<0$. Observe that
$D_{\bullet}^{+1}$ is a projective resolution of $\widetilde{D}_{1}$. We
define a chain map $\pi_{\bullet}^{n}:D_{\bullet}\rightarrow
D^{+1}[-1]_{\bullet}$ by setting

$\pi_{m}^{n}:=\begin{cases}(-1)^{m-1+n}\mathrm{id}_{D_{m}}:D_{m}\rightarrow
D_{m}&\text{if }m\geq 1\\\ {}\;\;0&\text{if }m\leq 0\end{cases}$

Pictorially

${{}\dots}$${D_{4}}$${D_{3}}$${D_{2}}$${D_{1}}$${D_{0}}$${{}\dots}$${D_{4}}$${D_{3}}$${D_{2}}$${D_{1}}$${0}$$\scriptstyle{d_{4}}$$\scriptstyle{(-1)^{n+3}\mathrm{id}}$$\scriptstyle{d_{3}}$$\scriptstyle{(-1)^{n+2}\mathrm{id}}$$\scriptstyle{d_{2}}$$\scriptstyle{(-1)^{n+1}\mathrm{id}}$$\scriptstyle{d_{1}}$$\scriptstyle{(-1)^{n}\mathrm{id}}$$\scriptstyle{-d_{4}}$$\scriptstyle{-d_{3}}$$\scriptstyle{-d_{2}}$
(6.2)

As is noted in [15, p. 166], we can equally well express the
$(n+1)^{\text{st}}$ Ext-group as

$\mathcal{E}xt_{\mathcal{C}}^{n+1}(A,D)=\mathrm{Hom}_{\mathrm{Ch}(\mathcal{C})}(A[n]_{\bullet},D[-1]_{\bullet})/\mathrm{Null}_{\mathrm{Ch}(\mathcal{C})}(A[n]_{\bullet},D[-1]_{\bullet})$

with an analogous statement true for
$\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n+1}(A,D)$. The following definition
bears similarity with the connecting homomorphisms
$\varepsilon^{-m}:S^{-m}T\rightarrow S^{-m+1}T$ established at the start of
Section 3 and with the morphisms used in the construction of the connecting
homomorphisms of the naïve construction found in Lemma 5.9.

######  Definition 6.5.

We set

$\displaystyle\Delta^{n}:\mathcal{E}xt_{\mathcal{C}}^{n}(A,D)$
$\displaystyle\rightarrow\mathcal{E}xt_{\mathcal{C}}^{n+1}(A,\widetilde{D}_{1})$
$\displaystyle\varphi_{\bullet+n}+\mathrm{Null}_{\mathrm{Ch}(\mathcal{C})}(A[n]_{\bullet},D_{\bullet})$
$\displaystyle\mapsto\pi_{\bullet}^{n}\circ\varphi_{\bullet+n}+\mathrm{Null}_{\mathrm{Ch}(\mathcal{C})}(A[n]_{\bullet},D^{+1}[-1]_{\bullet})$

to be the (proposed) connecting homomorphism associated too the short exact
sequence $0\rightarrow\widetilde{D}_{1}\rightarrow D_{0}\rightarrow
D\rightarrow 0$. Analogously, we define the (proposed) connecting homomorphism
$\widehat{\Delta}^{n}:\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A,D)\rightarrow\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n+1}(A,\widetilde{D}_{1})$.

If $0\rightarrow B\rightarrow C\rightarrow D\rightarrow 0$ is another short
exact sequence in $\mathcal{C}$, then consider the lift
$h^{\ast}:\widetilde{D}_{1}\rightarrow B$ of $\mathrm{id}_{D}:D\rightarrow D$
from Diagram 3.2 and a chain map
$h^{\ast}_{\bullet+1}:D_{\bullet}^{+1}\rightarrow B_{\bullet}$ as in [42,
Theorem 2.2.6]. We define the (proposed) connecting homomorphism for this
short exact sequence as

$\displaystyle\Delta^{n}:\mathcal{E}xt_{\mathcal{C}}^{n}(A,D)$
$\displaystyle\rightarrow\mathcal{E}xt_{\mathcal{C}}^{n+1}(A,B)$
$\displaystyle\varphi_{\bullet+n}+\mathrm{Null}_{\mathrm{Ch}(\mathcal{C})}(A[n]_{\bullet},D_{\bullet})$
$\displaystyle\mapsto
h^{\ast}[-1]_{\bullet+1}\circ\pi_{\bullet}^{n}\circ\varphi_{\bullet+n}+\mathrm{Null}_{\mathrm{Ch}(\mathcal{C})}(A[n]_{\bullet},B[-1]_{\bullet})$

where the definition of
$\widehat{\Delta}^{n}:\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A,D)\rightarrow\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n+1}(A,B)$
is analogous.

Because the maps in the above definition are defined by compositions of
(almost) chain maps, they are homomorphisms of abelian groups. However, they
are not well defined a priori because they depend on a choice of morphism
$h^{\ast}:\widetilde{D}_{1}\rightarrow B$.

######  Lemma 6.6.

The family $(\mathcal{E}xt_{\mathcal{C}}^{\bullet}(A,-),\Delta^{\bullet})$
forms a cohomological functor. Furthermore,
$\zeta_{\bullet}:(\mathrm{Ext}_{\mathcal{C}}^{\bullet}(A,-),\delta^{\bullet})\rightarrow(\mathcal{E}xt_{\mathcal{C}}^{\bullet}(A,-),\Delta^{\bullet})$
is an isomorphism of cohomological functors.

######  Proof.

It suffices to prove that
$\zeta_{n}:\mathrm{Ext}_{\mathcal{C}}^{n}(A,-)\rightarrow\mathcal{E}xt_{\mathcal{C}}^{\bullet}(A,-)$
commutes with the (proposed) connecting homomorphisms. Let $0\rightarrow
B\rightarrow C\rightarrow D\rightarrow 0$ be a short exact sequence in
$\mathcal{C}$ and let $h^{\ast}:\widetilde{D}_{1}\rightarrow B$ be a lift of
$\mathrm{id}_{D}$ as before. We have seen at the start of the proof of Lemma
5.9 that the associated connecting homomorphism factors through the one
associated to $0\rightarrow\widetilde{D}_{1}\rightarrow D_{0}\rightarrow
D\rightarrow 0$ as

$\delta^{n}:\mathrm{Ext}_{\mathcal{C}}^{n}(A,D)\xrightarrow{\delta^{n}}\mathrm{Ext}_{\mathcal{C}}^{n+1}(A,\widetilde{D}_{1})\xrightarrow{\mathrm{Ext}_{\mathcal{C}}^{n+1}(A,h^{\ast})}\mathrm{Ext}_{\mathcal{C}}^{n+1}(A,B)\,.$

Thus, the left hand triangle in the diagram

${\mathrm{Ext}_{\mathcal{C}}^{n}(A{,}\,D)}$${\mathcal{E}xt_{\mathcal{C}}^{n}(A{,}\,D)}$${\mathrm{Ext}_{\mathcal{C}}^{n+1}(A{,}\,\widetilde{D}_{1})}$${\mathcal{E}xt_{\mathcal{C}}^{n+1}(A{,}\,\widetilde{D}_{1})}$${\mathrm{Ext}_{\mathcal{C}}^{n+1}(A{,}\,B)}$${\mathcal{E}xt_{\mathcal{C}}^{n+1}(A{,}\,B)}$$\scriptstyle{\zeta_{n}}$$\scriptstyle{\delta^{n}}$$\scriptstyle{\delta^{n}}$$\scriptstyle{\Delta^{n}}$$\scriptstyle{\Delta^{n}}$$\scriptstyle{\zeta_{n+1}}$$\scriptstyle{\mathrm{Ext}_{\mathcal{C}}^{n+1}(A{,}\,h^{\ast})}$$\scriptstyle{\mathcal{E}xt_{\mathcal{C}}^{n+1}(A{,}\,h^{\ast})}$$\scriptstyle{\zeta_{n+1}}$
(6.3)

commutes. By Definition 6.2 and 6.5, the right hand triangle triangle also
commutes. According to Proposition 6.4, the bottom trapezium is commutative.
The top trapzium can be seen as a generalisation of Proposition 1.1 from [13]
where we use a different method to prove that it commutes. Let
$\varphi\in\mathrm{Ker}(\mathrm{Hom}_{\mathcal{C}}(a_{n},D))$. We explain how
one can combine Diagram 5.6 and 6.1 as

${{}\dots}$${A_{n+2}}$${A_{n+1}}$${\widetilde{A}_{n+1}}$${A_{n}}$${\widetilde{A}_{n}}$${{}\dots}$${D_{2}}$${D_{1}}$${\widetilde{D}_{1}}$${D_{0}}$${D}$$\scriptstyle{(-1)^{n}a_{n+2}}$$\scriptstyle{\varphi{[}n{]}_{2}}$$\scriptstyle{\pi_{n+1}}$$\scriptstyle{a_{n+1}}$$\scriptstyle{\varphi{[}n{]}_{1}}$$\scriptstyle{\Phi}$$\scriptstyle{\iota_{n+1}}$$\scriptstyle{\pi_{n}}$$\scriptstyle{\varphi}$$\scriptstyle{\varphi{[}n{]}_{0}}$$\scriptstyle{\alpha_{n+k}(B)(\varphi)}$$\scriptstyle{d_{2}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{d_{1}}$$\scriptstyle{\iota_{1}}$$\scriptstyle{\pi_{0}}$
(6.4)

The right most morphisms up to $\Phi:A_{n+1}\rightarrow\widetilde{B}_{1}$ are
taken from Diagram 5.6 while $\varphi:A_{n}\rightarrow D$ and
$\varphi[n]_{\bullet}:A[n]_{\bullet}\rightarrow D_{\bullet}$ are the morphisms
as they occur in Diagram 6.1. In particular,

$\delta^{n}\big{(}\varphi+\mathrm{Im}(\mathrm{Hom}_{\mathcal{C}}(a_{n},D))\big{)}=\Phi+\big{(}\varphi+\mathrm{Im}(\mathrm{Hom}_{\mathcal{C}}(a_{n},\widetilde{D}_{1}))\,.$

However, note that $\iota_{n+1}\circ\pi_{n+1}=a_{n+1}$ might not agree with
$(-1)^{n}a_{n+1}:A_{n+1}\rightarrow A_{n}$. This together with the
construction of the Diagram 5.6 and the proof of the Comparison from [42, p.
36] implies that $\Phi=(-1)^{n}\pi_{1}\circ\varphi[n]_{1}$. Therefore, we can
choose the representative
$\Phi[n+1]_{\bullet}:A[n+1]\rightarrow\widetilde{D}_{\bullet}^{+1}$ of a lift
of $\Phi$ as in Diagram 6.1 which is defined as

$\Phi[n+1]_{m}:=\begin{cases}(-1)^{n+m}\varphi[n]_{m+1}:A_{m+n+1}\rightarrow
D_{m+1}&\text{if }m\geq 0\\\ \;\;0&\text{if }m<0\end{cases}$

The alternating signs are due to the fact that the boundary maps of
$A[n]_{\bullet}$ and $A[n+1]$ have opposite signs. We conclude by Diagram 6.2,
Definition 6.5 and Diagram 6.4 that

$\zeta_{n+1}\Big{(}\delta^{n}\big{(}\varphi+\mathrm{Im}(\mathrm{Hom}_{\mathcal{C}}(a_{n},D))\big{)}\Big{)}=\Delta^{n}\Big{(}\zeta^{n}\big{(}\varphi+\mathrm{Im}(\mathrm{Hom}_{\mathcal{C}}(a_{n},D))\big{)}\Big{)}\,.$

In particular, the top trapezium and with it all of Diagram 6.3 commutes. As
for Theorem 5.10, we can now reiterate the proof of Theorem 4.5 with Diagram
6.3, the fact that
$(\mathrm{Ext}_{\mathcal{C}}^{\bullet}(A,-),\delta^{\bullet})$ is a
cohomological functor and that
$\zeta_{\bullet}:\mathrm{Ext}_{\mathcal{C}}^{\bullet}(A,-)\rightarrow\mathcal{E}xt_{\mathcal{C}}^{\bullet}(A,-)$
is a natural isomorphism by Proposition 6.4. $\square$

###  6.3 Isomorphism of cohomology groups

In this subsection we construct for every $n\in\mathbb{Z}$ a natural
isomorphism from the completed unenriched Ext-functor
$\mathrm{Ext}_{Res,\mathcal{C}}^{n}(A,-)$ defined through the resolution
construction to $\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A,-)$. In order to
do so, we first reformulate the former completed Ext-functor.

