Pair exact sequences with local coefficients

The inclusion of local chains on $A$ is injective, and the quotient is the relative local chain group by [F1], so $0\to C_{\ast }(A;\mathcal{L}{\mathrm{\mid }}_A)\to C_{\ast }(X;\mathcal{L})\to C_{\ast }(X,A;\mathcal{L})\to 0$ is degreewise exact. The standard chase in [F3] uses only representatives and ${\mathrm{\partial }}^2=0$, so it gives the first long exact sequence with connector $[c]\mapsto [\mathrm{\partial }c]$.

Restriction of local cochains from $X$ to $A$ is surjective: extend a simplex function by zero on every simplex not contained in $A$. Its kernel is the relative cochain group, so $0\to C^{\ast }(X,A;\mathcal{L})\to C^{\ast }(X;\mathcal{L})\to C^{\ast }(A;\mathcal{L}{\mathrm{\mid }}_A)\to 0$ is exact. The cochain chase of [F3] gives the second sequence; the explicit zero extension makes the stated connector well defined, and a different extension differs by a relative cochain and changes $\delta \tilde{a}$ by a relative coboundary.

The maps in [F2] commute with the inclusions, quotient maps, restrictions, and differentials in the two short exact sequences. Applying them to the representative formulas for the connectors proves commutativity of every naturality square, with $\mathcal{L}\to f^{\ast }\mathcal{K}$ in homology and $f^{\ast }\mathcal{K}\to \mathcal{L}$ in cohomology.

Exactness at degree zero includes the initial zero group because negative chain and cochain degrees vanish. If $A=\mathrm{\varnothing }$, the relative complex is the absolute complex and the $A$ terms are zero; if $A=X$, the relative complex is zero. Empty $X$, zero coefficients, points, degenerate simplices, and disconnected spaces require no modification. The extension by zero is a displayed function, so no AC is used.