6.3 Relative cohomology

The mapping cone construction for morphisms of simplicial sets works in the same way for simplicial objects in any category \mathscr{C} that has a final object e and finite sums. For u\colon Y_\bullet\to X_\bullet, the cone C(u) satisfies C(u)_n = X_n{\,\textstyle\coprod\,}\coprod_{i<n} Y_i{\,\textstyle\coprod\,}e. We take \mathscr{C} to be: a. the category of topological spaces and continuous maps, with final object e=P^t; b. the category of pairs (X,F) where X is a topological space and F is an abelian sheaf on X, with an arrow (u,f)\colon(Y,F)\to(X,G) consisting of a continuous map u\colon Y\to X and a u-morphism (5.1.5) f\colon G\to F, with final object e=(P^t,0).

Let u\colon Y_\bullet\to X_\bullet be a morphism of topological simplicial spaces, with cone C(u). Let F be an abelian sheaf on X_\bullet and G an abelian sheaf on Y_\bullet, and let f\colon G\to F be a u-morphism. The cone C(f) of f is an abelian sheaf on C(u), and we set \operatorname{H}^n(C(u),C(f)) = \operatorname{H}^n(X_\bullet\operatorname{mod}Y_\bullet, F\operatorname{mod}G). \tag{6.3.2.1} These are the relative cohomology groups. We can easily show that they fit into a long exact sequence \ldots \to \operatorname{H}^i(X_\bullet\operatorname{mod}Y_\bullet, F\operatorname{mod}G) \to \operatorname{H}^i(X_\bullet,F) \to \operatorname{H}^i(Y_\bullet,G) \to \ldots. \tag{6.3.2.2}

More generally, let L and K be bounded-below complexes of abelian sheaves on Y_\bullet and X_\bullet (respectively), and let f\colon K\to L be a u-morphism. We thus obtain a complex C(f) on C(u). We again define the hypercohomology \operatorname{\mathbb{H}}^n(C(u),C(f)) = \operatorname{\mathbb{H}}^n(X_\bullet\operatorname{mod}Y_\bullet, K\operatorname{mod}L). These groups appear in an exact sequence analogous to that of (6.3.2.2), coming, in the suitable derived category, from a distinguished triangle \mathrm{R}\Gamma(C(u),C(f)) \to \mathrm{R}\Gamma(X_\bullet,L) \to \mathrm{R}\Gamma(Y_\bullet,K) \xrightarrow{\partial} \mathrm{R}\Gamma(C(u),C(f))[+1] \tag{6.3.3.1}

The construction presented above is not the only one possible. It has the inconvenience that, even if we start with true topological spaces X and Y (i.e. constant simplicial spaces), we are led to consider non-constant simplicial spaces.