21.39 Homology on a category
In the case of a category over a point we will baptize the left derived lower shriek functors the homology functors.
Example 21.39.1 (Category over point). Let $\mathcal{C}$ be a category. Endow $\mathcal{C}$ with the chaotic topology (Sites, Example 7.6.6). Thus presheaves and sheaves agree on $\mathcal{C}$. The functor $p : \mathcal{C} \to *$ where $*$ is the category with a single object and a single morphism is cocontinuous and continuous. Let $\pi : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (*)$ be the corresponding morphism of topoi. Let $B$ be a ring. We endow $*$ with the sheaf of rings $B$ and $\mathcal{C}$ with $\mathcal{O}_\mathcal {C} = \pi ^{-1}B$ which we will denote $\underline{B}$. In this way
\[ \pi : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \underline{B}) \to (\mathop{\mathit{Sh}}\nolimits (*), B) \]
is an example of Situation 21.38.1. By Remark 21.38.6 we do not need to distinguish between $\pi _!$ on modules or abelian sheaves. By Lemma 21.38.8 we see that $\pi _!\mathcal{F} = \mathop{\mathrm{colim}}\nolimits _{\mathcal{C}^{opp}} \mathcal{F}$. Thus $L_ n\pi _!$ is the $n$th left derived functor of taking colimits. In the following, we write
\[ H_ n(\mathcal{C}, \mathcal{F}) = L_ n\pi _!(\mathcal{F}) \]
and we will name this the $n$th homology group of $\mathcal{F}$ on $\mathcal{C}$.
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.
Example 21.39.3. Let $u : \mathcal{C}' \to \mathcal{C}$ be a functor. Endow $\mathcal{C}'$ and $\mathcal{C}$ with the chaotic topology as in Example 21.39.1. The functors $u$, $\mathcal{C}' \to *$, and $\mathcal{C} \to *$ where $*$ is the category with a single object and a single morphism are cocontinuous and continuous. Let $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}') \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$, $\pi ' : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}') \to \mathop{\mathit{Sh}}\nolimits (*)$, and $\pi : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (*)$, be the corresponding morphisms of topoi. Let $B$ be a ring. We endow $*$ with the sheaf of rings $B$ and $\mathcal{C}'$, $\mathcal{C}$ with the constant sheaf $\underline{B}$. In this way
\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \underline{B}) \ar[rd]_{\pi '} \ar[rr]_ g & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \underline{B}) \ar[ld]^\pi \\ & (\mathop{\mathit{Sh}}\nolimits (*), B) } \]
is an example of Situation 21.38.3. Thus Lemma 21.38.5 applies to $g$ so we do not need to distinguish between $g_!$ on modules or abelian sheaves. In particular Remark 21.38.7 produces canonical maps
\[ H_ n(\mathcal{C}', \mathcal{F}') \longrightarrow H_ n(\mathcal{C}, \mathcal{F}) \]
whenever we have $\mathcal{F}$ in $\textit{Ab}(\mathcal{C})$, $\mathcal{F}'$ in $\textit{Ab}(\mathcal{C}')$, and a map $t : \mathcal{F}' \to g^{-1}\mathcal{F}$. In terms of the computation of homology given in Example 21.39.2 we see that these maps come from a map of complexes
\[ K_\bullet (\mathcal{F}') \longrightarrow K_\bullet (\mathcal{F}) \]
given by the rule
\[ (U'_ n \to \ldots \to U'_0, s') \longmapsto (u(U'_ n) \to \ldots \to u(U'_0), t(s')) \]
with obvious notation.
Lemma 21.39.5. Notation and assumptions as in Example 21.39.1. If $\mathcal{C}$ has either an initial or a final object, then $L\pi _! \circ \pi ^{-1} = \text{id}$ on $D(\textit{Ab})$, resp. $D(B)$.
Proof.
