# Compact generation and Grothendieck duality

We saw last semester that compact generation of a triangulated category was a very useful property. Here we establish that $D(QcohX)$ is compactly generated for a quasi-projective schemes $X$ over an affine base. In addition, we provide some useful replacements for deriving common functors.

**Lemma**. Let $R$ be a $Z$-graded ring. Then $R(i)$ for $i∈Z$ is a set of compact generators for $D(GrModR)$.

##
**Proof**. (Expand to view)

One checks that $Hom(R(i)[j],C)≅H_{−j}(C)_{i}$ Thus, if $C$ is right orthogonal to all $R(i)$ it must be acyclic.

▢**Proposition**. If $R$ is finitely-generated in degree $1$ over $R_{0}$,
then $D(QcohProjR)$ is compactly generated.

##
**Proof**. (Expand to view)

Let $X=ProjR$. Recall that we [have](2022_01_11.md) a SOD $D(GrModR)=⟨D(TorsR),D(QcohX)⟩$ Given by $(−) $ and $Γ $. Thus, $Hom(O(i)[j],F)≅Hom(R(i)[j],Gamma (F))$ This vanishes for all $i,j∈Z$ if and only if $Γ (F)$ is acyclic. As $Γ (F) ≅F$ we see that $F$ is acyclic also.

▢**Lemma**. If $D(QcohX)$ is compactly-generated and $U$ is an
open subscheme, then so is $U$.

##
**Proof**. (Expand to view)

Let $l:U→X$ denote the inclusion. Then the counit $l_{∗}Rl_{∗}F→F$ is an isomorphism. Thus, if $C_{i}$ is a set of compact generators for $X$, then $l_{∗}C_{i}$ is for $U$.

▢**Corollary**. Let $X$ be quasi-projective over an affine base. Then any compact
object is quasi-isomorphic to a bounded complex of locally-free sheaves.

##
**Proof**. (Expand to view)

We have seen that $O_{X}(i)$ form a set of compact generators. As we saw [last semester](https://738.f21.matthewrobertballard.com/notes/2021_10_05/), the compact objects in $D(QcohX)$ are summands of iterated cones over maps to objects from the set of compact generators. Such a complex is bounded and has locally-free components.

▢We saw last time that $E↦E(−n−1)[n]$ is the Serre
functor on $P_{k}$. In fact, this is a consequence of a more general
fact known as *Grothendieck duality*. One formulation of Grothendieck duality
is that for a map $f:X→Y$ of schemes possesses a right adjoint
$f_{!}:D(QcohY→QcohX)$
to $Rf_{∗}$. This is not automatic in full generality. But it is
true with mild conditions on $X$ and $Y$.

For the projection \pi : \mathbb{P}^n_k \to \operatorname{Spec} k} we have $π_{!}O_{Speck}≅O(−n−1)[n]$

Originally proof of the existence of a right adjoint was arduous. However, [Neeman](https://www.ams.org/journals/jams/1996-9-01/S0894-0347-96-00174- 9/S0894-0347-96-00174-9.pdf) demonstrated how Brown Representability can be used to give a clean proof.

Thanks to Brown Representability, we know that for $Rf_{∗}$ to have a right adjoint is equivalent to the natural map $i∈I⨁ Rf_{∗}E_{i}→Rf_{∗}(⨁E_{i})$ being an isomorphism for all sums. This question is local on $Y$ so we reduce to asking -- when does taking cohomology commute with direct sums?

One useful fact: formation of the Čech complex for an affine cover where all intersections are affine commutes with direct sums.

**Proposition**. For any map $f:X→Y$ between quasi-projective schemes over
an affine base, there is a right adjoint $f_{!}$ to $Rf_{∗}$.

##
**Proof**. (Expand to view)

From the discussion above, we reduce to the checking the case where $Y$ is affine. Here we are computing global sections over a quasi-affine scheme which can be done using a Čech complex.

▢**Remark**. Neeman proves the existence of $f_{!}$ whenver $X$ and $Y$ are
quasi-compact and quasi-separated.

## Deriving tensor and sheaf Hom

We have implicitly used both the tensor product $E⊗_{O_{X}}F$ and sheaf Hom $Hom_{O_{X}}(E,F)$ of $O_{X}$-modules.

Recall that $E$ is *locally-free* if each $p∈X$ has a neighborhood $U$ with
$E∣_{U}≅O_{U}$
Since exactness can be checked locally, $E⊗_{O_{X}}−$ preserves
exactness of complexes of $O_{X}$-modules if $E$ is locally-free. Thus, if we
can replace any complex with one whose components are locally-free, we can use
these adapted objects to derive $⊗_{O_{X}}$.

**Proposition**. For any quasi-projective schemes $X$ and any chain complex $E$
of quasi-coherent sheaves, there is complex $P$ with locally-free components
and a quasi-isomorphism $P→E$.

##
**Proof**. (Expand to view)

It suffices to prove this for projective $X$. Any complex of quasi-coherent sheaves on $X$ is of the form $C$ for some complex of graded modules $C$. There are enough K-projectives in K(\operatarname{GrMod} R). Indeed, the twists of $R$ suffice. Thus, we get a $K$-projective $P$ and a quasi-isomorphism $P→C$. Applying $(−) $ gives the result.

▢Note that despite the notation we cannot find enough K-projective complexes in $D(QcohX)$.

Using this result, we take $E⊗L −:=P⊗−$ for some such $P$.

Similarly, we can derive $Hom(E,F)$ by replacing $E$ by a complex of locally-frees or $F$ by a K-injective. The choices are naturally quasi-isomorphic so we denote either by $RHom(E,F).$