Hello,
I wonder if the techniques introduced in Neemans paper:
"The Grothendieck duality theorem via Bousfield's techniques and Brown representability "
can be used to establish Verdier duality. More precisely:
Consider the unbounded, derived category $D(M)$ of $\mathbb{Q}$ vector spaces on a compact complex manifold $M$ . I would like to show that $Rf_!$ has a right adjoint.
In order to use Brown representability one has to show that $D(M)$ is compactly generated.
i.e. there exists a set of objects $c_i$ that commutes with direct sums:
$$Hom(c_i,\bigoplus x_j)=\bigoplus Hom(c_i,x_j)$$
and generates $D(M)$:
$$\forall c_i Hom(c_i,x)=0 \Rightarrow x=0$$
My problem is that i can't find such a set of generators. I first tried shifts of
$$i_*\mathbb{Q}$$ where $i$ is the inclusion of an open subset. However these do neither commute with coproducts nor are they generators (they can not see sheaves without global sections).
My second try was shifts of
$$i_!\mathbb{Q}$$
these are generators, but again they do not seem to respect coproducts.
Can someone give a set of compact generators? Or is this approach to Verdier duality doomed anyway?
Best Answer
The category of sheaves of $\mathbb{Q}$ vector spaces on $M$ is a Grothendieck abelian category. It follows that the derived category of such, $D(M)$ in your notation, is a well generated triangulated category. A proof of this can be found in Neeman's paper "On The Derived Category of Sheaves on a Manifold". In particular $D(M)$ satisfies Brown representability.
I thought I'd also comment on the proof. The point is that by a result of Alonso, Jeremías and Souto one can extend the Gabriel-Popescu theorem to the level of derived categories. This realizes the derived category of any Grothendieck abelian category as a localization of the derived category of R-modules for some ring R. The kernel of this localization can be generated by a set of objects and so general nonsense gives the desired generating set of $\alpha$-small objects for some regular cardinal $\alpha$ for the localization.