If $\mathcal{C}$ has an initial object, then $\pi _!$ is computed by evaluating on this object and the statement is clear. If $\mathcal{C}$ has a final object, then $R\pi _*$ is computed by evaluating on this object, hence $R\pi _* \circ \pi ^{-1} \cong \text{id}$ on $D(\textit{Ab})$, resp. $D(B)$. This implies that $\pi ^{-1} : D(\textit{Ab}) \to D(\mathcal{C})$, resp. $\pi ^{-1} : D(B) \to D(\underline{B})$ is fully faithful, see Categories, Lemma 4.24.4. Then the same lemma implies that $L\pi _! \circ \pi ^{-1} = \text{id}$ as desired.
$\square$
Lemma 21.39.6. Notation and assumptions as in Example 21.39.1. Let $B \to B'$ be a ring map. Consider the commutative diagram of ringed topoi
\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \underline{B}) \ar[d]_\pi & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \underline{B'}) \ar[d]^{\pi '} \ar[l]^ h \\ (*, B) & (*, B') \ar[l]_ f } \]
Then $L\pi _! \circ Lh^* = Lf^* \circ L\pi '_!$.
Proof.
Both functors are right adjoint to the obvious functor $D(B') \to D(\underline{B})$.
$\square$
Lemma 21.39.7. Notation and assumptions as in Example 21.39.1. Let $U_\bullet $ be a cosimplicial object in $\mathcal{C}$ such that for every $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ the simplicial set $\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U_\bullet , U)$ is homotopy equivalent to the constant simplicial set on a singleton. Then
\[ L\pi _!(\mathcal{F}) = \mathcal{F}(U_\bullet ) \]
in $D(\textit{Ab})$, resp. $D(B)$ functorially in $\mathcal{F}$ in $\textit{Ab}(\mathcal{C})$, resp. $\textit{Mod}(\underline{B})$.
Proof.
As $L\pi _!$ agrees for modules and abelian sheaves by Lemma 21.38.5 it suffices to prove this when $\mathcal{F}$ is an abelian sheaf. For $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ the abelian sheaf $j_{U!}\mathbf{Z}_ U$ is a projective object of $\textit{Ab}(\mathcal{C})$ since $\mathop{\mathrm{Hom}}\nolimits (j_{U!}\mathbf{Z}_ U, \mathcal{F}) = \mathcal{F}(U)$ and taking sections is an exact functor as the topology is chaotic. Every abelian sheaf is a quotient of a direct sum of $j_{U!}\mathbf{Z}_ U$ by Modules on Sites, Lemma 18.28.8. Thus we can compute $L\pi _!(\mathcal{F})$ by choosing a resolution
\[ \ldots \to \mathcal{G}^{-1} \to \mathcal{G}^0 \to \mathcal{F} \to 0 \]
whose terms are direct sums of sheaves of the form above and taking $L\pi _!(\mathcal{F}) = \pi _!(\mathcal{G}^\bullet )$. Consider the double complex $A^{\bullet , \bullet } = \mathcal{G}^\bullet (U_\bullet )$. The map $\mathcal{G}^0 \to \mathcal{F}$ gives a map of complexes $A^{0, \bullet } \to \mathcal{F}(U_\bullet )$. Since $\pi _!$ is computed by taking the colimit over $\mathcal{C}^{opp}$ (Lemma 21.38.8) we see that the two compositions $\mathcal{G}^ m(U_1) \to \mathcal{G}^ m(U_0) \to \pi _!\mathcal{G}^ m$ are equal. Thus we obtain a canonical map of complexes
\[ \text{Tot}(A^{\bullet , \bullet }) \longrightarrow \pi _!(\mathcal{G}^\bullet ) = L\pi _!(\mathcal{F}) \]
To prove the lemma it suffices to show that the complexes
\[ \ldots \to \mathcal{G}^ m(U_1) \to \mathcal{G}^ m(U_0) \to \pi _!\mathcal{G}^ m \to 0 \]
are exact, see Homology, Lemma 12.25.4. Since the sheaves $\mathcal{G}^ m$ are direct sums of the sheaves $j_{U!}\mathbf{Z}_ U$ we reduce to $\mathcal{G} = j_{U!}\mathbf{Z}_ U$. The complex $j_{U!}\mathbf{Z}_ U(U_\bullet )$ is the complex of abelian groups associated to the free $\mathbf{Z}$-module on the simplicial set $\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U_\bullet , U)$ which we assumed to be homotopy equivalent to a singleton. We conclude that
\[ j_{U!}\mathbf{Z}_ U(U_\bullet ) \to \mathbf{Z} \]
is a homotopy equivalence of abelian groups hence a quasi-isomorphism (Simplicial, Remark 14.26.4 and Lemma 14.27.1). This finishes the proof since $\pi _!j_{U!}\mathbf{Z}_ U = \mathbf{Z}$ as was shown in the proof of Lemma 21.38.5.
