# The projection formula and standard adjunctions

The main functors in play for sheaves of modules are

- $f_{∗}$ for a map $f:X→Y$
- $f_{∗}$ for $f:X→Y$
- $⊗_{O_{X}}$ and
- $Hom_{O_{X}}$

We will also use the notation $E_{∨}:=Hom_{O_{X}}(E,O_{X})$ for both the underived and derived duals over $O_{X}$.

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

**Lemma**. We have an adjunction $f_{∗}⊣f_{∗}$.

**Lemma**. We also have an adjunction
$⊗_{O_{X}}E⊣Hom_{O_{X}}(E,−)$

**Lemma**. Tensor products commute with pullbacks
$f_{∗}(E⊗_{O_{X}}F)≅f_{∗}E⊗_{O_{X}}f_{∗}F$

From these, we can derive some less immediate ones.

**Lemma**. We have an adjunction
$f_{∗}(−⊗_{O_{Y}}F)⊣Hom_{O_{Y}}(F,f_{∗}(−))$

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**Proof**. (Expand to view)

We have natural isomorphisms $Hom_{X}(f_{∗}(E⊗F),G) ≅Hom_{Y}(E⊗F,f_{∗}G)≅Hom_{Y}(E,Hom(F,f_{∗}G)) $

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

**Lemma**. There is a natural isomorphism
$Hom_{O_{Y}}(E,f_{∗}F)≅f_{∗}Hom_{O_{X}}(f_{∗}E,F)$

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**Proof**. (Expand to view)

We have $Hom_{X}(f_{∗}E⊗f_{∗}F,G) ≅Hom_{X}(f_{∗}E,Hom(f_{∗}F,G)≅Hom_{Y}(E,f_{∗}Hom_{O_{X}}(f_{∗}F,G))) $ From the previous lemma, we have two right adjoints. Thus, they must be isomorphic.

▢One of the most useful relationships between these functors is one that is not true.
Thanks to the map
$f_{∗}(f_{∗}E⊗F)≅f_{∗}f_{∗}E⊗f_{∗}F→E⊗f_{∗}F$
coming from the counit and adjunction we have a natural map
$f_{∗}E⊗F→f_{∗}(E⊗f_{∗}F)$
However, this is *not* an isomorphism of quasi-coherent sheaves in general.

For what $E$ and $F$ is the map $f_{∗}E⊗F→f_{∗}(E⊗f_{∗}F)$ an isomorphism? Taking $F=O_{Y}$ works. Since isomorphism can be detected locally, it suffices that $F$ is locally isomorphic to $O_{Y}$.

Moreover, if $f_{∗}E⊗F_{a}→f_{∗}(E⊗f_{∗}F_{a})$ for some $a∈A$, then $f_{∗}E⊗a∈A⨁ F_{a}→f_{∗}(E⊗f_{∗}(⨁F_{a}))$ will also be an isomorphism if $f_{∗}$ commutes with coproducts which is true for quasi-projective schemes (and more generally).

**Proposition**. Let $f:X→Y$ be a map of quasi-projective schemes. Then for
any $E∈D(QcohX)$ and $F∈D(QcohY)$, the natural map
$Rf_{∗}E⊗L F→Rf_{∗}(E⊗L Lf_{∗}F)$
is a quasi-isomorphism.

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**Proof**. (Expand to view)

We can replace $E$ with a $K$-injective complex and $F$ with complexe of locally-free sheaves. Then, from the discussion, above we know that the induced map on the components is an isomorphism.

▢The previous result is often called the *projection formula*.