Definition 25.4.1. Let $\mathcal{C}$ be a site. Let $K$ be a simplicial object of $\textit{PSh}(\mathcal{C})$. By the above we get a simplicial object $\mathbf{Z}_ K^\# $ of $\textit{Ab}(\mathcal{C})$. We can take its associated complex of abelian presheaves $s(\mathbf{Z}_ K^\# )$, see Simplicial, Section 14.23. The homology of $K$ is the homology of the complex of abelian sheaves $s(\mathbf{Z}_ K^\# )$.
25.4 Acyclicity
Let $\mathcal{C}$ be a site. For a presheaf of sets $\mathcal{F}$ we denote $\mathbf{Z}_\mathcal {F}$ the presheaf of abelian groups defined by the rule
\[ \mathbf{Z}_\mathcal {F}(U) = \text{free abelian group on }\mathcal{F}(U). \]
We will sometimes call this the free abelian presheaf on $\mathcal{F}$. Of course the construction $\mathcal{F} \mapsto \mathbf{Z}_\mathcal {F}$ is a functor and it is left adjoint to the forgetful functor $\textit{PAb}(\mathcal{C}) \to \textit{PSh}(\mathcal{C})$. Of course the sheafification $\mathbf{Z}_\mathcal {F}^\# $ is a sheaf of abelian groups, and the functor $\mathcal{F} \mapsto \mathbf{Z}_\mathcal {F}^\# $ is a left adjoint as well. We sometimes call $\mathbf{Z}_\mathcal {F}^\# $ the free abelian sheaf on $\mathcal{F}$.
For an object $X$ of the site $\mathcal{C}$ we denote $\mathbf{Z}_ X$ the free abelian presheaf on $h_ X$, and we denote $\mathbf{Z}_ X^\# $ its sheafification.
In other words, the $i$th homology $H_ i(K)$ of $K$ is the sheaf of abelian groups $H_ i(K) = H_ i(s(\mathbf{Z}_ K^\# ))$. In this section we worry about the homology in case $K$ is a hypercovering of an object $X$ of $\mathcal{C}$.
Lemma 25.4.2. Let $\mathcal{C}$ be a site. Let $\mathcal{F} \to \mathcal{G}$ be a morphism of presheaves of sets. Denote $K$ the simplicial object of $\textit{PSh}(\mathcal{C})$ whose $n$th term is the $(n + 1)$st fibre product of $\mathcal{F}$ over $\mathcal{G}$, see Simplicial, Example 14.3.5. Then, if $\mathcal{F} \to \mathcal{G}$ is surjective after sheafification, we have The isomorphism in degree $0$ is given by the morphism $H_0(K) \to \mathbf{Z}_\mathcal {G}^\# $ coming from the map $(\mathbf{Z}_ K^\# )_0 = \mathbf{Z}_\mathcal {F}^\# \to \mathbf{Z}_\mathcal {G}^\# $.
\[ H_ i(K) = \left\{ \begin{matrix} 0
& \text{if}
& i > 0
\\ \mathbf{Z}_\mathcal {G}^\#
& \text{if}
& i = 0
\end{matrix} \right. \]
Proof. Let $\mathcal{G}' \subset \mathcal{G}$ be the image of the morphism $\mathcal{F} \to \mathcal{G}$. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Set $A = \mathcal{F}(U)$ and $B = \mathcal{G}'(U)$. Then the simplicial set $K(U)$ is equal to the simplicial set with $n$-simplices given by
\[ A \times _ B A \times _ B \ldots \times _ B A\ (n + 1 \text{ factors)}. \]
By Simplicial, Lemma 14.32.3 the morphism $K(U) \to B$ is a trivial Kan fibration. Thus it is a homotopy equivalence (Simplicial, Lemma 14.30.8). Hence applying the functor “free abelian group on” to this we deduce that
\[ \mathbf{Z}_ K(U) \longrightarrow \mathbf{Z}_ B \]
is a homotopy equivalence. Note that $s(\mathbf{Z}_ B)$ is the complex
\[ \ldots \to \bigoplus \nolimits _{b \in B}\mathbf{Z} \xrightarrow {0} \bigoplus \nolimits _{b \in B}\mathbf{Z} \xrightarrow {1} \bigoplus \nolimits _{b \in B}\mathbf{Z} \xrightarrow {0} \bigoplus \nolimits _{b \in B}\mathbf{Z} \to 0 \]
see Simplicial, Lemma 14.23.3. Thus we see that $H_ i(s(\mathbf{Z}_ K(U))) = 0$ for $i > 0$, and $H_0(s(\mathbf{Z}_ K(U))) = \bigoplus _{b \in B}\mathbf{Z} = \bigoplus _{s \in \mathcal{G}'(U)} \mathbf{Z}$. These identifications are compatible with restriction maps.
We conclude that $H_ i(s(\mathbf{Z}_ K)) = 0$ for $i > 0$ and $H_0(s(\mathbf{Z}_ K)) = \mathbf{Z}_{\mathcal{G}'}$, where here we compute homology groups in $\textit{PAb}(\mathcal{C})$. Since sheafification is an exact functor we deduce the result of the lemma. Namely, the exactness implies that $H_0(s(\mathbf{Z}_ K))^\# = H_0(s(\mathbf{Z}_ K^\# ))$, and similarly for other indices. $\square$
Lemma 25.4.3. Let $\mathcal{C}$ be a site. Let $f : L \to K$ be a morphism of simplicial objects of $\textit{PSh}(\mathcal{C})$. Let $n \geq 0$ be an integer. Assume that
For $i < n$ the morphism $L_ i \to K_ i$ is an isomorphism.