$\square$
Lemma 21.39.8. Notation and assumptions as in Example 21.39.3. If there exists a cosimplicial object $U'_\bullet $ of $\mathcal{C}'$ such that Lemma 21.39.7 applies to both $U'_\bullet $ in $\mathcal{C}'$ and $u(U'_\bullet )$ in $\mathcal{C}$, then we have $L\pi '_! \circ g^{-1} = L\pi _!$ as functors $D(\mathcal{C}) \to D(\textit{Ab})$, resp. $D(\mathcal{C}, \underline{B}) \to D(B)$.
Proof.
Follows immediately from Lemma 21.39.7 and the fact that $g^{-1}$ is given by precomposing with $u$.
$\square$
Lemma 21.39.9. Let $\mathcal{C}_ i$, $i = 1, 2$ be categories. Let $u_ i : \mathcal{C}_1 \times \mathcal{C}_2 \to \mathcal{C}_ i$ be the projection functors. Let $B$ be a ring. Let $g_ i : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_1 \times \mathcal{C}_2), \underline{B}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ i), \underline{B})$ be the corresponding morphisms of ringed topoi, see Example 21.39.3. For $K_ i \in D(\mathcal{C}_ i, B)$ we have
\[ L(\pi _1 \times \pi _2)_!( g_1^{-1}K_1 \otimes _{\underline{B}}^\mathbf {L} g_2^{-1}K_2) = L\pi _{1, !}(K_1) \otimes _ B^\mathbf {L} L\pi _{2, !}(K_2) \]
in $D(B)$ with obvious notation.
Proof.
As both sides commute with colimits, it suffices to prove this for $K_1 = j_{U!}\underline{B}_ U$ and $K_2 = j_{V!}\underline{B}_ V$ for $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_1)$ and $V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_2)$. See construction of $L\pi _!$ in Lemma 21.37.2. In this case
\[ g_1^{-1}K_1 \otimes _{\underline{B}}^\mathbf {L} g_2^{-1}K_2 = g_1^{-1}K_1 \otimes _{\underline{B}} g_2^{-1}K_2 = j_{(U, V)!}\underline{B}_{(U, V)} \]
Verification omitted. Hence the result follows as both the left and the right hand side of the formula of the lemma evaluate to $B$, see construction of $L\pi _!$ in Lemma 21.37.2.
$\square$
Lemma 21.39.10. Notation and assumptions as in Example 21.39.1. If there exists a cosimplicial object $U_\bullet $ of $\mathcal{C}$ such that Lemma 21.39.7 applies, then
\[ L\pi _!(K_1 \otimes ^\mathbf {L}_{\underline{B}} K_2) = L\pi _!(K_1) \otimes ^\mathbf {L}_ B L\pi _!(K_2) \]
for all $K_ i \in D(\underline{B})$.
Proof.
