The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

21.38 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 21.38.1 (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 21.37.1. By Remark 21.37.6 we do not need to distinguish between $\pi _!$ on modules or abelian sheaves. By Lemma 21.37.8 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 21.38.2 (Computing homology). In Example 21.38.1 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 4.14.11. 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 12.11.1 and Lemma 12.12.12. 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

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

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 12.11.4 and Derived Categories, Lemma 13.17.6.

Example 21.38.3. Let $u : \mathcal{C}' \to \mathcal{C}$ be a functor. Endow $\mathcal{C}'$ and $\mathcal{C}$ with the chaotic topology as in Example 21.38.1. 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 21.37.3. Thus Lemma 21.37.5 applies to $g$ so we do not need to distinguish between $g_!$ on modules or abelian sheaves. In particular Remark 21.37.7 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 21.38.2 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 21.38.4. Notation and assumptions as in Example 21.38.1. 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 21.38.5. Notation and assumptions as in Example 21.38.1. 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 4.24.3. Then the same lemma implies that $L\pi _! \circ \pi ^{-1} = \text{id}$ as desired. $\square$

Lemma 21.38.6. Notation and assumptions as in Example 21.38.1. 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 21.38.7. Notation and assumptions as in Example 21.38.1. 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 $\mathop{Mor}\nolimits _\mathcal {C}(U_\bullet , U)$ 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 21.37.5 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 18.28.7. 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 21.37.8) 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 12.22.7. 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 $\mathop{Mor}\nolimits _\mathcal {C}(U_\bullet , U)$ 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 21.37.5. $\square$

Lemma 21.38.8. Notation and assumptions as in Example 21.38.3. If there exists a cosimplicial object $U'_\bullet $ of $\mathcal{C}'$ such that Lemma 21.38.7 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 21.38.7 and the fact that $g^{-1}$ is given by precomposing with $u$. $\square$

Lemma 21.38.9. 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 21.38.3. 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 21.36.2. 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 21.36.2. $\square$

Lemma 21.38.10. Notation and assumptions as in Example 21.38.1. If there exists a cosimplicial object $U_\bullet $ of $\mathcal{C}$ such that Lemma 21.38.7 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

\[ \mathop{Mor}\nolimits _{\mathcal{C} \times \mathcal{C}}(u(U_\bullet ), (U_1, U_2)) = \mathop{Mor}\nolimits _\mathcal {C}(U_\bullet , U_1) \times \mathop{Mor}\nolimits _\mathcal {C}(U_\bullet , U_2) \]

which is homotopy equivalent to a point by Simplicial, Lemma 14.26.10. Thus Lemma 21.38.8 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 21.19.4). To finish we apply Lemma 21.38.9. $\square$

Remark 21.38.11 (Simplicial modules). Let $\mathcal{C} = \Delta $ and let $B$ be any ring. This is a special case of Example 21.38.1 where the assumptions of Lemma 21.38.7 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 $\Delta [m] : n \mapsto \mathop{Mor}\nolimits _\Delta ([n], [m])$ 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 21.38.10 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 21.38.12. 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 21.38.7, 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 13.28.2 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 18.28.7. Thus it suffices to prove the lemma for $j_{U!}\mathcal{O}_ U$. By assumption

\[ S_\bullet = \mathop{Mor}\nolimits _\mathcal {C}(U_\bullet , U) \]

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 21.38.7. 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 21.38.7). We conclude that $X_\bullet $ and $X'_\bullet $ are quasi-isomorphic as desired. $\square$

Remark 21.38.13. Let $\mathcal{C}$ and $B$ be as in Example 21.38.1. Assume there exists a cosimplicial object as in Lemma 21.38.7. 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 21.38.12. 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 (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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 08RW. Beware of the difference between the letter 'O' and the digit '0'.