######  Definition 6.7.

For any $B\in\mathrm{obj}(\mathcal{C})$, $f:B\rightarrow C$ and
$n\in\mathbb{Z}$ define

$\mathcal{E}xt_{Res,\mathcal{C}}^{n}(A,B):=\varinjlim_{k\in\mathbb{N}_{0}}(\mathcal{E}xt_{\mathcal{C}}^{n+k}(A,\widetilde{B}_{k}),\Delta^{n+k})$

and

$\mathcal{E}xt_{Res,\mathcal{C}}^{n}(A,f):=\varinjlim_{k\in\mathbb{N}_{0}}\mathcal{E}xt_{\mathcal{C}}^{n+k}(A,\widetilde{f}_{k}):\mathcal{E}xt_{\mathcal{C}}^{n}(A,B)\rightarrow\mathcal{E}xt_{\mathcal{C}}^{n}(A,C)$

where $\widetilde{f}_{k}:\widetilde{B}_{k}\rightarrow\widetilde{C}_{k}$ are
taken as in Proposition 4.2. For any short exact sequence $0\rightarrow
B\rightarrow C\rightarrow D\rightarrow 0$ we set

$\widetilde{\Delta}^{n}:=\varinjlim_{k\in\mathbb{N}_{0}}(-1)^{k}\Delta^{n+k}:\mathcal{E}xt_{Res,\mathcal{C}}^{n}(A,D)\rightarrow\mathcal{E}xt_{Res,\mathcal{C}}^{n+1}(A,B)$

where
$\Delta^{n+k}:\mathcal{E}xt_{\mathcal{C}}^{n+k}(A,\widetilde{D}_{k})\rightarrow\mathcal{E}xt_{\mathcal{C}}^{n+k+1}(A,\widetilde{B}_{k})$
are taken as in Diagram 4.10. Using the isomorphisms of cohomological functors
$\zeta_{n+k}:\mathrm{Ext}_{\mathcal{C}}^{n+k}(A,-)\rightarrow\mathcal{E}xt_{\mathcal{C}}^{n+k}(A,-)$
from Lemma 6.6, we can extend the natural isomorphisms

$\zeta^{n}:=\varinjlim_{k\in\mathbb{N}_{0}}\zeta_{n+k}:\mathrm{Ext}_{Res,\mathcal{C}}^{n}(A,-)\rightarrow\mathcal{E}xt_{Res,\mathcal{C}}^{n}(A,-)$

to
$\zeta^{\bullet}:(\mathrm{Ext}_{Res,\mathcal{C}}^{\bullet}(A,-),\widetilde{\delta}^{\bullet})\rightarrow(\mathcal{E}xt_{Res,\mathcal{C}}^{\bullet}(A,-),\widetilde{\Delta}^{\bullet})$
which is an isomorphism of cohomological functors.

We only consider the cofinal system
$(\mathcal{E}xt_{\mathcal{C}}^{n+2k}(A,\widetilde{B}_{2k}),\Delta^{n+2k+1}\circ\Delta^{n+2k})_{k\in\mathbb{N}_{0}}$
and specifically form $\mathcal{E}xt_{Res,\mathcal{C}}^{n}(A,B)$ as its direct
limit in order to ease notation when constructing a homomorphism from any
cohomology group $\mathcal{E}xt_{Res,\mathcal{C}}^{n}(A,B)$ to
$\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A,B)$. Nonetheless, we require

######  Notation 6.8.

If $(B_{\bullet},b_{\bullet})$ is a projective resolution of $B$ and
$k\in\mathbb{N}_{0}$, then we define the projective resolution
$(B_{\bullet}^{+k},b_{\bullet}^{+k})_{m\in\mathbb{Z}}$ of $\widetilde{B}_{k}$
by declaring for $m\in\mathbb{Z}$ that

$(B_{m}^{+k},b_{m+1}^{+k}):=\begin{cases}(B_{m+k},\,b_{m+1+k}:B_{m+1+k}\rightarrow
B_{m+k})&\text{if }m\geq 0\\\ \;\;0&\text{if }m<0\end{cases}$

######  Lemma 6.9.

There is an isomorphism
$\vartheta^{n}(B):\mathcal{E}xt_{Res,\mathcal{C}}^{n}(A,B)\rightarrow\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A,B)$
for any $n\in\mathbb{Z}$.

######  Proof.

For any $k\in\mathbb{N}_{0}$ we construct a map

$\displaystyle\vartheta_{n}^{2k}(B):\mathcal{E}xt_{\mathcal{C}}^{n+2k}(A,\widetilde{B}_{2k})$
$\displaystyle\rightarrow\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A,B)$
$\displaystyle\varphi_{\bullet+n+2k}+\mathrm{Null}_{\mathrm{Ch}(\mathcal{C})}(A[n+2k]_{\bullet},B_{\bullet}^{+2k})$
$\displaystyle\mapsto\Phi_{\bullet+n}+\widehat{\mathrm{Null}}_{\mathrm{Ch}(\mathcal{C})}(A[n]_{\bullet},B_{\bullet})$

as follows. For a representative
$\varphi_{\bullet+n+2k}\in\mathrm{Hom}_{\mathrm{Ch}(\mathcal{C})}(A[n+2k]_{\bullet},B_{\bullet}^{+2k})$
of an element in $\mathcal{E}xt_{\mathcal{C}}^{n+2k}(A,\widetilde{B}_{2k})$ we
define $\Phi_{\bullet+n}:A[n]_{\bullet}\rightarrow B_{\bullet}$ by

$\Phi_{m+n}:=\begin{cases}\varphi_{m+n}:A_{m+n}\rightarrow B_{m}&\text{if
}m\geq 2k\\\ \;\;0&\text{if }m<2k\end{cases}$

As the boundary maps of $A[n]$ and $A[n+2k]$ agree modulo a shift of $2k$ in
their indices, all squares of the form

${A_{m+n}}$${B_{m}}$${A_{m-1+n}}$${B_{m-1}}$$\scriptstyle{\Phi_{m+n}}$$\scriptstyle{(-1)^{n}a_{m+n}}$$\scriptstyle{b_{m}}$$\scriptstyle{\Phi_{n+m-1}}$

commute for $m\in\mathbb{Z}\setminus\\{2k\\}$. We conclude that
$\Phi_{\bullet+n}\in\widehat{\mathrm{Hom}}_{\mathrm{Ch}(\mathcal{C})}(A[n]_{\bullet},B_{\bullet})$.
If the chain map $\varphi_{\bullet+n+2k}$ is nullhomotopic, then
$\Phi_{\bullet+n}$ is genuinely nullhomotopic in any degree
$m\in\mathbb{Z}\setminus\\{2k-1\\}$, hence nullhomotopic as an almost chain
map. Thus, $\vartheta_{n}^{2k}$ is a well defined map. It follows by
construction that it is also a homomorphism of abelian groups.

We argue that the triangle

${\mathcal{E}xt_{\mathcal{C}}^{n+2k}(A{,}\,\widetilde{B}_{2k})}$${\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A{,}\,B)}$${\mathcal{E}xt_{\mathcal{C}}^{n+2k+2}(A{,}\,\widetilde{B}_{2k+2)})}$$\scriptstyle{\vartheta_{n}^{2k}(B)}$$\scriptstyle{\Delta^{n+2k+1}\circ\Delta^{n+2k}}$$\scriptstyle{\vartheta_{n}^{2k+2)}(B)}$
(6.5)

commutes. By Definition 6.5, $\Delta^{n+2k}$ and $\Delta^{n+2k+2}$ are
constructed using the chain maps
$\pi_{\bullet}^{n}:B_{\bullet}^{+2k}\rightarrow B^{+2k+1}[-1]_{\bullet}$ and
$\pi^{n+1}[-1]_{\bullet}:B^{+2k+1}[-1]_{\bullet}\rightarrow
B^{+2k+2}[-2]_{\bullet}$ which are pictorially rendered in Diagram 6.2. Then
$\pi^{n+1}[-1]_{\bullet}\circ\pi_{\bullet}^{n}$ is given by

$\pi^{n+1}[-1]_{m}\circ\pi_{m}^{n}=\begin{cases}\mathrm{id}_{B_{m}}:B_{m}\rightarrow
B_{m}&\text{if }m\geq 2\\\ \;\;0&\text{if }m\leq 1\end{cases}$

since $(-1)^{(m-1)-1+(n+1)}\cdot(-1)^{m-1+n}=1$. Hence, the $m^{th}$ degree of
an almost chain map representative of the image of an element
$\varphi_{\bullet+n+2k}+\mathrm{Null}_{\mathrm{Ch}(\mathcal{C})}(A[n+2k],B_{\bullet}^{+2k})$
in $\mathcal{E}xt_{\mathcal{C}}^{n+2k}(A,\widetilde{B}_{2k})$ under
$\Delta^{n+2k+1}\circ\Delta^{n+2k}$ is

$\begin{cases}\varphi_{m+n+2k+2}:A_{m+n+2k+2}\rightarrow B_{m+2k+2}&\text{if
}m\geq 0\\\ \;\;0&\text{if }m<0\end{cases}$ (6.6)

We infer by definition of $\vartheta_{n}^{2k}(B)$ and
$\vartheta_{n}^{2k+2}(B)$ that

$\displaystyle\vartheta_{n}^{2k}(B$
$\displaystyle)\big{(}\varphi_{\bullet+n+2k}+\mathrm{Null}_{\mathrm{Ch}(\mathcal{C})}(A[n+2k],B_{\bullet}^{+2k})\big{)}=$
$\displaystyle\big{(}\vartheta_{n}^{2k+2}\circ\Delta^{n+2k+1}\circ\Delta^{n+2k}\big{)}\big{(}\varphi_{\bullet+n+2k}+\mathrm{Null}_{\mathrm{Ch}(\mathcal{C})}(A[n+2k],B_{\bullet}^{+2k})\
\big{)}$

The resulting homomorphism in the direct limit of the above commuting
triangles is

$\vartheta^{n}(B):=\varinjlim_{k\in\mathbb{N}_{0}}\vartheta_{n}^{2k}(B):\;\mathcal{E}xt_{Res,\mathcal{C}}^{n}(A,B)\rightarrow\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A,B)\,.$

To show that it is surjective, let
$\Phi_{\bullet+n}+\widehat{\mathrm{Null}}_{\mathrm{Ch}(\mathcal{C})}(A[n]_{\bullet},B_{\bullet})$
be in $\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A,B)$. Since it is an almost
chain map, there is $k\in\mathbb{N}_{0}$ such that
$(\Phi_{m+n}:A_{m+n}\rightarrow B_{m})_{m\geq 2k}$ is a chain map. Defining
$\varphi_{\bullet+n+2k}:A[n+2k]_{\bullet}\rightarrow B_{\bullet}^{+2k}$ by
setting

$\varphi_{m+n+2k}:=\begin{cases}\Phi_{m+n+2k}:A_{m+n+2k}\rightarrow
B_{m+2k}&\text{if }m\geq 0\\\ \;\;0&\text{if }m<0\end{cases}$

gives an element in $\mathcal{E}xt_{\mathcal{C}}^{n+2k}(A,\widetilde{B}_{2k})$
that $\vartheta_{n}^{2k}(B)$ maps to
$\Phi_{\bullet+n}+\widehat{\mathrm{Null}}_{\mathrm{Ch}(\mathcal{C})}(A[n]_{\bullet},B_{\bullet})$.
To prove that $\vartheta^{n}(B)$ is injective, let
$x\in\varinjlim\mathcal{E}xt_{\mathcal{C}}^{n+2k}(A,\widetilde{B}_{2k})$ with
$\vartheta^{n}(B)(x)=0$. According to Proposition 5.6, there exists
$k\in\mathbb{N}_{0}$ and an element

