Singular UCT extension from cycle projections

The differential of these cochains squares to zero because $d_{j+1}{d}_{j+2}=0$. A cocycle $\phi$ kills $B_n$, so its restriction to $Z_n$ factors uniquely through $H_n$. A coboundary $h{d}_n$ vanishes on $Z_n$. Thus $\beta$ is well-defined and linear, including on both kinds of representatives. By [F1], $0\to B_{n-1}\to Z_{n-1}\to H_{n-1}\to 0$ is a free resolution. By [F2], its Ext group is $\mathrm{Hom}(B_{n-1},G)$ modulo the restrictions of maps $Z_{n-1}\to G$.

For $\psi :B_{n-1}\to G$, the cochain $\psi d_n$ is closed. If $\psi$ changes by $g{\mathrm{\mid }}_{B_{n-1}}$, with $g:Z_{n-1}\to G$, its cochain changes by $g{\pi }_{n-1}{d}_n=\delta (g{\pi }_{n-1})$, since $d_n$ lands in $Z_{n-1}$ and ${\pi }_{n-1}$ is the identity there. Thus $\iota$ descends to a linear map on Ext. If $\psi d_n=h{d}_n$ for $h:C_{n-1}\to G$, surjectivity of $d_n:C_n\to B_{n-1}$ gives $\psi =h{\mathrm{\mid }}_{B_{n-1}}$. The map $h{\mathrm{\mid }}_{Z_{n-1}}$ then represents the required restriction, so $[\psi ]=0$. This proves injectivity.

For $u:H_n\to G$, write $q:Z_n\to H_n$ for the quotient and set $\phi =uq{\pi }_n$. This kills $B_n$, since ${\pi }_n$ is the identity on $Z_n$ and $q$ kills $B_n$. It is a cocycle with evaluation $u$, so $\beta$ is onto. The construction is linear in $u$ for a fixed projection. It does not assert that the chosen projection is natural.

Every $\psi d_n$ vanishes on $Z_n$, hence has zero evaluation. Conversely, if a cocycle $\phi$ has zero evaluation, then $\phi (z)=0$ for every $z\in Z_n$: the zero map on $H_n$ is zero on every class. Define $\psi (d_{n}c)=\phi (c)$. If $d_{n}c=d_n{c}^{\mathrm{\prime }}$, then $c-c^{\mathrm{\prime }}\in Z_n$, so the definition is independent of the preimage, is linear, and gives $\phi =\psi d_n$. This proves both containments of $\mathrm{im}\iota =\ker \beta$.

Let $f:C\to D$ be a chain map. It restricts to maps $Z_{j}C\to Z_{j}D$ and $B_{j}C\to B_{j}D$ commuting with inclusion and quotient, hence induces the contravariant map on the displayed Ext cokernels by precomposition on $B_{n-1}$. A restricted map from $Z_{n-1}D$ pulls back to a restricted map from $Z_{n-1}C$, so this is well-defined. The equality $(\psi d_n^D)f_n=(\psi f_{n-1}{\mathrm{\mid }}_{B_{n-1}C})d_n^C$ proves naturality of $\iota$. On cycles, $(\phi f_n)(z)=\phi (f_{n}z)$ proves naturality of $\beta$. A coefficient map commutes with every formula by postcomposition. Identity and composition laws follow directly from composition of maps.

For completeness let $0\to P_1\stackrel{a}{\to }{P}_0\stackrel{q}{\to }M\to 0$ and $0\to Q_1\stackrel{b}{\to }{Q}_0\stackrel{r}{\to }N\to 0$ be length-one projective resolutions, and $v:M\to N$. Projectivity lifts $vq$ to $f_0:P_0\to Q_0$ through $r$. Since $r{f}_{0}a=0$, there is a unique $f_1:P_1\to Q_1$ with $b{f}_1=f_{0}a$. Any second lift $g_0$ has $f_0-g_0=bh$ for a unique $h:P_0\to Q_1$, and injectivity of $b$ gives $f_1-g_1=ha$. For $\psi :Q_1\to G$, the difference $\psi f_1-\psi g_1=(\psi h)a$ is zero in the cokernel defining Ext. Thus the induced map is independent of the lift. Composites of lifts lift composites; identity lifts induce identities. Taking $M=N$ and $v=1$ in both directions yields inverse canonical isomorphisms between the two cokernels. This also proves compatibility with the restricted-resolution maps of step 3.2 and with coefficient maps. For free resolutions the requisite projectivity follows from [F4]; for supplied projective resolutions it is their defining property.

Steps 2.1, 3.1 and 2.2 give the asserted short exact sequence, and steps 3.2 and 4.1 give its natural interpretation. At $n=0$, $B_{-1}=Z_{-1}=H_{-1}=0$, so Ext is zero and evaluation is an isomorphism. For the zero complex or $G=0$ every displayed map is the unique zero map. For a single nonzero free term in degree $n$, evaluation is literally the identity on $\mathrm{Hom}(C_n,G)$; no boundary term survives. AC is used in [F1] for arbitrary-rank freeness and sections, and in [F4] for free comparison lifts; the quotient and exactness chases introduce no further choices. No injective-derived or balanced Ext theorem has been used.