Example 21.39.2 (08PG): Computing homology

Example 21.39.2 (Computing homology). In Example 21.39.1 we can compute the functors $H_ n(\mathcal{C}, -)$ as follows. Let $\mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\textit{Ab}(\mathcal{C}))$. Consider the chain complex

\[ K_\bullet (\mathcal{F}) : \ \ldots \to \bigoplus \nolimits _{U_2 \to U_1 \to U_0} \mathcal{F}(U_0) \to \bigoplus \nolimits _{U_1 \to U_0} \mathcal{F}(U_0) \to \bigoplus \nolimits _{U_0} \mathcal{F}(U_0) \]

where the transition maps are given by

\[ (U_2 \to U_1 \to U_0, s) \longmapsto (U_1 \to U_0, s) - (U_2 \to U_0, s) + (U_2 \to U_1, s|_{U_1}) \]

and similarly in other degrees. By construction

\[ H_0(\mathcal{C}, \mathcal{F}) = \mathop{\mathrm{colim}}\nolimits _{\mathcal{C}^{opp}} \mathcal{F} = H_0(K_\bullet (\mathcal{F})), \]

see Categories, Lemma 4.14.12. The construction of $K_\bullet (\mathcal{F})$ is functorial in $\mathcal{F}$ and transforms short exact sequences of $\textit{Ab}(\mathcal{C})$ into short exact sequences of complexes. Thus the sequence of functors $\mathcal{F} \mapsto H_ n(K_\bullet (\mathcal{F}))$ forms a $\delta $-functor, see Homology, Definition 12.12.1 and Lemma 12.13.12. For $\mathcal{F} = j_{U!}\mathbf{Z}_ U$ the complex $K_\bullet (\mathcal{F})$ is the complex associated to the free $\mathbf{Z}$-module on the simplicial set $X_\bullet $ with terms

\[ X_ n = \coprod \nolimits _{U_ n \to \ldots \to U_1 \to U_0} \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U_0, U) \]

This simplicial set is homotopy equivalent to the constant simplicial set on a singleton $\{ *\} $. Namely, the map $X_\bullet \to \{ *\} $ is obvious, the map $\{ *\} \to X_ n$ is given by mapping $*$ to $(U \to \ldots \to U, \text{id}_ U)$, and the maps

\[ h_{n, i} : X_ n \longrightarrow X_ n \]

(Simplicial, Lemma 14.26.2) defining the homotopy between the two maps $X_\bullet \to X_\bullet $ are given by the rule

\[ h_{n, i} : (U_ n \to \ldots \to U_0, f) \longmapsto (U_ n \to \ldots \to U_ i \to U \to \ldots \to U, \text{id}) \]

for $i > 0$ and $h_{n, 0} = \text{id}$. Verifications omitted. This implies that $K_\bullet (j_{U!}\mathbf{Z}_ U)$ has trivial cohomology in negative degrees (by the functoriality of Simplicial, Remark 14.26.4 and the result of Simplicial, Lemma 14.27.1). Thus $K_\bullet (\mathcal{F})$ computes the left derived functors $H_ n(\mathcal{C}, -)$ of $H_0(\mathcal{C}, -)$ for example by (the duals of) Homology, Lemma 12.12.4 and Derived Categories, Lemma 13.16.6.

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