$\psi_{\bullet+n+2k}\in\mathrm{Hom}_{\mathrm{Ch}(\mathcal{C})}(A[n+2k]_{\bullet},B_{\bullet}^{+2k})$

mapping to $x$ in the direct limit that lies in the kernel of the homomorphism
$\vartheta_{n}^{2k}(B)$. If we construct an almost chain map representative
$\Psi_{\bullet+n}:A[n]_{\bullet}\rightarrow B_{\bullet}$ from
$\psi_{\bullet+n+k}$ as above, then
$\Psi_{\bullet+n}\in\widehat{\mathrm{Null}}_{\mathrm{Ch}(\mathcal{C})}(A[n]_{\bullet},B_{\bullet})$.
Therefore, there is $K\geq k$ such that $(\Psi_{m})_{m\geq 2K}$ is
nullhomotopic as a chain map and not just as an almost chain map. The chain
map $\overline{\psi}_{\bullet+n+2K}:A[n+2K]_{\bullet}\rightarrow
B_{\bullet}^{+2K}$ that we define by

$\overline{\psi}_{m+n+2K}:=\begin{cases}\Psi_{m+n+2K}=\psi_{m+n+2K}:A_{m+n+2K}\rightarrow
B_{m+2K}&\text{if }m\geq 0\\\ \;\;0&\text{if }m<0\end{cases}$

lies in
$\mathrm{Null}_{\mathrm{Ch}(\mathcal{C})}(A[n+2K]_{\bullet},B_{\bullet}^{+2K})$.
By Equation 6.6, $\overline{\psi}_{\bullet+n+2K}$ is a representative of the
image of
$\psi_{\bullet+n+2k}+\mathrm{Null}_{\mathrm{Ch}(\mathcal{C})}(A[n+2k]_{\bullet},B_{\bullet}^{+2k})$
under $\Delta^{n+2K-1}\circ{}\dots\circ\Delta^{n+2k}$, which also maps to the
element $x$ in the direct limit. As $x=0$, this implies that
$\vartheta^{n}(B)$ is injective and thus an isomorphism. $\square$

######  Proposition 6.10.

For every $n\in\mathbb{Z}$ the isomorphism $\vartheta^{n}(-)$ is natural. That
is, for every morphism $f:B\rightarrow C$ in $\mathcal{C}$ the square

${\mathcal{E}xt_{Res,\mathcal{C}}^{n}(A{,}\,B)}$${\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A{,}\,B)}$${\mathcal{E}xt_{Res,\mathcal{C}}^{n}(A{,}\,C)}$${\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A{,}\,C)}$$\scriptstyle{\vartheta^{n}(B)}$$\scriptstyle{\mathcal{E}xt_{Res,\mathcal{C}}^{n}(A{,}\,f)}$$\scriptstyle{\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A{,}\,f)}$$\scriptstyle{\vartheta^{n}(C)}$

commutes.

######  Proof.

Let
$(\widetilde{f}_{k}:\widetilde{M}_{k}\rightarrow\widetilde{N}_{k})_{k\in\mathbb{N}_{0}}$
be a sequence of morphism as arising from Diagram 4.8. By Definition 6.7 it
suffices to consider the diagram

${{\scriptscriptstyle\mathcal{E}xt_{\mathcal{C}}^{n+2k}(A{,}\,\widetilde{C}_{2k})}}$${{\scriptscriptstyle\mathcal{E}xt_{\mathcal{C}}^{n+2k}(A{,}\,\widetilde{B}_{2k})}}$${{\scriptscriptstyle\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A{,}\,C)}}$${{\scriptscriptstyle\mathcal{E}xt_{\mathcal{C}}^{n+2k+2}(A{,}\,\widetilde{C}_{2k+2})}}$${{\scriptscriptstyle\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A{,}\,B)}}$${{\scriptscriptstyle\mathcal{E}xt_{\mathcal{C}}^{n+2k+2}(A{,}\,\widetilde{B}_{2k+2})}}$$\scriptstyle{\scriptscriptstyle\vartheta_{n}^{2k}(C)}$$\scriptstyle{\scriptscriptstyle\Delta^{n+2k+1}\circ\Delta^{n+2k}}$$\scriptstyle{\scriptscriptstyle\mathcal{E}xt_{\mathcal{C}}^{n+2k}(A{,}\,\widetilde{f}_{2k})}$$\scriptstyle{\scriptscriptstyle\Delta^{n+2k+1}\circ\Delta^{n+2k}}$$\scriptstyle{\scriptscriptstyle\vartheta_{n}^{2k+2}(C)}$$\scriptstyle{\scriptscriptstyle\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A{,}\,f)}$$\scriptstyle{\scriptscriptstyle\vartheta_{n}^{2k}(B)}$$\scriptstyle{\scriptscriptstyle\mathcal{E}xt_{\mathcal{C}}^{n+2k+2}(A{,}\,\widetilde{f}_{2k+2})}$$\scriptstyle{\scriptscriptstyle\vartheta_{n}^{2k+2}(B)}$
(6.7)

The left hand side of the prism commutes because $\Delta^{n+2k}$ and
$\Delta^{n+2k+1}$ are connecting homomorphisms. The triangles in the front and
the back commute by Diagram 6.5. In Definitino 6.2, we have constructed
$\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A,f)$ by lifting $f:B\rightarrow C$
to a chain map $f_{\bullet}:B_{\bullet}\rightarrow C_{\bullet}$ through the
Comparison Theorem [42, Theorem 2.2.6] and by composing any almost chain map
representative of an element in
$\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A,B)$ with $f_{\bullet}$. Note that
we can lift
$\widetilde{f}_{2k}:\widetilde{B}_{2k}\rightarrow\widetilde{C}_{2k}$ to a
chain map $f_{\bullet}^{+2k}:B_{\bullet}^{+2k}\rightarrow C_{\bullet}^{+2k}$
given by

$f_{m}^{+2k}:=\begin{cases}f_{m+2k}:B_{m+2k}\rightarrow C_{m+2k}&\text{if
}m\geq 0\\\ \;\;0&\text{if }m<0\end{cases}$

In particular, the square

${\mathcal{E}xt_{\mathcal{C}}^{n+2k}(A{,}\,\widetilde{B}_{2k})}$${\mathcal{E}xt_{\mathcal{C}}^{n+2k}(A{,}\,\widetilde{C}_{2k})}$${\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A{,}\,B)}$${\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A{,}\,C)}$$\scriptstyle{\mathcal{E}xt_{\mathcal{C}}^{n+2k}(A{,}\,\widetilde{f}_{2k})}$$\scriptstyle{\vartheta_{n}^{2k}(B)}$$\scriptstyle{\vartheta_{n}^{2k}(C)}$$\scriptstyle{\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A{,}\,f)}$

commutes and with it Diagram 6.7. The top right and bottom right squares of
the latter diagram form a direct system in whose direct limit we obtain the
commuting square in the statement of the proposition. $\square$

######  Corollary 6.11.

For every $n\in\mathbb{Z}$ we define the natural isomorphism

$\sigma^{n}:=\vartheta^{n}\circ\zeta^{n}:\widehat{\mathrm{Ext}}_{\mathcal{C}}^{n}(A,-)\rightarrow\mathcal{E}xt_{Res,\mathcal{C}}^{n}(A,-)\rightarrow\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A,-)$

where $\zeta^{n}$ is taken from Definition 6.7.

###  6.4 Isomorphism of cohomological functors

In order to prove that the natural isomorphisms
$\vartheta^{n}:\mathcal{E}xt_{Res,\mathcal{C}}^{n}(A,-)\rightarrow\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A,-)$
extend to an isomorphism of cohomological functors, we need the following
technical result.

######  Proposition 6.12.

Let $0\rightarrow B\xrightarrow{f}C\xrightarrow{g}D\rightarrow 0$ be a short
exact sequence in $\mathcal{C}$. If $h^{\ast}:\widetilde{D}_{1}\rightarrow B$
denotes a lift of $\mathrm{id}_{D}:D\rightarrow D$ as in Diagram 3.2 and
$D_{\bullet}$ a projective resolution of $D$, consider a lift of $h^{\ast}$ to
a chain map $h_{\bullet+1}:D_{\bullet}^{+1}\rightarrow B_{\bullet}$ given by
[42, Theorem 2.2.6]. Denote by
$\widetilde{h}_{k+1}^{\ast}:\widetilde{D}_{k+1}\rightarrow\widetilde{B}_{k}$
the induced morphism between the corresponding syzygies where
$\widetilde{h}_{1}^{\ast}=h^{\ast}$. Then there is a projective resolution
$\underline{D}_{\bullet}^{\prime}$ of $D$ such that for any
$k\in\mathbb{N}_{0}$, $\widetilde{D}_{k}=\widetilde{\underline{D}}_{k}$ and
there is a commutative diagram

${0}$${\widetilde{D}_{k+1}}$${\underline{D}_{k}}$${\widetilde{D}_{k}}$${0}$${0}$${\widetilde{B}_{k}}$${\widetilde{C}_{k}}$${\widetilde{D}_{k}}$${0}$$\scriptstyle{\iota_{k+1}}$$\scriptstyle{\widetilde{h}_{k+1}^{\ast}}$$\scriptstyle{\pi_{k}}$$\scriptstyle{h_{k}}$$\scriptstyle{\mathrm{id}}$$\scriptstyle{\widetilde{f}_{k}}$$\scriptstyle{\widetilde{g}_{k}}$
(6.8)

where the morphisms $\iota_{k+1}$, $\pi_{k}$ stem from
$\underline{D}_{\bullet}$. In particular, any $\widetilde{h}_{k+1}^{\ast}$ is
a lift of $\mathrm{id}_{\widetilde{D}_{k}}$ as in Diagram 3.2.

Let us explain the context in which we need this result. In Definition 6.5, we
have first constructed a connecting homomorphism
$\widehat{\Delta}^{n}:\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A,D)\rightarrow\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n+1}(A,\widetilde{D}_{1})$
associated to the short exact sequence
$0\rightarrow\widetilde{D}_{1}\rightarrow D_{0}\rightarrow D\rightarrow 0$.
Then we defined

$\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A,D)\xrightarrow{\widehat{\Delta}^{n}}\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n+1}(A,\widetilde{D}_{1})\xrightarrow{\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n+1}(A,h^{\ast})}\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n+1}(A,B)$

to be the connecting homomorphism associated to $0\rightarrow B\rightarrow
C\rightarrow D\rightarrow 0$. Later, we shall compare this to

$\mathcal{E}xt_{\mathcal{C}}^{n+2k}(A,\widetilde{D}_{2k})\xrightarrow{\Delta^{n}}\mathcal{E}xt_{\mathcal{C}}^{n+1+2k}(A,\widetilde{D}_{2k+1})\xrightarrow{\mathcal{E}xt_{\mathcal{C}}^{n+1+2k}(A,\widetilde{h}_{2k+1}^{\ast})}\mathcal{E}xt_{\mathcal{C}}^{n+1+2k}(A,\widetilde{B}_{2k})$

which we would like to be the connecting homomorphism associated to the short
exact sequence
$0\rightarrow\widetilde{B}_{2k}\rightarrow\widetilde{C}_{2k}\rightarrow\widetilde{D}_{2k}\rightarrow
0$. This is where Diagram 6.8 comes into play.

