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21.15 The base change map

In this section we construct the base change map in some cases; the general case is treated in Remark 21.19.3. The discussion in this section avoids using derived pullback by restricting to the case of a base change by a flat morphism of ringed sites. Before we state the result, let us discuss flat pullback on the derived category. Suppose $g : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ is a flat morphism of ringed topoi. By Modules on Sites, Lemma 18.31.2 the functor $g^* : \textit{Mod}(\mathcal{O}_\mathcal {D}) \to \textit{Mod}(\mathcal{O}_\mathcal {C})$ is exact. Hence it has a derived functor

\[ g^* : D(\mathcal{O}_\mathcal {D}) \to D(\mathcal{O}_\mathcal {C}) \]

which is computed by simply pulling back an representative of a given object in $D(\mathcal{O}_\mathcal {D})$, see Derived Categories, Lemma 13.16.9. It preserved the bounded (above, below) subcategories. Hence as indicated we indicate this functor by $g^*$ rather than $Lg^*$.

Lemma 21.15.1. Let

\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}) \ar[r]_{g'} \ar[d]_{f'} & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \ar[d]^ f \\ (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}'), \mathcal{O}_{\mathcal{D}'}) \ar[r]^ g & (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D}) } \]

be a commutative diagram of ringed topoi. Let $\mathcal{F}^\bullet $ be a bounded below complex of $\mathcal{O}_\mathcal {C}$-modules. Assume both $g$ and $g'$ are flat. Then there exists a canonical base change map

\[ g^*Rf_*\mathcal{F}^\bullet \longrightarrow R(f')_*(g')^*\mathcal{F}^\bullet \]

in $D^{+}(\mathcal{O}_{\mathcal{D}'})$.

Proof. Choose injective resolutions $\mathcal{F}^\bullet \to \mathcal{I}^\bullet $ and $(g')^*\mathcal{F}^\bullet \to \mathcal{J}^\bullet $. By Lemma 21.14.2 we see that $(g')_*\mathcal{J}^\bullet $ is a complex of injectives representing $R(g')_*(g')^*\mathcal{F}^\bullet $. Hence by Derived Categories, Lemmas 13.18.6 and 13.18.7 the arrow $\beta $ in the diagram

\[ \xymatrix{ (g')_*(g')^*\mathcal{F}^\bullet \ar[r] & (g')_*\mathcal{J}^\bullet \\ \mathcal{F}^\bullet \ar[u]^{adjunction} \ar[r] & \mathcal{I}^\bullet \ar[u]_\beta } \]

exists and is unique up to homotopy. Pushing down to $\mathcal{D}$ we get

\[ f_*\beta : f_*\mathcal{I}^\bullet \longrightarrow f_*(g')_*\mathcal{J}^\bullet = g_*(f')_*\mathcal{J}^\bullet \]

By adjunction of $g^*$ and $g_*$ we get a map of complexes $g^*f_*\mathcal{I}^\bullet \to (f')_*\mathcal{J}^\bullet $. Note that this map is unique up to homotopy since the only choice in the whole process was the choice of the map $\beta $ and everything was done on the level of complexes. $\square$

Comments (2)

Comment #2178 by Kestutis Cesnavicius on

In the first display of the section the indices C and D seem to be mixed up.

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