Lemma 20.17.1. Let

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

in $D^{+}(S')$.

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 17.19.2 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 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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