######  Proof (Proposition 6.12).

We proceed by induction over $k\in\mathbb{N}_{0}$. If $k=0$, then we take
$\underline{D}_{0}:=D_{0}$ where Diagram 3.2 yields the desired diagram for
this case. Assume that $\underline{D}_{0},\dots,\underline{D}_{k}$ have been
defined and that there is a diagram of the required form for a number $k\geq
0$. In order to construct $\underline{D}_{k+1}$ and establish Diagram 6.8 with
$h_{k+1}:\underline{D}_{k+1}\rightarrow\widetilde{C}_{k}$ and
$\widetilde{h}_{k+2}^{\ast}:\widetilde{D}_{k+2}\rightarrow\widetilde{B}_{k+1}$,
we consider two instance of Diagram 4.9 from the Horseshoe Lemma. For this, we
follow the construction found in [42, p. 37–38] that yields the proof of this
so-called lemma. We explain how to obtain the diagram

${0}$${0}$${0}$${0}$${\widetilde{D}_{k+2}}$${\mathrm{Ker}(\varepsilon)}$${\widetilde{D}_{k+1}}$${0}$${0}$${D_{k+1}}$${D_{k+1}\oplus\underline{D}_{k}}$${\underline{D}_{k}}$${0}$${0}$${\widetilde{D}_{k+1}}$${\underline{D}_{k}}$${\widetilde{D}_{k}}$${0}$${0}$${0}$${0}$$\scriptstyle{\iota_{k+2}^{\prime\prime}}$$\scriptstyle{\iota_{k+2}^{\prime}}$$\scriptstyle{\pi_{k+1}^{\prime\prime}}$$\scriptstyle{\iota_{k+1}}$$\scriptstyle{i_{1}^{1}}$$\scriptstyle{\iota_{k+1}\circ\pi_{k+1}^{\prime}}$$\scriptstyle{\pi_{k+1}^{\prime}}$$\scriptstyle{p_{2}^{1}}$$\scriptstyle{p_{1}^{1}}$$\scriptstyle{\epsilon}$$\scriptstyle{i_{2}^{1}}$$\scriptstyle{\mathrm{id}}$$\scriptstyle{\pi_{k}}$$\scriptstyle{\iota_{k+1}}$$\scriptstyle{\pi_{k}}$
(6.9)

The morphisms $\pi_{k}$, $\iota_{k+1}$ stem from the projective resolution
$(\underline{D}_{m})_{m=1}^{k}$ and the morphisms $\pi_{k+1}^{\prime}$,
$\iota_{k+2}^{\prime}$ from $D_{\bullet}$. In particular, the triangle in the
bottom left corner and the one in the bottom right corner commute. The
morphisms of the middle row are the ones of a direct product as they can be
found in [27, p. 250]. Moreover,
$\varepsilon=(\iota_{k+1}\circ\pi_{k+1}^{\prime})\circ
p_{1}^{1}+\mathrm{id}\circ p_{2}^{1}$ by [27, p. 250] and [42, p. 37]. Since
$\iota_{k+2}^{\prime}$ and $\iota_{k+1}$ can be taken as kernels of
$\pi_{k+1}^{\prime}$ and $\pi_{k}$, the morphisms $\iota_{k+2}^{\prime}$ and
$\pi_{k+1}^{\prime\prime}$ are induced by the morphisms $i_{1}^{1}$ and
$p_{2}^{1}$. All rows and columns are exact. Because $\underline{D}_{k}$ is
projective, the middle column is a split exact sequence by [38, Tag 010G].
This together with $D_{k+1}\oplus\underline{D}_{k}$ being projective implies
that $\mathrm{Ker}(\varepsilon)$ is also projective. Similarly, we construct
with the morphisms $\widetilde{f}_{k}$ and $\widetilde{g}_{k}$ from Diagram
6.8 the diagram

${0}$${0}$${0}$${0}$${\widetilde{B}_{k+1}}$${\widetilde{C}_{k+1}}$${\widetilde{D}_{k+1}}$${0}$${0}$${B_{k}}$${B_{k}\oplus\underline{D}_{k}}$${\underline{D}_{k}}$${0}$${0}$${\widetilde{B}_{k}}$${C_{k}}$${\widetilde{D}_{k}}$${0}$${0}$${0}$${0}$$\scriptstyle{\widetilde{f}_{k+1}}$$\scriptstyle{\widetilde{g}_{k+1}}$$\scriptstyle{\iota_{k+1}}$$\scriptstyle{i_{1}^{2}=f_{k}}$$\scriptstyle{\widetilde{f}_{k}\circ\check{\pi}_{k}}$$\scriptstyle{\check{\pi}_{k}}$$\scriptstyle{p_{2}^{2}=g_{k}}$$\scriptstyle{p_{1}^{2}}$$\scriptstyle{\eta}$$\scriptstyle{i_{2}^{2}}$$\scriptstyle{h_{k}}$$\scriptstyle{\pi_{k}}$$\scriptstyle{\widetilde{f}_{k}}$$\scriptstyle{\widetilde{g}_{k}}$
(6.10)

Note that $\eta=(\widetilde{f}_{k}\circ\check{\pi}_{k})\circ
p_{1}^{2}+h_{k}\circ p_{2}^{2}$ and that the triangle at the bottom right
corner commutes by Diagram 6.8. Now consider the diagram

${{\scriptstyle D_{k+1}}}$${{\scriptstyle
D_{k+1}\oplus\underline{D}_{k}}}$${{\scriptstyle\underline{D}_{k}}}$${{\scriptstyle
B_{k}}}$${{\scriptstyle
B_{k}\oplus\underline{D}_{k}}}$${{\scriptstyle\underline{D}_{k}}}$${{\scriptstyle\widetilde{D}_{k+1}}}$${{\scriptstyle\underline{D}_{k}}}$${{\scriptstyle\widetilde{D}_{k}}}$${{\scriptstyle\widetilde{B}_{k}}}$${{\scriptstyle
C_{k}}}$${{\scriptstyle\widetilde{D}_{k}}}$$\scriptstyle{\scriptstyle
i_{1}^{1}}$$\scriptstyle{\scriptstyle\iota_{k+1}\circ\pi_{k+1}^{\prime}}$$\scriptstyle{\scriptstyle
h_{k+1}^{\ast}}$$\scriptstyle{\scriptstyle\pi_{k+1}^{\prime}}$$\scriptstyle{\scriptstyle
p_{2}^{1}}$$\scriptstyle{\scriptstyle
h_{k+1}^{\ast}\oplus\mathrm{id}}$$\scriptstyle{\scriptstyle\epsilon}$$\scriptstyle{\scriptstyle\mathrm{id}}$$\scriptstyle{\scriptstyle\mathrm{id}}$$\scriptstyle{\scriptstyle\pi_{k}}$$\scriptstyle{\scriptstyle
i_{1}^{2}}$$\scriptstyle{\scriptstyle\check{\pi}_{k}}$$\scriptstyle{\scriptstyle
p_{2}^{2}}$$\scriptstyle{\scriptstyle\iota_{k+1}}$$\scriptstyle{\scriptstyle\widetilde{h}_{k+1}^{\ast}}$$\scriptstyle{\scriptstyle\pi_{k}}$$\scriptstyle{\scriptstyle
h_{k}}$$\scriptstyle{\scriptstyle\mathrm{id}}$$\scriptstyle{\scriptstyle\widetilde{f}_{k}}$$\scriptstyle{\scriptstyle\widetilde{g}_{k}}$$\scriptstyle{\scriptstyle\widetilde{f}_{k}\circ\check{\pi}_{k}}$$\scriptstyle{\scriptstyle\eta}$$\scriptstyle{\scriptstyle
h_{k}}$$\scriptstyle{\scriptstyle\pi_{k}}$ (6.11)

The bottom side stems from Diagram 6.8, the back side from the lower half of
Diagram 6.9 and the front side from the lower half of Diagram 6.10. The left
hand side arises from the chain map
$h_{\bullet}^{\ast}:D_{\bullet}^{+1}\rightarrow B_{\bullet}$ while the right
hand side commutes trivially. The morphism
$h_{k+1}^{\ast}\oplus\mathrm{id}:D_{k+1}\oplus\underline{D}_{k}\rightarrow
B_{k}\oplus\underline{D}_{k}$ is constructed as in [27, p. 251] and renders
the top side commutative. In particular, all squares on the outside of Diagram
6.11 commute. The slanted squares in the interior give rise to the commuting
diagram

${D_{k+1}}$${\underline{D}_{k}}$${\underline{D}_{k}}$${B_{k}}$${C_{k}}$${\underline{D}_{k}}$$\scriptstyle{\iota_{k+1}\circ\pi_{k+1}^{\prime}}$$\scriptstyle{h_{k+1}^{\ast}}$$\scriptstyle{h_{k}}$$\scriptstyle{\mathrm{id}}$$\scriptstyle{\mathrm{id}}$$\scriptstyle{\widetilde{f}_{k}\circ\check{\pi}_{k}}$$\scriptstyle{h_{k}}$

We can use it together with Diagram 6.9, Diagram 6.10 and the content of [27,
p. 251] to prove that that the middle square in the interior of Diagram 6.11
also commutes. Namely,

$\displaystyle\eta\circ(h_{k+1}^{\ast}\oplus\mathrm{id})$
$\displaystyle=\big{(}(\widetilde{f}_{k}\circ\check{\pi}_{k})\circ
p_{1}^{2}+h_{k}\circ p_{2}^{2}\big{)}\circ(h_{k+1}^{\ast}\oplus\mathrm{id})$
$\displaystyle=(\widetilde{f}_{k}\circ\check{\pi}_{k})\circ
p_{1}^{2}\circ(h_{k+1}^{\ast}\oplus\mathrm{id})+h_{k}\circ
p_{2}^{2}\circ(h_{k+1}^{\ast}\oplus\mathrm{id})$
$\displaystyle=(\widetilde{f}_{k}\circ\check{\pi}_{k})\circ
h_{k+1}^{\ast}\circ p_{1}^{1}+h_{k}\circ\mathrm{id}\circ p_{2}^{1}$
$\displaystyle=h_{k}\circ(\iota_{k+1}\circ\pi_{k+1}^{\prime})\circ
p_{1}^{1}+h_{k}\circ\mathrm{id}\circ p_{2}^{1}$
$\displaystyle=h_{k}\circ\big{(}(\iota_{k+1}\circ\pi_{k+1}^{\prime})\circ
p_{1}^{1}+\mathrm{id}\circ p_{2}^{1}\big{)}$
$\displaystyle=h_{k}\circ\varepsilon\,.$

Therefore, all squares in Diagram 6.11 commute. Its top side connected by
epimorphisms to its bottom side according to Diagram 6.9 and 6.10. Still using
the latter two diagrams, we can take kernels to obtain the commuting diagram

${0}$${\widetilde{D}_{k+2}}$${\mathrm{Ker}(\varepsilon)}$${\widetilde{D}_{k+1}}$${0}$${0}$${\widetilde{B}_{k+1}}$${\widetilde{C}_{k+1}}$${\widetilde{D}_{k+1}}$${0}$$\scriptstyle{\iota_{k+2}^{\prime\prime}}$$\scriptstyle{\widetilde{h}_{k+2}^{\ast}}$$\scriptstyle{\pi_{k+1}^{\prime\prime}}$$\scriptstyle{H}$$\scriptstyle{\mathrm{id}}$$\scriptstyle{\widetilde{f}_{k+1}}$$\scriptstyle{\widetilde{g}_{k+1}}$

where the rows are short exact sequences. Having seen that
$\mathrm{Ker}(\varepsilon)$ is projective, we complete the inductive step by
setting $\underline{D}_{k+1}:=\mathrm{Ker}(\varepsilon)$,
$\iota_{k+2}:=\iota_{k+2}^{\prime\prime}$,
$\pi_{k+1}:=\pi_{k+1}^{\prime\prime}$ and $h_{k+1}:=H$. $\square$

######  Theorem 6.13.

For any short exact sequence $0\rightarrow B\rightarrow C\rightarrow
D\rightarrow 0$ in $\mathcal{C}$ the square

${\mathcal{E}xt_{Res,\mathcal{C}}^{n}(A{,}\,D)}$${\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A{,}\,D)}$${\mathcal{E}xt_{Res,\mathcal{C}}^{n+1}(A{,}\,B)}$${\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n+1}(A{,}\,B)}$$\scriptstyle{\vartheta^{n}}$$\scriptstyle{\widetilde{\Delta}^{n}}$$\scriptstyle{\widehat{\Delta}^{n}}$$\scriptstyle{\vartheta^{n+1}}$

commutes for every $n\in\mathbb{Z}$. In particular,
$(\widehat{\mathcal{E}xt}_{\mathcal{C}}^{\bullet}(A,-),\widehat{\Delta}^{\bullet})$
forms a cohomological functor and
$\vartheta^{\bullet}:\mathcal{E}xt_{Res,\mathcal{C}}^{\bullet}(A,-)\rightarrow\widehat{\mathcal{E}xt}_{\mathcal{C}}^{\bullet}(A,-)$
an isomorphism of cohomological functors.

