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The Stacks project

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 nth 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 nth 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.

Remark 21.39.4. Notation and assumptions as in Example 21.39.1. Let \mathcal{F}^\bullet be a bounded complex of abelian sheaves on \mathcal{C}. For any object U of \mathcal{C} there is a canonical map

\mathcal{F}^\bullet (U) \longrightarrow L\pi _!(\mathcal{F}^\bullet )

in D(\textit{Ab}). If \mathcal{F}^\bullet is a complex of \underline{B}-modules then this map is in D(B). To prove this, note that we compute L\pi _!(\mathcal{F}^\bullet ) by taking a quasi-isomorphism \mathcal{P}^\bullet \to \mathcal{F}^\bullet where \mathcal{P}^\bullet is a complex of projectives. However, since the topology is chaotic this means that \mathcal{P}^\bullet (U) \to \mathcal{F}^\bullet (U) is a quasi-isomorphism hence can be inverted in D(\textit{Ab}), resp. D(B). Composing with the canonical map \mathcal{P}^\bullet (U) \to \pi _!(\mathcal{P}^\bullet ) coming from the computation of \pi _! as a colimit we obtain the desired arrow.

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

Remark 21.39.11 (Simplicial modules). Let \mathcal{C} = \Delta and let B be any ring. This is a special case of Example 21.39.1 where the assumptions of Lemma 21.39.7 hold. Namely, let U_\bullet be the cosimplicial object of \Delta given by the identity functor. To verify the condition we have to show that for [m] \in \mathop{\mathrm{Ob}}\nolimits (\Delta ) the simplicial set \Delta [m] : n \mapsto \mathop{\mathrm{Mor}}\nolimits _\Delta ([n], [m]) is homotopy equivalent to a point. This is explained in Simplicial, Example 14.26.7.

In this situation the category \textit{Mod}(\underline{B}) is just the category of simplicial B-modules and the functor L\pi _! sends a simplicial B-module M_\bullet to its associated complex s(M_\bullet ) of B-modules. Thus the results above can be reinterpreted in terms of results on simplicial modules. For example a special case of Lemma 21.39.10 is: if M_\bullet , M'_\bullet are flat simplicial B-modules, then the complex s(M_\bullet \otimes _ B M'_\bullet ) is quasi-isomorphic to the total complex associated to the double complex s(M_\bullet ) \otimes _ B s(M'_\bullet ). (Hint: use flatness to convert from derived tensor products to usual tensor products.) This is a special case of the Eilenberg-Zilber theorem which can be found in [Eilenberg-Zilber].

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

  1. there exists a cosimplicial object U_\bullet in \mathcal{C} as in Lemma 21.39.7, and

  2. 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

Remark 21.39.13. Let \mathcal{C} and B be as in Example 21.39.1. Assume there exists a cosimplicial object as in Lemma 21.39.7. Let \mathcal{O} \to \underline{B} be a map sheaf of rings on \mathcal{C} which induces an isomorphism L\pi _!\mathcal{O} \to L\pi _!\underline{B}. In this case we obtain an exact functor of triangulated categories

L\pi _! : D(\mathcal{O}) \longrightarrow D(B)

Namely, for any object K of D(\mathcal{O}) we have L\pi ^{\textit{Ab}}_!(K) = L\pi ^{\textit{Ab}}_!(K \otimes _{\mathcal{O}}^\mathbf {L} \underline{B}) by Lemma 21.39.12. Thus we can define the displayed functor as the composition of - \otimes ^\mathbf {L}_\mathcal {O} \underline{B} with the functor L\pi _! : D(\underline{B}) \to D(B). In other words, we obtain a B-module structure on L\pi _!(K) coming from the (canonical, functorial) identification of L\pi _!(K) with L\pi _!(K \otimes _\mathcal {O}^\mathbf {L} \underline{B}) of the lemma.


Comments (1)

Comment #9539 by S on

The example 08PG is not clear. What is the reference for this section?


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