Lemma 3.13 (0NWS) · ScalingStacks

0NWS

Lemma 3.13. Let $M\in\qc(X)$ , and let $N\in\qc(Y)\simeq\Mod_{A}$ . Then the natural projection map $f_{*}M\otimes N\rightarrow f_{*}(M\otimes f^{*}N)$ is an equivalence.

0NWT

Proof. Tensor products and pullbacks always preserve colimits, and in our setting $f_{*}$ is colimit preserving as well. Therefore for any $M$ the functors $f_{*}M\otimes(-)$ and $f_{*}(M\otimes f^{*}-)$ define colimit preserving endofunctors of $\Mod_{A}$ . Hence both functors are determined by their value on $A$ , and canonically take the value $f_{*}M$ . We find that the natural map is an equivalence. ∎

David Ben-Zvi, John Francis, David Nadler

Original source: arXiv:0805.0157v5