######  Proof.

Analogously to the proof of Proposition 6.10, it suffices by Definition 6.7 to
show that the diagram

${{\scriptscriptstyle\mathcal{E}xt_{\mathcal{C}}^{n+1+2k}(A{,}\,\widetilde{B}_{2k})}}$${{\scriptscriptstyle\mathcal{E}xt_{\mathcal{C}}^{n+2k}(A{,}\,\widetilde{D}_{2k})}}$${{\scriptscriptstyle\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n+1}(A{,}\,B)}}$${{\scriptscriptstyle\mathcal{E}xt_{\mathcal{C}}^{(n+1)+2k+2}(A{,}\,\widetilde{B}_{2k+2})}}$${{\scriptscriptstyle\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A{,}\,D)}}$${{\scriptscriptstyle\mathcal{E}xt_{\mathcal{C}}^{n+2k+2}(A{,}\,\widetilde{D}_{2k+2}}}$$\scriptstyle{\scriptscriptstyle\vartheta_{n+1}^{2k}(B)}$$\scriptstyle{\scriptscriptstyle\Delta^{(n+1)+2k+1}\circ\Delta^{n+1+2k}}$$\scriptstyle{\scriptscriptstyle\Delta^{n+2k}}$$\scriptstyle{\scriptscriptstyle\Delta^{n+2k+1}\circ\Delta^{n+2k}}$$\scriptstyle{\scriptscriptstyle\vartheta_{n+1}^{2k+2}(B)}$$\scriptstyle{\scriptscriptstyle\widehat{\Delta}^{n}}$$\scriptstyle{\scriptscriptstyle\vartheta_{n}^{2k}(D)}$$\scriptstyle{\scriptscriptstyle\Delta^{n+2k+2}}$$\scriptstyle{\scriptscriptstyle\vartheta_{n}^{2k+2)}(D)}$
(6.12)

is commutative in order to conclude that the square in the statement of the
theorem commutes. If we write $N:=n+2k$ and $K:=2k$, we can reformulate the
above prism in form of the diagram

${{\scriptscriptstyle\mathcal{E}xt_{\mathcal{C}}^{N}(A{,}\,\widetilde{D}_{K})}}$${{\scriptscriptstyle\mathcal{E}xt_{\mathcal{C}}^{N+1}(A{,}\,\widetilde{D}_{K+1})}}$${{\scriptscriptstyle\mathcal{E}xt_{\mathcal{C}}^{N+1}(A{,}\,\widetilde{B}_{K})}}$${{\scriptscriptstyle\mathcal{E}xt_{\mathcal{C}}^{N+1}(A{,}\,\widetilde{D}_{K+1})}}$${{\scriptscriptstyle\mathcal{E}xt_{\mathcal{C}}^{N+2}(A{,}\,\widetilde{D}_{K+2})}}$${{\scriptscriptstyle\mathcal{E}xt_{\mathcal{C}}^{N+2}(A{,}\,\widetilde{B}_{K+1})}}$${{\scriptscriptstyle\mathcal{E}xt_{\mathcal{C}}^{N+2}(A{,}\,\widetilde{D}_{K+2})}}$${{\scriptscriptstyle\mathcal{E}xt_{\mathcal{C}}^{N+3}(A{,}\,\widetilde{D}_{K+3})}}$${{\scriptscriptstyle\mathcal{E}xt_{\mathcal{C}}^{N+3}(A{,}\,\widetilde{B}_{K+2})}}$${{\scriptscriptstyle\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A{,}\,D)}}$${{\scriptscriptstyle\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n+1}(A{,}\,\widetilde{D}_{1})}}$${{\scriptscriptstyle\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n+1}(A{,}\,B)}}$$\scriptstyle{\scriptscriptstyle\Delta^{N}}$$\scriptstyle{\Delta^{N}}$$\scriptstyle{\scriptscriptstyle\Delta^{N}}$$\scriptstyle{\scriptscriptstyle\mathcal{E}xt_{\mathcal{C}}^{N+1}(A{,}\,\widetilde{h}_{K+1}^{\ast})}$$\scriptstyle{\scriptscriptstyle\Delta^{N+1}}$$\scriptstyle{\scriptscriptstyle\Delta^{N+1}}$$\scriptstyle{\scriptscriptstyle\vartheta_{n+1}^{K}(B)}$$\scriptstyle{\scriptscriptstyle-\Delta^{N+1}}$$\scriptstyle{-\Delta^{N+1}}$$\scriptstyle{\scriptscriptstyle\Delta^{N+1}}$$\scriptstyle{\scriptscriptstyle\mathcal{E}xt_{\mathcal{C}}^{N+2}(A{,}\,\widetilde{h}_{K+2}^{\ast})}$$\scriptstyle{\scriptscriptstyle\Delta^{N+2}}$$\scriptstyle{\scriptscriptstyle\Delta^{N+2}}$$\scriptstyle{\scriptscriptstyle\Delta^{N+2}}$$\scriptstyle{\Delta^{N+2}}$$\scriptstyle{\scriptscriptstyle\vartheta_{n}^{K+2}(D)}$$\scriptstyle{\scriptscriptstyle\mathcal{E}xt_{\mathcal{C}}^{N+2}(A{,}\,\widetilde{h}_{K+3}^{\ast})}$$\scriptstyle{\scriptscriptstyle\vartheta_{n+1}^{K+2}(\widetilde{D}_{1})}$$\scriptstyle{\scriptscriptstyle\vartheta_{n+1}^{K+2}(B)}$$\scriptstyle{\scriptscriptstyle\widehat{\Delta}^{n}}$$\scriptstyle{\widehat{\Delta}^{n}}$$\scriptstyle{\scriptscriptstyle\vartheta_{n}^{K}(D)}$$\scriptstyle{\scriptscriptstyle\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n+1}(A{,}\,h^{\ast})}$$\scriptstyle{\scriptscriptstyle\vartheta_{n+1}^{K}(\widetilde{D}_{1})}$
(6.13)

The left hand side of Diagram 6.12 corresponds to the top four squares of
Diagram 6.13 while the bottom right hand side of the former corresponds to the
bottom two squares of the latter. In Diagram 6.13, the top and bottom row
together with the morphisms connecting them directly retrieve the top right
side of Diagram 6.12.

Let us explain how to construct Diagram 6.13 by making use of Proposition 6.12
and its proof. We take the morphism $h^{\ast}:\widetilde{D}_{1}\rightarrow B$
as in that proposition, meaning that any term $\widetilde{h}_{2k+1}^{\ast}$
originates from a chain map $h_{\bullet+1}^{\ast}:D_{\bullet}^{+1}\rightarrow
B_{\bullet}$. By Definition 6.5, the connecting homomorphism
$\widehat{\Delta}^{n}:\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A,D)\rightarrow\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n+1}(A,B)$
associated to the short exact sequence $0\rightarrow B\rightarrow C\rightarrow
D\rightarrow 0$ is rendered in the very bottom row. The top left square arises
from Diagram 4.10 and 6.9. More specifically, its top connecting homomorphism
is associated to the short exact sequence at the bottom of Diagram 6.9, its
right hand connecting homomorphism to the short exact sequence at the left
hand side of the latter diagram, its bottom one to the top short exact
sequence and its left hand one to the right hand short exact sequence. Because
the connecting homomorphism at the right hand side of the top right square is
associated to the short exact sequences at the left hand side of Diagram 6.10,
the top two squares commute. By Proposition 6.12, the homomorphism

$\mathcal{E}xt_{\mathcal{C}}^{n+2k}(A,\widetilde{D}_{2k})\xrightarrow{\Delta^{n+2k}}\mathcal{E}xt_{\mathcal{C}}^{n+2k+1}(A,\widetilde{D}_{2k+1})\xrightarrow{\mathcal{E}xt_{\mathcal{C}}^{n+2k+1}(A,\widetilde{h}_{2k+1}^{\ast})}\mathcal{E}xt_{\mathcal{C}}^{n+2k+1}(A,\widetilde{B}_{2k})$

at the very top of the diagram is the connecting homomorphism $\Delta^{n+2k}$
associated to the short exact sequence
$0\rightarrow\widetilde{B}_{k}\rightarrow\widetilde{C}_{k}\rightarrow\widetilde{D}_{k}\rightarrow
0$. This does not only imply that the entire top third of the diagram
commutes, but also that the middle third does so too. Due to Proposition 6.10
the entire right hand side of the diagram commutes. Regarding the left hand
side, since we are in the same situation as in the proof of the latter
proposition, we are left to show that the diagram

${\mathcal{E}xt_{\mathcal{C}}^{n+2k}(A{,}\,\widetilde{D}_{2k})}$${\mathcal{E}xt_{\mathcal{C}}^{n+1+2k}(A{,}\,\widetilde{D}_{2k+1})}$${\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A{,}\,D)}$${\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n+1}(A{,}\,\widetilde{D}_{1})}$$\scriptstyle{\Delta^{n+2k}}$$\scriptstyle{\vartheta_{n}^{2k}(D)}$$\scriptstyle{\vartheta_{n+1}^{2k}(\widetilde{D}_{1})}$$\scriptstyle{\widehat{\Delta}^{n}}$

commutes. If
$\varphi_{\bullet+n+2k}+\mathrm{Hom}_{\mathrm{Ch}(\mathcal{C})}(A[n+2k],D_{\bullet}^{+2k})$
is a an element in $\mathcal{E}xt_{\mathcal{C}}^{n+2k}(A,\widetilde{D}_{2k})$.
Then

$\begin{cases}(-1)^{m-1+n}\varphi_{m+n}:A_{m+n}\rightarrow D_{m}&\text{if
}m\geq 2k+1\\\ \;\;0&\text{if }m\leq 2k\end{cases}$

is an almost chain map representative of the image under the homomorphism
$\widehat{\Delta}^{n}\circ\vartheta_{n}^{2k}(D)$ while

$\begin{cases}(-1)^{m-2k-1+n}\varphi_{m+n}:A_{m+n}\rightarrow D_{m}&\text{if
}m\geq 2k+1\\\ \;\;0&\text{if }m\leq 2k\end{cases}$

is a representative of the image under
$\vartheta_{n+1}^{2k}\circ\Delta^{n+2k}$. Thus, Diagram 6.12 and 6.13 commute
as well as the square in the statement of the theorem.

By Definition 6.7,
$(\mathcal{E}xt_{Res,\mathcal{C}}^{\bullet}(A,-),\widetilde{\Delta}^{\bullet})$
forms a cohomological functor and by Lemma 6.9 and Proposition 6.10,
$\vartheta^{n}$ is a natural isomorphism for any $n\in\mathbb{Z}$. As for
Theorem 5.10 and Lemma 6.6, we can reiterate the proof of Theorem 4.5 to
demonstrate that
$(\widehat{\mathcal{E}xt}_{\mathcal{C}}^{\bullet}(A,-),\widehat{\Delta}^{\bullet})$
is a cohomological functor and that
$\vartheta^{\bullet}:\mathcal{E}xt_{Res,\mathcal{C}}^{\bullet}(A,-)\rightarrow\widehat{\mathcal{E}xt}_{\mathcal{C}}^{\bullet}(A,-)$
is an isomorphism of cohomological functors. $\square$

######  Remark 6.14.

Given any short exact sequence $0\rightarrow B\rightarrow C\rightarrow
D\rightarrow 0$ in $\mathcal{C}$, we conclude by means of Theorem 6.13 that
every associated connecting homomorphism
$\widehat{\Delta}^{n}:\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A,D)\rightarrow\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n+1}(A,B)$
is well defined in the sense that it does not depend on a choice of a morphism
$h^{\ast}:\widetilde{D}_{1}\rightarrow B$ as it occurs in Definition 6.5. We
have not been able to prove this directly.

######  Corollary 6.15.

We can extend the natural isomorphisms $\sigma^{n}$ from Corollary 6.11 to
isomorphism of cohomological functors

$\sigma^{\bullet}=\vartheta^{\bullet}\circ\zeta^{\bullet}:\big{(}\widehat{\mathrm{Ext}}_{\mathcal{C}}^{n}(A,-),\widehat{\delta}^{\bullet}\big{)}\rightarrow\big{(}\mathcal{E}xt_{Res,\mathcal{C}}^{n}(A,-),\widetilde{\Delta}^{\bullet}\big{)}\rightarrow\big{(}\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A,-),\widehat{\Delta}^{\bullet}\big{)}\,.$

In [19, p. 21–22], S. Guo and L. Liang establish an isomorphism
$\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A,B)\rightarrow
BC_{\mathcal{C}}^{n}(A,B)$ from the cohomology groups of the hypercohomology
construction to the ones of the naïve construction for every $n\in\mathbb{Z}$.
As it is used in the construction of Yoneda products in [16, Theorem 6.6], we
present it below and prove that it is an isomorphism of cohomological
functors.

