The projection formula and standard adjunctions

The main functors in play for sheaves of modules are

• $f_{\ast }$ for a map $f:X\to Y$
• $f^{\ast }$ for $f:X\to Y$
• ${\otimes }_{{\mathcal{O}}_X}$ and
• $\mathcal{H}o{m}_{{\mathcal{O}}_X}$

We will also use the notation $$E^{\vee }:=\mathcal{H}o{m}_{{\mathcal{O}}_X}(E,{\mathcal{O}}_X)$$ for both the underived and derived duals over ${\mathcal{O}}_X$.

We record some back facts about them and we omit the details for the most part.

Lemma. We have an adjunction $f^{\ast }⊣f_{\ast }$.

Lemma. We also have an adjunction $${\otimes }_{{\mathcal{O}}_X}E⊣\mathcal{H}o{m}_{{\mathcal{O}}_X}(E,-)$$

Lemma. Tensor products commute with pullbacks $$f^{\ast }(E{\otimes }_{{\mathcal{O}}_X}F)\cong f^{\ast }E{\otimes }_{{\mathcal{O}}_X}{f}^{\ast }F$$

From these, we can derive some less immediate ones.

Lemma. We have an adjunction $$f^{\ast }(-{\otimes }_{{\mathcal{O}}_Y}F)⊣\mathcal{H}o{m}_{{\mathcal{O}}_Y}(F,f_{\ast }(-))$$

We also have a sheafy version of pull-push adjunction.

Lemma. There is a natural isomorphism $$\mathcal{H}o{m}_{{\mathcal{O}}_Y}(E,f_{\ast }F)\cong f_{\ast }\mathcal{H}o{m}_{{\mathcal{O}}_X}(f^{\ast }E,F)$$

One of the most useful relationships between these functors is one that is not true. Thanks to the map $$f^{\ast }(f_{\ast }E\otimes F)\cong f^{\ast }{f}_{\ast }E\otimes f^{\ast }F\to E\otimes f^{\ast }F$$ coming from the counit and adjunction we have a natural map $$f_{\ast }E\otimes F\to f_{\ast }(E\otimes f^{\ast }F)$$ However, this is not an isomorphism of quasi-coherent sheaves in general.

For what $E$ and $F$ is the map $$f_{\ast }E\otimes F\to f_{\ast }(E\otimes f^{\ast }F)$$ an isomorphism? Taking $F={\mathcal{O}}_Y$ works. Since isomorphism can be detected locally, it suffices that $F$ is locally isomorphic to ${\mathcal{O}}_Y$.

Moreover, if $$f_{\ast }E\otimes F_a\to f_{\ast }(E\otimes f^{\ast }{F}_a)$$ for some $a\in A$, then $$f_{\ast }E\otimes \underset{a\in A}{⨁}{F}_a\to f_{\ast }\left(E\otimes f^{\ast }\left(⨁F_a\right)\right)$$ will also be an isomorphism if $f_{\ast }$ commutes with coproducts which is true for quasi-projective schemes (and more generally).

Proposition. Let $f:X\to Y$ be a map of quasi-projective schemes. Then for any $E\in D(\mathrm{Qcoh}X)$ and $F\in D(\mathrm{Qcoh}Y)$, the natural map $$\mathbf{R}{f}_{\ast }E\stackrel{\mathbf{L}}{\otimes }F\to \mathbf{R}{f}_{\ast }\left(E\stackrel{\mathbf{L}}{\otimes }\mathbf{L}{f}^{\ast }F\right)$$ is a quasi-isomorphism.

The previous result is often called the projection formula.