Consider the diagram of categories and functors
\[ \xymatrix{ & & \mathcal{C} \\ \mathcal{C} \ar[r]^-u & \mathcal{C} \times \mathcal{C} \ar[rd]^{u_2} \ar[ru]_{u_1} \\ & & \mathcal{C} } \]
where $u$ is the diagonal functor and $u_ i$ are the projection functors. This gives morphisms of ringed topoi $g$, $g_1$, $g_2$. For any object $(U_1, U_2)$ of $\mathcal{C}$ we have
\[ \mathop{\mathrm{Mor}}\nolimits _{\mathcal{C} \times \mathcal{C}}(u(U_\bullet ), (U_1, U_2)) = \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U_\bullet , U_1) \times \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U_\bullet , U_2) \]
which is homotopy equivalent to a point by Simplicial, Lemma 14.26.10. Thus Lemma 21.39.8 gives $L\pi _!(g^{-1}K) = L(\pi \times \pi )_!(K)$ for any $K$ in $D(\mathcal{C} \times \mathcal{C}, B)$. Take $K = g_1^{-1}K_1 \otimes _ B^\mathbf {L} g_2^{-1}K_2$. Then $g^{-1}K = K_1 \otimes ^\mathbf {L}_{\underline{B}} K_2$ because $g^{-1} = g^* = Lg^*$ commutes with derived tensor product (Lemma 21.18.4). To finish we apply Lemma 21.39.9.
$\square$
Lemma 21.39.12. Let $\mathcal{C}$ be a category (endowed with chaotic topology). Let $\mathcal{O} \to \mathcal{O}'$ be a map of sheaves of rings on $\mathcal{C}$. Assume
there exists a cosimplicial object $U_\bullet $ in $\mathcal{C}$ as in Lemma 21.39.7, and
$L\pi _!\mathcal{O} \to L\pi _!\mathcal{O}'$ is an isomorphism.
For $K$ in $D(\mathcal{O})$ we have
\[ L\pi _!(K) = L\pi _!(K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}') \]
in $D(\textit{Ab})$.
Proof.
Note: in this proof $L\pi _!$ denotes the left derived functor of $\pi _!$ on abelian sheaves. Since $L\pi _!$ commutes with colimits, it suffices to prove this for bounded above complexes of $\mathcal{O}$-modules (compare with argument of Derived Categories, Proposition 13.29.2 or just stick to bounded above complexes). Every such complex is quasi-isomorphic to a bounded above complex whose terms are direct sums of $j_{U!}\mathcal{O}_ U$ with $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, see Modules on Sites, Lemma 18.28.8. Thus it suffices to prove the lemma for $j_{U!}\mathcal{O}_ U$. By assumption
\[ S_\bullet = \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U_\bullet , U) \]
is a simplicial set homotopy equivalent to the constant simplicial set on a singleton. Set $P_ n = \mathcal{O}(U_ n)$ and $P'_ n = \mathcal{O}'(U_ n)$. Observe that the complex associated to the simplicial abelian group
\[ X_\bullet : n \longmapsto \bigoplus \nolimits _{s \in S_ n} P_ n \]
computes $L\pi _!(j_{U!}\mathcal{O}_ U)$ by Lemma 21.39.7. Since $j_{U!}\mathcal{O}_ U$ is a flat $\mathcal{O}$-module we have $j_{U!}\mathcal{O}_ U \otimes ^\mathbf {L}_\mathcal {O} \mathcal{O}' = j_{U!}\mathcal{O}'_ U$ and $L\pi _!$ of this is computed by the complex associated to the simplicial abelian group
\[ X'_\bullet : n \longmapsto \bigoplus \nolimits _{s \in S_ n} P'_ n \]
As the rule which to a simplicial set $T_\bullet $ associated the simplicial abelian group with terms $\bigoplus _{t \in T_ n} P_ n$ is a functor, we see that $X_\bullet \to P_\bullet $ is a homotopy equivalence of simplicial abelian groups. Similarly, the rule which to a simplicial set $T_\bullet $ associates the simplicial abelian group with terms $\bigoplus _{t \in T_ n} P'_ n$ is a functor. Hence $X'_\bullet \to P'_\bullet $ is a homotopy equivalence of simplicial abelian groups. By assumption $P_\bullet \to P'_\bullet $ is a quasi-isomorphism (since $P_\bullet $, resp. $P'_\bullet $ computes $L\pi _!\mathcal{O}$, resp. $L\pi _!\mathcal{O}'$ by Lemma 21.39.7). We conclude that $X_\bullet $ and $X'_\bullet $ are quasi-isomorphic as desired.
$\square$
Comments (1)
Comment #9539 by S on