######  Definition 6.16.

([19, p. 21–22]) For any $A,B\in\mathrm{obj}(\mathcal{C})$ and
$n\in\mathbb{Z}$ let the map

$\rho^{n}:\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A,B)\rightarrow
BC_{\mathcal{C}}^{n}(A,B)$

be given as follows. If
$x=\varphi_{\bullet+n}+\widehat{\mathrm{Null}}_{\mathrm{Ch}(\mathcal{C})}(A[n]_{\bullet},B_{\bullet})$
is an element in $\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A,B)$, then there
exists $K\in\mathbb{N}_{0}$ such that $(\varphi_{m+n}:A_{m+n}\rightarrow
B_{m})_{m\geq 2K}$ forms a chain map. Denote by
$\widetilde{\varphi}_{2K+n}:\widetilde{A}_{2K+n}\rightarrow\widetilde{B}_{2K}$
the morphism induced between the corresponding syzygies. We set $\rho^{n}(x)$
to be the element in $BC_{\mathcal{C}}^{n}(A,B)$ to which the element
$\widetilde{\varphi}_{2K}+\mathcal{P}_{\mathcal{C}}(\widetilde{A}_{2K+n},\widetilde{B}_{2K})$
in $[\widetilde{A}_{2K+n},B_{2K}]_{\mathcal{C}}$ is mapped by the
corresponding homomorphism going to the direct limit.

######  Lemma 6.17.

The maps $\rho^{n}$ form an isomorphism of cohomological functors such that

$\rho^{\bullet}=\beta^{\bullet}\circ(\sigma^{\bullet})^{-1}:(\widehat{\mathcal{E}xt}_{\mathcal{C}}^{\bullet}(A,-),\widehat{\Delta}^{\bullet})\rightarrow(\mathrm{Ext}_{Res,\mathcal{C}}^{n}(A,-),\widetilde{\delta}^{\bullet})\rightarrow(BC_{\mathcal{C}}^{\bullet}(A,-),\tau^{\bullet})$

where $\beta^{\bullet}$ is taken from Theorem 5.10.

######  Proof.

Since $\beta^{\bullet}$ and $\sigma^{\bullet}$ are already isomorphisms of
cohomological functors, it suffices to prove that
$\rho^{n}=\beta^{n}\circ(\sigma^{n})^{-1}$ for every $n\in\mathbb{Z}$. Let
$\varphi_{\bullet+n}+\widehat{\mathrm{Null}}_{\mathrm{Ch}(\mathcal{C})}(A[n]_{\bullet},B_{\bullet})$
be an element in $\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A,B)$. If
$K\in\mathbb{N}_{0}$ is such that $(\varphi_{m+n})_{m\geq 2K}$ is chain map,
then for any $k\geq K$ we define
$\varphi_{\bullet+n+2k}^{\\{2k\\}}:A[n+2k]_{\bullet}\rightarrow
B_{\bullet}^{+2k}$ by setting

$\varphi_{m+n+2k}^{\\{2k\\}}:=\begin{cases}\varphi_{m+n+2k}:A_{m+n+2k}\rightarrow
B_{m+2k}&\text{if }m\geq 0\\\ \,\,0&\text{if }m<0\end{cases}$

By Corollary 6.11,
$(\sigma^{n})^{-1}=(\zeta^{n})^{-1}\circ(\vartheta^{n})^{-1}$. As
$\mathrm{Ext}_{Res,\mathcal{C}}^{n}(A,B)$ is formed as a direct limit of
abelian groups, we introduce the following notation. Let
$(A_{i})_{i\in\mathbb{N}}$ be a direct system of abelian groups with
homomorphisms given by $a_{i,j}:A_{i}\rightarrow A_{j}$ for $i\leq j$. We
write $(x_{k})_{k\geq K}$ if $x_{k}\in A_{k}$ and $a_{i,j}(x_{i})=x_{j}$ for
any $j\geq i\geq K$. This results in an element in
$\varinjlim_{i\in\mathbb{N}}A_{i}$ that we identify with $(x_{k})_{k\geq K}$.
Then we see that

$(\vartheta^{n})^{-1}\big{(}\varphi_{\bullet+n}+\widehat{\mathrm{Null}}_{\mathrm{Ch}(\mathcal{C})}(A[n]_{\bullet},B_{\bullet})\big{)}=\big{(}\varphi_{\bullet+n+2k}^{\\{2k\\}}+\mathrm{Null}_{\mathrm{Ch}(\mathcal{C})}(A[n+2k]_{\bullet},B_{\bullet}^{+2k})\big{)}_{k\geq
K}\,.$

Denote by $a_{m}:A_{m}\rightarrow A_{m-1}$ the boundary maps in the projective
resolution $A_{\bullet}$ and by $\pi_{m}:B_{m}\rightarrow\widetilde{B}_{m}$
the morphisms to the $m^{\text{th}}$ syzygy. According to the construction of
$\zeta_{m}$ in the proof of Lemma 6.6 and of $\sigma^{n}$ in Definition 6.7,
we infer that

$(\sigma^{n})^{-1}\big{(}\varphi_{\bullet+n}+\widehat{\mathrm{Null}}_{\mathrm{Ch}(\mathcal{C})}(A[n]_{\bullet},B_{\bullet})\big{)}=\Big{(}\pi_{n+2k}\circ\varphi_{n+2k}+\mathrm{Im}\big{(}\mathrm{Hom}_{\mathcal{C}}(a_{n+2k},\widetilde{B}_{2k})\big{)}\Big{)}_{k\geq
K}\,.$

We deduce from the construction of $\beta^{n}$ in Lemma 5.7 that

$\beta^{n}\circ(\sigma^{n})^{-1}\big{(}\varphi_{\bullet+n}+\widehat{\mathrm{Null}}_{\mathrm{Ch}(\mathcal{C})}(A[n]_{\bullet},B_{\bullet})\big{)}=\big{(}\alpha_{n+2k}(\widetilde{B}_{2k})(\varphi_{n+2k})+[\widetilde{A}_{2k+n},\widetilde{B}_{2k}]_{\mathcal{C}}\big{)}_{k\geq
K}$

where $\alpha_{n+2k}(\widetilde{B}_{2k})$ denotes the homomorphism from
Diagram 5.4. Because

$\beta^{n}\circ(\sigma^{n})^{-1}\big{(}\varphi_{\bullet+n}+\widehat{\mathrm{Null}}_{\mathrm{Ch}(\mathcal{C})}(A[n]_{\bullet},B_{\bullet})\big{)}=\big{(}\widetilde{\varphi}_{n+2k}+[\widetilde{A}_{2k+n},\widetilde{B}_{2k}]_{\mathcal{C}}\big{)}_{k\geq
K}$

by Diagram 6.1, we conclude that $\rho^{n}=\beta^{n}\circ(\sigma^{n})^{-1}$.
$\square$

We close our account on constructions of Mislin completions by posing a
question. In Notation 6.1 we have remarked that any completed Ext-group
$\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A,B)$ can be defined as the
$n^{\text{th}}$ cohomology group of the Vogel complex
$\mathrm{Vog}_{\mathcal{C}}^{n}(A_{\bullet},B_{\bullet})_{\bullet}$ whence one
obtains

######  Lemma 6.18.

([19, Proposition 4.8]) If $0\rightarrow B\rightarrow C\rightarrow
D\rightarrow 0$ is a short exact sequence in $\mathcal{C}$, one can lift it to
a short exact sequence $0\rightarrow B_{\bullet}\rightarrow
C_{\bullet}\rightarrow D_{\bullet}\rightarrow 0$ of chain complexes by the
Horseshoe Lemma [42, p. 37]. Then, by the Snake Lemma [42, p. 11–12], there
are associated connecting homomorphisms
$\overline{\delta}^{n}:\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(A,D)\rightarrow\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n+1}(A,B)$
and
$\overline{\delta}^{n}:\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n}(B,A)\rightarrow\widehat{\mathcal{E}xt}_{\mathcal{C}}^{n+1}(D,A)$.
In particular,
$\big{(}\widehat{\mathcal{E}xt}_{\mathcal{C}}^{\bullet}(A,-),\overline{\delta}^{\bullet}\big{)}$
and
$\big{(}\widehat{\mathcal{E}xt}_{\mathcal{C}}^{\bullet}(-,A),\overline{\delta}^{\bullet}\big{)}$
form cohomological functors.

######  Question 6.19.

Does the connecting homomorphism $\widehat{\Delta}^{\bullet}$ constructed in
Definition 6.5 agree with $\overline{\delta}^{\bullet}$? More specifically, is
$\big{(}\widehat{\mathcal{E}xt}_{\mathcal{C}}^{\bullet}(A,-),\overline{\delta}^{\bullet}\big{)}$
a Mislin completion of the unenriched Ext-functors
$\big{(}\mathrm{Ext}_{\mathcal{C}}^{\bullet}(A,-),\delta^{\bullet}\big{)}$?

##  7 Complete cohomology in condensed mathematics

As we implement complete cohomology in condensed mathematics, we provide
hereby more detail on the latter. First, we define condensed sets following
D. Clausen and P. Scholze’s account [9, Lecture I and II]. After giving a
definition that is sheaf-theoretic in nature, we provide a more down-to-earth
definition. A finite collection of (continuous) maps
$\\{f_{i}:S_{i}\rightarrow S\\}_{i=1}^{n}$ in the category $\mathbf{Pro}$ of
profinite spaces is called a covering if the induced map
$\bigsqcup_{i=1}^{n}S_{i}\rightarrow S$ is surjective. Although this turns
$\mathbf{Pro}$ into a Grothendieck site, it is not well suited for sheaf
theoretic purposes as it is not essentially small. A cardinal number $\kappa$
is called a strong limit cardinal if for every $\lambda<\kappa$ we also have
$2^{\lambda}<\kappa$. Then we define the site of $\kappa$-small profinite
spaces $\mathbf{Pro}_{\kappa}$ for an uncountable strong limit cardinal
$\kappa$ as the full subcategory of $\mathbf{Pro}$ consisting of all profinite
spaces with less than $\kappa$ clopen subsets and restrict the Grothendieck
topology to it. Note that the Grothendieck site $\mathbf{Pro}_{\kappa}$ is
essentially small. Then a $\kappa$-condensed set is a sheaf of sets on
$\mathbf{Pro}_{\kappa}$. In more down-to-earth terms, a (contravariant)
functor

$X:\mathbf{Pro}_{\kappa}^{\mathrm{op}}\rightarrow\mathbf{Set}$

is a $\kappa$-condensed set if the following two conditions are satisfied.

  1. 1.

For every finite collection $\\{S_{i}\\}_{i=1}^{n}$ of objects in the category
$\mathbf{Pro}_{\kappa}$ there is a bijection
$X(\bigsqcup_{i=1}^{n}S_{i})\rightarrow\prod_{i=1}^{n}X(S_{i})$.

  2. 2.

