Processing math: 100%

The Stacks project

Lemma 21.19.6. Let f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D}) be a morphism of ringed topoi. Let \mathcal{K}^\bullet be a complex of \mathcal{O}_\mathcal {C}-modules. The diagram

\xymatrix{ Lf^*f_*\mathcal{K}^\bullet \ar[r] \ar[d] & f^*f_*\mathcal{K}^\bullet \ar[d] \\ Lf^*Rf_*\mathcal{K}^\bullet \ar[r] & \mathcal{K}^\bullet }

coming from Lf^* \to f^* on complexes, f_* \to Rf_* on complexes, and adjunction Lf^* \circ Rf_* \to \text{id} commutes in D(\mathcal{O}_\mathcal {C}).

Proof. We will use the existence of K-flat resolutions and K-injective resolutions, see Lemmas 21.17.11, 21.18.2, and 21.18.1 and the discussion above. Choose a quasi-isomorphism \mathcal{K}^\bullet \to \mathcal{I}^\bullet where \mathcal{I}^\bullet is K-injective as a complex of \mathcal{O}_\mathcal {C}-modules. Choose a quasi-isomorphism \mathcal{Q}^\bullet \to f_*\mathcal{I}^\bullet where \mathcal{Q}^\bullet is a K-flat complex of \mathcal{O}_\mathcal {D}-modules with flat terms. We can choose a K-flat complex of \mathcal{O}_\mathcal {D}-modules \mathcal{P}^\bullet with flat terms and a diagram of morphisms of complexes

\xymatrix{ \mathcal{P}^\bullet \ar[r] \ar[d] & f_*\mathcal{K}^\bullet \ar[d] \\ \mathcal{Q}^\bullet \ar[r] & f_*\mathcal{I}^\bullet }

commutative up to homotopy where the top horizontal arrow is a quasi-isomorphism. Namely, we can first choose such a diagram for some complex \mathcal{P}^\bullet because the quasi-isomorphisms form a multiplicative system in the homotopy category of complexes and then we can choose a resolution of \mathcal{P}^\bullet by a K-flat complex with flat terms. Taking pullbacks we obtain a diagram of morphisms of complexes

\xymatrix{ f^*\mathcal{P}^\bullet \ar[r] \ar[d] & f^*f_*\mathcal{K}^\bullet \ar[d] \ar[r] & \mathcal{K}^\bullet \ar[d] \\ f^*\mathcal{Q}^\bullet \ar[r] & f^*f_*\mathcal{I}^\bullet \ar[r] & \mathcal{I}^\bullet }

commutative up to homotopy. The outer rectangle witnesses the truth of the statement in the lemma. \square


Comments (1)

Comment #7138 by Hao Peng on

I think this lemma can be genralized to the case of two adjoint functor between Abelian categories. I can prove it using the universal properties of derived functors in the sence of Gelfand's book "methods in homological algebra", assuming both functor have enough acyclic objects. Not sure about the most genral case.


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.