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 $D(\mathrm{Qcoh}X)$ 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 $\mathbb{Z}$-graded ring. Then $R(i)$ for $i\in \mathbb{Z}$ is a set of compact generators for $D(\mathrm{GrMod}R)$.

Proposition. If $R$ is finitely-generated in degree $1$ over $R_0$, then $D(\mathrm{Qcoh}\mathrm{Proj}R)$ is compactly generated.

Lemma. If $D(\mathrm{Qcoh}X)$ is compactly-generated and $U$ is an open subscheme, then so is $U$.

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.

We saw last time that $E\mapsto E(-n-1)[n]$ is the Serre functor on ${\mathbb{P}}_k^n$. 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 $f:X\to Y$ of schemes possesses a right adjoint $$f^{!}:D(\mathrm{Qcoh}Y\to \mathrm{Qcoh}X)$$ to $\mathbf{R}{f}_{\ast }$. 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 $${\pi }^{!}{\mathcal{O}}_{\mathrm{Spec}k}\cong \mathcal{O}(-n-1)[n]$$

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 $\mathbf{R}{f}_{\ast }$ to have a right adjoint is equivalent to the natural map $$\underset{i\in I}{⨁}\mathbf{R}{f}_{\ast }{E}_i\to \mathbf{R}{f}_{\ast }\left(⨁E_i\right)$$ 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 $f:X\to Y$ between quasi-projective schemes over an affine base, there is a right adjoint $f^{!}$ to $\mathbf{R}{f}_{\ast }$.

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 $$E{\otimes }_{{\mathcal{O}}_X}F$$ and sheaf Hom $$\mathcal{H}o{m}_{{\mathcal{O}}_X}(E,F)$$ of ${\mathcal{O}}_X$-modules.

Recall that $E$ is locally-free if each $p\in X$ has a neighborhood $U$ with $$E{|}_U\cong {\mathcal{O}}_U^{\oplus I}$$ Since exactness can be checked locally, $E{\otimes }_{{\mathcal{O}}_X}-$ preserves exactness of complexes of ${\mathcal{O}}_X$-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 ${\otimes }_{{\mathcal{O}}_X}$.

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 $P\to E$.

Note that despite the notation we cannot find enough K-projective complexes in $D(\mathrm{Qcoh}X)$.

Using this result, we take $$E\stackrel{\mathbf{L}}{\otimes }-:=P\otimes -$$ for some such $P$.

Similarly, we can derive $\mathcal{H}om(E,F)$ 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 $$\mathbf{R}\mathcal{H}om(E,F).$$