ScalingStacks

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. ∎

Proof. Since $g$ is affine and hence $g^{\prime}$ is as well, the fiber product $X\times_{A}\Spec B$ can be identified with the relative spectrum $\Spec_{X}C$ of a commutative algebra object $C\in{\rm Alg}(\qc(X))$, and $g^{\prime}_{*}$ induces an equivalence $\qc({X\times_{A}\Spec B})\simeq\Mod_{C}(\qc(X))$. Furthermore, the pullback $g^{\prime*}$ can be described as tensoring with $C$, and thus in particular $f^{\prime}_{*}g^{\prime*}M\simeq f_{*}(C\otimes M)$. However, the global sections functor takes fiber products to tensor products, so we can identify $C\simeq f^{*}g_{*}B$. Applying the previously established projection formula twice, we can now compute $f_{*}(f^{*}g_{*}B\otimes M)\simeq f_{*}M\otimes_{A}g_{*}B\simeq g^{*}f_{*}M\otimes_{B}B\simeq g^{*}f_{*}M$, completing the proof. ∎

Proof. Since $\qc(Y)\simeq\lim_{\it{Aff}/Y}\Mod_{B}$, the claim that $f_{+}M$ forms a quasi-coherent sheaf on $Y$ is equivalent to the claim that for any diagram of the form

$f_{+}M(g\circ h)$ is canonically equivalent to $h^{*}f_{+}M(g)$. Unraveling these formulas, by definition we have that $f_{+}M(g\circ h)=f^{\prime\prime}_{*}h^{\prime*}g^{\prime*}M$ and that $h^{*}f_{+}M(g)\simeq h^{*}f^{\prime}_{*}g^{\prime*}M$. By the previous lemma, these are equivalent by base change in the left hand square: $f^{\prime}:X\times_{Y}\Spec B\rightarrow\Spec B$ is perfect, and so $h^{*}f^{\prime}_{*}\simeq f^{\prime\prime}_{*}h^{\prime*}$. ∎

Proof. Take any $N\in\qc(Y)$ and $M\in\qc(X)$. First note that for any quasi-coherent sheaf $K\in\qc(Z)$, there is an equivalence $K\simeq\lim_{g\in\it{Aff}/Z}g_{*}g^{*}K$. Thus we calculate

To prove the lemma, it thus suffices to show that the natural map $M\rightarrow\lim g^{\prime}_{*}g^{\prime*}M$ is an equivalence.

This is a consequence of the stronger claim that the functor $\qc(X)\rightarrow\lim_{\it{Aff}/Y}\qc({X\times_{Y}B})$ is an equivalence. Since the functor $\qc(-)$ takes all colimits of stacks to limits, it therefore suffices to show that the natural map $X\rightarrow\lim_{\it{Aff}/Y}(X\times_{Y}B)$ is an equivalence. This limit can be calculated by picking an affine cover $U\rightarrow Y$, and realizing $Y$ as the geometric realization of the usual simplicial object $U_{*}\to Y$. Finally, since geometric realization commutes with fiber products we are done. ∎

Proof of Proposition 3.10. The first assertions were proved above. Since base change is local in the target, one can prove the final statement for an arbitrary $Y^{\prime}\rightarrow Y$ by choosing a cover of $Y^{\prime}$ by an affine $\Spec A\rightarrow Y^{\prime}$, thus reducing to the case which was proved above. ∎