The Stacks project

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. $\square$

Comments (2)

Comment #5457 by Du on

It looks like there is not need to assume to be flat. I went over the proof and the lemmas it uses, only Lemma 02N5 needs to be flat. shows up at the ending part of the proof, but no place indicates that needs to be flat.

Comment #5675 by on

Dear Du, yes what you say is correct, except for the following problem: the "correct" version of the base change map has as its source the object . And if is not flat, then this isn't computed by . So the proof in the case where is not flat would give you a map from an object that doesn't have a well defined meaning (in some sense).

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