The morphism $L_ n \to K_ n$ is surjective after sheafification.
The canonical map $L \to \text{cosk}_ n \text{sk}_ n L$ is an isomorphism.
The canonical map $K \to \text{cosk}_ n \text{sk}_ n K$ is an isomorphism.
Then $H_ i(f) : H_ i(L) \to H_ i(K)$ is an isomorphism.
Proof. This proof is exactly the same as the proof of Lemma 25.4.2 above. Namely, we first let $K_ n' \subset K_ n$ be the sub presheaf which is the image of the map $L_ n \to K_ n$. Assumption (2) means that the sheafification of $K_ n'$ is equal to the sheafification of $K_ n$. Moreover, since $L_ i = K_ i$ for all $i < n$ we see that get an $n$-truncated simplicial presheaf $U$ by taking $U_0 = L_0 = K_0, \ldots , U_{n - 1} = L_{n - 1} = K_{n - 1}, U_ n = K'_ n$. Denote $K' = \text{cosk}_ n U$, a simplicial presheaf. Because we can construct $K'_ m$ as a finite limit, and since sheafification is exact, we see that $(K'_ m)^\# = K_ m$. In other words, $(K')^\# = K^\# $. We conclude, by exactness of sheafification once more, that $H_ i(K) = H_ i(K')$. Thus it suffices to prove the lemma for the morphism $L \to K'$, in other words, we may assume that $L_ n \to K_ n$ is a surjective morphism of presheaves!
In this case, for any object $U$ of $\mathcal{C}$ we see that the morphism of simplicial sets
\[ L(U) \longrightarrow K(U) \]
satisfies all the assumptions of Simplicial, Lemma 14.32.1. Hence it is a trivial Kan fibration. In particular it is a homotopy equivalence (Simplicial, Lemma 14.30.8). Thus
\[ \mathbf{Z}_ L(U) \longrightarrow \mathbf{Z}_ K(U) \]
is a homotopy equivalence too. This for all $U$. The result follows. $\square$
Lemma 25.4.4. Let $\mathcal{C}$ be a site. Let $K$ be a simplicial presheaf. Let $\mathcal{G}$ be a presheaf. Let $K \to \mathcal{G}$ be an augmentation of $K$ towards $\mathcal{G}$. Assume that
The morphism of presheaves $K_0 \to \mathcal{G}$ becomes a surjection after sheafification.
The morphism
becomes a surjection after sheafification.
For every $n \geq 1$ the morphism
turns into a surjection after sheafification.
\[ (d^1_0, d^1_1) : K_1 \longrightarrow K_0 \times _\mathcal {G} K_0 \]
\[ K_{n + 1} \longrightarrow (\text{cosk}_ n \text{sk}_ n K)_{n + 1} \]
Then $H_ i(K) = 0$ for $i > 0$ and $H_0(K) = \mathbf{Z}_\mathcal {G}^\# $.
Proof. Denote $K^ n = \text{cosk}_ n \text{sk}_ n K$ for $n \geq 1$. Define $K^0$ as the simplicial object with terms $(K^0)_ n$ equal to the $(n + 1)$-fold fibred product $K_0 \times _\mathcal {G} \ldots \times _\mathcal {G} K_0$, see Simplicial, Example 14.3.5. We have morphisms
\[ K \longrightarrow \ldots \to K^ n \to K^{n - 1} \to \ldots \to K^1 \to K^0. \]
The morphisms $K \to K^ i$, $K^ j \to K^ i$ for $j \geq i \geq 1$ come from the universal properties of the $\text{cosk}_ n$ functors. The morphism $K^1 \to K^0$ is the canonical morphism from Simplicial, Remark 14.20.4. We also recall that $K^0 \to \text{cosk}_1 \text{sk}_1 K^0$ is an isomorphism, see Simplicial, Lemma 14.20.3.
By Lemma 25.4.2 we see that $H_ i(K^0) = 0$ for $i > 0$ and $H_0(K^0) = \mathbf{Z}_\mathcal {G}^\# $.
Pick $n \geq 1$. Consider the morphism $K^ n \to K^{n - 1}$. It is an isomorphism on terms of degree $< n$. Note that $K^ n \to \text{cosk}_ n \text{sk}_ n K^ n$ and $K^{n - 1} \to \text{cosk}_ n \text{sk}_ n K^{n - 1}$ are isomorphisms. Note that $(K^ n)_ n = K_ n$ and that $(K^{n - 1})_ n = (\text{cosk}_{n - 1} \text{sk}_{n - 1} K)_ n$. Hence by assumption, we have that $(K^ n)_ n \to (K^{n - 1})_ n$ is a morphism of presheaves which becomes surjective after sheafification. By Lemma 25.4.3 we conclude that $H_ i(K^ n) = H_ i(K^{n - 1})$. Combined with the above this proves the lemma. $\square$
Lemma 25.4.5. Let $\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\mathcal{C}$. Let $K$ be a hypercovering of $X$. The homology of the simplicial presheaf $F(K)$ is $0$ in degrees $> 0$ and equal to $\mathbf{Z}_ X^\# $ in degree $0$.
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)