Let $f:T\rightarrow S$ be a (continuous) surjection in $\mathbf{Pro}_{\kappa}$
and denote the projections from the fibre product by
$p_{1},p_{2}:T\times_{S}T\rightarrow T$. Then the function
$X(f):X(S)\rightarrow X(T)$ is injective with image

$\mathrm{Im}(X(f))=\\{x\in X(T)\mid X(p_{1})(x)=X(p_{2})(x)\in
X(T\times_{S}T)\\}\,.$

One can think of $X$ as encoding a topological space and of $X(S)$ as the
continuous maps “$S\rightarrow X$” from a $\kappa$-profinite space. If
$\kappa<\kappa^{\prime}$ is another strong limit cardinal, then there is a
fully faithful functor

$\mathrm{Cond}_{\kappa}(\mathbf{Set})\rightarrow\mathrm{Cond}_{\kappa^{\prime}}(\mathbf{Set})$

due to our choice of the cardinal numbers. This means that one can extend any
$\kappa$-condensed set to a $\kappa^{\prime}$-condensed set. If $K$ denotes
the class of uncountable strong limit cardinals, then the category of
condensed sets is defined as the colimit category

$\mathrm{Cond}(\mathbf{Set}):=\varinjlim_{\kappa\in
K}\mathrm{Cond}_{\kappa}(\mathbf{Set})\,.$

Hence, a condensed set is an equivalence class of $\kappa$-condensed sets.
According to [35, p. 15/16 and Proposition 1.7] we can establish the following
relation with topological spaces. If $T1\text{-}\mathbf{Top}$ denotes the
category of $T1$ topological spaces, then there is a well-defined faithful
functor $T1\text{-}\mathbf{Top}\rightarrow\mathrm{Cond}(\mathbf{Set})$ mapping
any $Y$ to its so-called condensate

$\underline{Y}:\mathbf{Pro}_{\kappa}^{\mathrm{op}}\rightarrow\mathbf{Set},\;S\mapsto\\{f:S\rightarrow
Y\text{ continuous}\\}$

where $\kappa$ is chosen to be a sufficiently large strong limit cardinal.
This functor is fully faithful when restricted to $T1$ compactly generated
topological spaces. It follows from the proofs of Proposition 2.9 and
Proposition 2.15 in [35] that the functor does not depend on $\kappa$. Namely,
for every strong limit cardinal $\kappa<\kappa^{\prime}$ and
$\kappa^{\prime}$-profinite space $S=\varprojlim_{i\in I}S_{i}$ written as a
$\kappa$-inverse limit of $\kappa$-profinite spaces $S_{i}$, the function
$\varinjlim_{i\in I}\underline{Y}(S_{i})\rightarrow\underline{Y}(S)$ is a
bijection. Although $\mathrm{Cond}(\mathbf{Set})$ does not form the category
of sheaves over any site [35, Remark 2.12], it nevertheless inherits good
sheaf-theoretic properties from the categories
$\mathrm{Cond}_{\kappa}(\mathbf{Set})$ [9, p. 13].

Condensed groups (resp. rings, modules etc.) are defined analogously [35, p.
7] and condensates of $T1$ topological groups are condensed groups (resp.
rings, modules etc) [35, p. 8]. In the category of condensed abelian groups
$\mathrm{Cond}(\mathbf{Ab})$ all limits and colimits exist where arbitrary
products, arbitrary direct sums and direct limits are exact [35, Theorem
1.10]. According to [35, p. 12], $\mathrm{Cond}(\mathbf{Ab})$ has enough
projectives. Following the source, these assertions also hold for the category
of condensed $\mathcal{R}$-modules $\mathrm{Cond}(Mod(\mathcal{R}))$ where
$\mathcal{R}$ is a condensed ring. If $S$ is a projective object in
$\mathbf{Pro}$, then one can form the free condensed $\mathcal{R}$-module
$\mathcal{R}[\underline{S}]$ as the sheafification of the presheaf of
$\mathcal{R}$-modules

$\mathbf{Pro}_{\kappa}^{\mathrm{op}}\rightarrow\mathbf{Ab},T\mapsto\mathcal{R}(T)[\underline{S}(T)]$

where $\mathcal{R}(T)[\underline{S}(T)]$ denotes the free
$\mathcal{R}(T)$-module over the set $\underline{S}(T)$ and $\kappa$ is a
sufficiently large strong limit cardinal [35, p. 12]. Since the Grothendieck
site $\mathbf{Pro}_{\kappa}$ is essentially small, sheafification exists and
again by the proof of [35, Proposition 2.9], the construction of
$\mathcal{R}[\underline{S}]$ is independent of $\kappa$. Moreover, the
condensed $\mathcal{R}$-modules $\mathcal{R}[\underline{S}]$ are compact
projective generators, meaning that for every condensed $\mathcal{R}$-module
$A$ there is a collection of projective profinite spaces $S_{i}$ such that
there is an epimorphism $\bigoplus_{i\in
I}\mathcal{R}[\underline{S_{i}}]\rightarrow A$ [35, p. 12]. For condensed
$\mathcal{R}$-modules $A$, $B$ one does not only have the “usual” Hom-set
$\mathrm{Hom}_{\mathrm{Cond}(Mod(\mathcal{R}))}(A,B)\in\mathrm{obj}(\mathbf{Ab})$,
but also an internal Hom-set which is constructed in [35, p. 13]. Namely, for
any $C\in\mathrm{obj}(\mathrm{Cond}(Mod(\mathcal{R})))$ one can form the
tensor product $C\otimes_{\mathcal{R}}A\in\mathrm{Cond}(Mod(\mathcal{R}))$ as
the sheafification of

$\mathbf{Pro}_{\kappa}^{\mathrm{op}}\rightarrow\mathbf{Ab},T\mapsto
C(T)\otimes_{\mathcal{R}(T)}A(T)\,.$

Then define the internal Hom-set
$\underline{\mathrm{Hom}}_{\mathrm{Cond}(Mod(\mathcal{R}))}(A,B)\in\mathrm{obj}(\mathrm{Cond}(\mathbf{Ab}))$
by

$\mathbf{Pro}_{\kappa}^{\mathrm{op}}\rightarrow\mathbf{Ab},T\mapsto\mathrm{Hom}_{\mathrm{Cond}(Mod(\mathcal{R}))}(A\otimes_{\mathcal{R}}\mathcal{R}[\underline{T}],B)\,.$

By definition, it satisfies the adjunction

$\mathrm{Hom}_{\mathrm{Cond}(Mod(\mathcal{R}))}(C,\underline{\mathrm{Hom}}_{\mathrm{Cond}(Mod(\mathcal{R}))}(A,B))\cong\mathrm{Hom}_{\mathrm{Cond}(Mod(\mathcal{R}))}(C\otimes_{\mathcal{R}}A,B)\,.$

Because the category $\mathrm{Cond}(Mod(\mathcal{R}))$ has enough projectives,
we can define the unenriched Ext-functors

$\mathrm{Ext}_{\mathrm{Cond}(Mod(\mathcal{R}))}^{\bullet}(A,-):\mathrm{Cond}(Mod(\mathcal{R}))\rightarrow\mathbf{Ab}$

and the internal Ext-functors

$\underline{\mathrm{Ext}}_{\mathrm{Cond}(Mod(\mathcal{R}))}^{\bullet}(A,-):\mathrm{Cond}(Mod(\mathcal{R}))\rightarrow\mathrm{Cond}(\mathbf{Ab})$

to be the derived functors of
$\mathrm{Hom}_{\mathrm{Cond}(Mod(\mathcal{R}))}(A,-)$ and
$\underline{\mathrm{Hom}}_{\mathrm{Cond}(Mod(\mathcal{R}))}(A,-)$. This can be
used to define cohomology of a condensed group $\mathcal{G}$. Following [2, p.
2–3] one can form the condensed group ring $\mathcal{R}[\mathcal{G}]$ such
that the category $\mathrm{Cond}(Mod(\mathcal{R}[\mathcal{G}]))$ is equivalent
to the subcategory of $\mathrm{Cond}(\mathcal{R})$ of all
$\mathcal{R}$-modules $M$ with a $\mathcal{G}$-action $\mathcal{G}\times
M\rightarrow M$. By [2, p. 5/8], one can define condensed unenriched group
cohomology $H_{\mathcal{R}}^{\bullet}(\mathcal{G},-)$ and the condensed
internal group cohomology
$\underline{H}_{\mathcal{R}}^{\bullet}(\mathcal{G},-)$ as the Ext-functors
$\mathrm{Ext}_{\mathrm{Cond}(Mod(\mathcal{R}[\mathcal{G}]))}^{\bullet}(\mathcal{R},-)$
and
$\underline{\mathrm{Ext}}_{\mathrm{Cond}(Mod(\mathcal{R}[\mathcal{G}]))}^{\bullet}(\mathcal{R},-)$.
By the above, we conclude

######  Theorem 7.1.

Let $\mathcal{R}$ be a condensed ring and $A$ a condensed
$\mathcal{R}$-module. Then there are completed condensed unenriched Ext-
functors
$\widehat{\mathrm{Ext}}_{\mathrm{Cond}(Mod(\mathcal{R}))}^{\bullet}(A,-)$ and
completed condensed internal Ext-functors
$\widehat{\underline{\mathrm{Ext}}}_{\mathrm{Cond}(Mod(\mathcal{R}))}^{\bullet}(A,-)$.
If $\mathcal{G}$ is a condensed group, then complete condensed unenriched
group cohomology $\widehat{H}_{\mathcal{R}}^{\bullet}(\mathcal{G},-)$ and
complete condensed internal group cohomology
$\widehat{\underline{H}}_{\mathcal{R}}^{\bullet}(\mathcal{G},-)$ can be
defined.

######  Remark 7.2.

Because both
$\widehat{\underline{\mathrm{Ext}}}_{\mathrm{Cond}(Mod(\mathcal{R}))}^{\bullet}(A,-)$
and $\widehat{\underline{H}}_{\mathcal{R}}^{\bullet}(\mathcal{G},-)$ have as
codomain category $\mathrm{Cond}(\mathbf{Ab})$, they are an example of
completed Ext-functors that are not covered under the previous frameworks such
as the one by A. Beligiannis and I. Reiten in [4], the one by S. Guo and
L. Liang’s in [19] and the one by J. Hu et al. in [22].

Having discussed condensed modules, let us consider solid modules. We have
already mentioned in Section 1 that one can think of these as being a form of
completed condensed modules where a condensed ring $\mathcal{A}$ is called
analytic if it admits a choice of solid modules. More formally, let
$\mathbf{ProjPro}$ denote the full subcategory of $\mathbf{Pro}$ consisting of
projective objects. Then an analytic ring $\mathcal{A}$ comes with a
particular functor

$c_{\mathcal{A}}:\mathbf{ProjPro}\rightarrow\mathrm{Cond}(Mod(\mathcal{A})),\;S\mapsto\mathcal{A}[S]_{\blacksquare}\,.$

and a condensed $\mathcal{A}$-module homomorphism
$C_{\mathcal{A}}(S):\mathcal{A}[\underline{S}]\rightarrow\mathcal{A}[\underline{S}]_{\blacksquare}$
for every projective profinite space $S$ [35, p. 44]. One can think of
$c_{\mathcal{A}}$ as assigning to every projective profinite space $S$ a
canonical choice of free solid $\mathcal{A}$-module
$\mathcal{A}[S]_{\blacksquare}$. According to [35, Proposition 7.5], analytic
rings have the following properties. The full subcategory
$\mathrm{Solid}(\mathcal{A})$ of $\mathrm{Cond}(Mod(\mathcal{A}))$ with
objects $M$ such that

$\mathrm{Hom}_{\mathrm{Cond}(Mod(\mathcal{A}))}(C_{\mathcal{A}}(S),M):\mathrm{Hom}_{\mathrm{Cond}(Mod(\mathcal{A}))}(\mathcal{A}[S]_{\blacksquare},M)\rightarrow
M(S)$

is an isomorphism for all $S\in\mathbf{ProjPro}$ is an abelian category stable
under limits, colimits and extensions. The objects
$\mathcal{A}[S]_{\blacksquare}$ are compact projective generators in
$\mathrm{Solid}(\mathcal{A})$. Moreover, there is an extension of the
morphisms
$C_{\mathcal{A}}(S):\mathcal{A}[\underline{S}]\rightarrow\mathcal{A}[\underline{S}]_{\blacksquare}$
to a functor
$\mathrm{Cond}(Mod(\mathcal{A}))\rightarrow\mathrm{Solid}(\mathcal{A})$ left
adjoint to the inclusion of categories that we call the solidification
functor. Lastly, if $\mathcal{A}$ is commutative, then there exists a unique
symmetric monoidal tensor product on $\mathrm{Solid}(\mathcal{A})$ rendering
the solidification functor symmetric monoidal. In particular, since the
category $\mathrm{Solid}(\mathcal{A})$ has enough projectives, we can define
the Ext functors

$\mathrm{Ext}_{\mathrm{Solid}(\mathcal{A})}^{\bullet}(M,-):\mathrm{Solid}(\mathcal{A})\rightarrow\mathbf{Ab}$

as the derived functors of $\mathrm{Hom}_{\mathrm{Solid}(\mathcal{A})}(M,-)$.

######  Lemma 7.3.

Let $\mathcal{A}$ be an analytic ring and $M$ a solid $\mathcal{A}$-module.
Then there are completed solid Ext-functors
$\widehat{\mathrm{Ext}}_{\mathrm{Solid}(\mathcal{A})}^{\bullet}(M,-)$.

