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20.17 The base change map

We will need to know how to construct the base change map in some cases. Since we have not yet discussed derived pullback we only discuss this in the case of a base change by a flat morphism of ringed spaces. Before we state the result, let us discuss flat pullback on the derived category. Namely, suppose that $g : X \to Y$ is a flat morphism of ringed spaces. By Modules, Lemma 17.20.2 the functor $g^* : \textit{Mod}(\mathcal{O}_ Y) \to \textit{Mod}(\mathcal{O}_ X)$ is exact. Hence it has a derived functor

\[ g^* : D^{+}(Y) \to D^{+}(X) \]

which is computed by simply pulling back an representative of a given object in $D^{+}(Y)$, see Derived Categories, Lemma 13.16.9. Hence as indicated we indicate this functor by $g^*$ rather than $Lg^*$.

Lemma 20.17.1. Let

\[ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ S' \ar[r]^ g & S } \]

be a commutative diagram of ringed spaces. Let $\mathcal{F}^\bullet $ be a bounded below complex of $\mathcal{O}_ X$-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 $D^{+}(S')$.

Proof. Choose injective resolutions $\mathcal{F}^\bullet \to \mathcal{I}^\bullet $ and $(g')^*\mathcal{F}^\bullet \to \mathcal{J}^\bullet $. By Lemma 20.11.11 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 $S$ 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. $\square$

Remark 20.17.2. The “correct” version of the base change map is the map

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

The construction of this map involves unbounded complexes, see Remark 20.28.3.

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