The projection formula and standard adjunctions

The main functors in play for sheaves of modules are

  • for a map
  • for
  • and

We will also use the notation for both the underived and derived duals over .

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

Lemma. We have an adjunction .

Lemma. We also have an adjunction

Lemma. Tensor products commute with pullbacks

From these, we can derive some less immediate ones.

Lemma. We have an adjunction

Proof. (Expand to view)

We have natural isomorphisms


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

Lemma. There is a natural isomorphism

Proof. (Expand to view)

We have 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 coming from the counit and adjunction we have a natural map However, this is not an isomorphism of quasi-coherent sheaves in general.

For what and is the map an isomorphism? Taking works. Since isomorphism can be detected locally, it suffices that is locally isomorphic to .

Moreover, if for some , then will also be an isomorphism if commutes with coproducts which is true for quasi-projective schemes (and more generally).

Proposition. Let be a map of quasi-projective schemes. Then for any and , the natural map is a quasi-isomorphism.

Proof. (Expand to view)

We can replace with a -injective complex and 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.