######  Example 7.4 ([2, p. 2]).

Let $S=\varprojlim_{i\in I}$ be a profinite space and define the condensed
$\underline{\mathbb{Z}}$-module
$\mathbb{Z}[S]^{\blacksquare}:=\varprojlim_{i\in
I}\underline{\mathbb{Z}}[\underline{S_{i}}]$. Then the condensed ring
$\underline{\mathbb{Z}}$ together with the assignment
$S\mapsto\mathbb{Z}[S]^{\blacksquare}$ for every projective profinite space
$S$ forms an analytic ring. By extension, we denote by
$(-)^{\blacksquare}:\mathrm{Cond}(\mathbf{Ab})\rightarrow\mathrm{Solid}(\underline{\mathbb{Z}})$
the solidification functor. If $\Lambda$ denotes one of the profinite rings
$\mathbb{Z}_{p}$, $\mathbb{Z}/p\mathbb{Z}$ or $\mathbb{Z}/p^{n}\mathbb{Z}$,
then also
$(\underline{\Lambda},S\mapsto(\underline{\Lambda}[\underline{S}])^{\blacksquare})$
is an analytic ring.

######  Lemma 7.5.

Let $G$ be a profinite group and $(L,S\mapsto L[S]^{\blacksquare})$ an
analytic ring where $(-)^{\blacksquare}$ denotes the solidification functor of
the analytic ring $\underline{\mathbb{Z}}$ from Example 7.4. Then the group
ring $(L[\underline{G}],S\mapsto L[\underline{G}][S]^{\blacksquare})$ is also
an analytic ring and one can thus define complete solid cohomology
$\widehat{H}_{\mathrm{Solid}(L)}^{\bullet}(\underline{G},-):\mathrm{Solid}(L[\underline{G}])\rightarrow\mathbf{Ab}$
and complete solid internal cohomology
$\widehat{\underline{H}}_{\mathrm{Solid}(L)}^{\bullet}(\underline{G},-):\mathrm{Solid}(L[\underline{G}])\rightarrow\mathrm{Solid}(\underline{\mathbb{Z}})$.

######  Proof.

It follows from Corollary 5.5 and Corollary 6.1 of [35] that
$L[S]^{\blacksquare}$ is a projective solid $L$-module for every profinite
(but not necessarily projective) space $S$. Then, by [2, Lemma 1.3], the group
ring $(L[\underline{G}],S\mapsto L[\underline{G}][S]^{\blacksquare})$ is
analytic. It follows from [2, p. 3–4] that the category of solid
$L[\underline{G}]$ is equivalent to the category of solid $L$-modules together
with an action $\underline{G}\times M\rightarrow M$ in the category of solid
$L$-modules. In particular, there is an unenriched Hom-functor
$\mathrm{Hom}_{\mathrm{Solid}(L[\underline{G}])}(L,-)$ from which we can form
the solid group cohomology $H_{\mathrm{Solid}(L)}^{\bullet}(\underline{G},-)$
[2, p. 5] and thus the corresponding complete solid cohomology. On the other
hand, the internal Hom-functor of condensed $L[\underline{G}]$-modules
restricted onto the full subcategory $\mathrm{Solid}(L[\underline{G}])$

$\mathrm{Solid}(L[\underline{G}])\rightarrow\mathrm{Cond}(\mathbf{Ab}),M\rightarrow\underline{\mathrm{Hom}}_{\mathrm{Cond}(L[\underline{G}])}(L,M)$

maps into solid $\underline{\mathbb{Z}}$-modules according to [2, p. 8].
Therefore, we can also form the complete solid internal cohomology. $\square$

## Acknowledgements

A special thanks goes to Peter H. Kropholler for sharing his expertise with
me. He has discussed examples of complete cohomology groups of discrete groups
with me and pointed out his joint paper with Jonathan Cornick “On Complete
Resolutions”. In particular, he introduced me to condensed mathematics. I
would also like to express my gratitude to Emma Brink for her great help in
condensed mathematics.

I would like to acknowledge Alejandro Adem, Jon F. Carlson, Joel Friedman,
Kalle Karu, Zinovy Reichstein and Ben Williams who assessed my PhD thesis and
whose invaluable suggestions have led to an improvement of the introduction.
Moreover, I am thankful to Andrew Fisher for pointing out the paper “Complete
Homology over Associative Rings” by Olgur Celikbas et al. Lastly, I am
grateful to Ged Corob Cook for insightful discussions.

## References

  * [1] A. Adem and R. J. Milgram.  Cohomology of Finite Groups, volume 309 of Grundlehren der mathematischen Wissenschaften / A Series of Comprehensive Studies in Mathematics.  Springer-Verlag Berlin Heidelberg New York, 2nd edition, 2004.
  * [2] J. Anschütz and A.-C. Le Bras.  Solid Group Cohomology.  https://www.math.uni-bonn.de/people/ja/homology_profinite.pdf, September 2020.  This website has been consulted on 15 July 2023.
  * [3] J. Asadollahi and S. Salarian.  Cohomology Theories for Complexes.  Journal of Pure and Applied Algebra, 210(3):771–787, 2007.
  * [4] A. Beligiannis and I. Reiten.  Homological and Homotopical Aspects of Torsion Theories, volume 188 of Memoirs of the American Mathematical Society.  American Mathematical Society, 2007.
  * [5] D. J. Benson and J. F. Carlson.  Products in Negative Cohomology.  Journal of Pure and Applied Algebra, 82(2):107–129, 1992.
  * [6] K. S. Brown.  Cohomology of Groups, volume 87 of Graduate Texts in Mathematics.  Springer-Verlag New York, Inc., 1982.
  * [7] H. Cartan and S. Eilenberg.  Homological Algebra, volume 19 of Princeton Mathematical Series.  Princeton University Press, 1956.
  * [8] O. Celikbas, L. W. Christensen, L. Liang, and G. Piepmeyer.  Complete Homology over Associative Rings.  Israel Journal of Mathematics, 221(1):1–24, 2017.
  * [9] D. Clausen and P. Scholze.  Condensed Mathematics and Complex Geometry.  https://people.mpim-bonn.mpg.de/scholze/Complex.pdf, April 2022\.  This website has been consulted on 8 April 2023.
  * [10] J. Cornick and P. H. Kropholler.  On Complete Resolutions.  Topology and its Applications, 78(3):235––250, 1997.
  * [11] F. Dembegioti.  On the Zeroeth Complete Cohomology.  Journal of Pure and Applied Algebra, 203(1–3):119–132, 2005.
  * [12] F. Dembegioti.  On the Zeroeth Complete Cohomology of Certain Polycyclic Groups.  Communications in Algebra, 36(5):1927–1941, 2008.
  * [13] I. Emmanouil and P. Manousaki.  On the Stable Homology of Modules.  Journal of Pure and Applied Algebra, 221(9):2198–2219, 2017.
  * [14] F. T. Farrell.  An Extension of Tate Cohomology to a Class of Infinite Groups.  Journal of Pure and Applied Algebra, 10(2):153–161, 1977.
  * [15] S. I. Gelfand and Yu. I. Manin.  Methods of Homological Algebra.  Springer Monographs in Mathematics. Springer-Verlag Berlin Heidelberg, 2nd edition, 2003.
  * [16] M. Gheorghiu.  On a Completion of Cohomological Functors Generalising Tate Cohomology II.  arXiv e-prints, page arXiv:2405.03634v1, May 2024.
  * [17] F. Goichot.  Homologie de Tate-Vogel Équivariante.  Journal of Pure and Applied Algebra, 82(1):39–64, 1992.
  * [18] A. Grothendieck.  Sur Quelques Points d’Algèbre Homologique.  Tohoku Mathematical Journal, 9(2):119–221, 1957.
  * [19] S. Guo and L. Liang.  Stable Functors and Cohomology Theory in Abelian Categories.  arXiv e-prints, page arXiv:2006.13562v3, October 2023.
  * [20] S. Guo, L. Liang, and X. Yang.  Balancedness of Complete Cohomology with respect to Subcategories.  https://sites.google.com/site/lliangnju/publications, 2023.  This website has been consulted on 7 June 2024.
  * [21] J. Hu, D. Zhang, T. Zhao, and P. Zhou.  Balance of Complete Cohomology in Extriangulated Categories.  Electronic Research Archive, 29(5):3341–3359, 2021.
  * [22] J. Hu, D. Zhang, T. Zhao, and P. Zhou.  Complete Cohomology for Extriangulated Categories.  Algebra Colloquium, 28(4):701–720, 2021.
  * [23] J. H. Jo.  Complete Homology and Related Dimensions of Groups.  Journal of Group Theory, 12(3):431–448, 2009.
  * [24] J. R. Klein.  Axioms for Generalized Farrell–Tate Cohomology.  Journal of Pure and Applied Algebra, 172(2–3):225–238, 2002.
  * [25] P. H. Kropholler.  Hierarchical Decompositions, Generalized Tate Cohomology, and Groups of Type $(\mathrm{FP})_{\infty}$.  In Combinatorial and Geometric Group Theory, Edinburgh 1993, volume 204 of London Mathematical Society Lecture Note Series, pages 190–216. Cambridge University Press, January 1995.
  * [26] M. Land.  Condensed Mathematics.  https://www.markus-land.de/teaching/, December 2022.  This website has been consulted on 22 September 2023.
  * [27] S. Mac Lane.  Homology.  Classics in Mathematics. Springer-Verlag Berlin Heidelberg, 1995.
  * [28] G. Mislin.  Tate Cohomology for Arbitrary Groups via Satellites.  Topology and its Applications, 56:293–300, 1994.
  * [29] B. E. A. Nucinkis.  Complete Cohomology for Arbitrary Rings Uusing Injectives.  Journal of Pure and Applied Algebra, 131(3):297–318, 1998.
  * [30] M. S. Osborne.  Basic Homological Algebra, volume 196 of Graduate Texts in Mathematics.  Springer-Verlag New York, Inc., 2000.
  * [31] M. Paganin.  On Some Generalizations of Tate Cohomology: an Overview.  Pro Mathematica, 29(58):63–115, 2016.
  * [32] B. Poonen.  Why All Rings should have a $1$.  https://math.mit.edu/~poonen/papers/ring.pdf, June 2018.  This website has been consulted on 25 October 2023.
  * [33] L. Ribes.  Profinite Graphs and Groups, volume 66 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics.  Springer Nature, 2017.
  * [34] L. Ribes and P. Zalesskii.  Profinite Groups, volume 40 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics.  Springer-Verlag Berlin Heidelberg, 2nd edition, 2010.
  * [35] P. Scholze.  Lectures on Condensed Mathematics.  https://www.math.uni-bonn.de/people/scholze/Condensed.pdf, May 2019\.  This website has been consulted on 21 February 2023.
  * [36] P. Scholze.  Liquid Tensor Experiment.  https://xenaproject.wordpress.com/2020/12/05/liquid-tensor-experiment/, December 2020.  This website has been consulted on 25 September 2023.
  * [37] J.-P. Serre.  Cohomologie Galoisienne, volume 5 of Lecture Notes in Mathematics.  Springer-Verlag Berlin Heidelberg, 5th edition, 1994.
  * [38] The Stacks project authors.  The Stacks Project.  https://stacks.math.columbia.edu, 2024.
  * [39] P. Symonds.  The Tate-Farrell Cohomology of the Morava Stabilizer Group $s_{p-1}$ with Coefficients in $e_{p-1}$.  In Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic $K$-Theory, volume 346 of Contemporary Mathematics, pages 485–402. American Mathematical Society, 2004\.
  * [40] P. Symonds.  Permutation Complexes for Profinite Groups.  Commentarii Mathematici Helvetici, 82(1):1–37, 2007.
  * [41] P. Symonds and T. Weigel.  Cohomology of $p$-adic Analytic Groups.  In New Horizons in Pro-p Groups, volume 184 of Progress in Mathematics, pages 347–408. Birkhäuser, Boston, 2000.
  * [42] C. A. Weibel.  An Introduction to Homological Algebra, volume 38 of Cambridge Studies in Advanced Mathematics.  Cambridge University Press, 1994.
  * [43] J. S. Wilson.  Profinite Groups, volume 19 of London Mathematical Society Monographs New Series.  Oxford University Press Inc., New York, 1998.