# When is D(Qcoh X) compactly generated by a single perfect complex?

Because I just keep on forgetting the conditions under which it is known when $\mathbf{D}(\mathop{\mathrm{Qcoh}}X)$ is generated by a single perfect complex (where $X$ is allowed to be an algebraic stack) I decided to write a quick summary post about the state of the art.

I phrase the question in a specific way: for the sake of this blogpost I am only interested in $\mathbf{D}(\mathop{\mathrm{Qcoh}}X)$. One can also consider $\mathbf{D}_{\mathrm{qc}}(\mathcal{O}_X)$, the derived category of $\mathcal{O}_X$-modules whose cohomologies are all quasicoherent, which under suitable hypotheses is equivalent to the derived category of quasicoherent sheaves via the natural functor $\mathbf{D}(\mathop{\mathrm{Qcoh}}X)\to\mathbf{D}_{\mathrm{qc}}(\mathcal{O}_X)$. But for the sake of this post, $\mathbf{D}(\mathop{\mathrm{Qcoh}}X)$ is the main player.

## When $X$ is a scheme

In this case the result holds if $X$ is **quasicompact** and **has affine diagonal**.

It is already classical, and due to Neeman (1996), although for the statement that there is a single perfect complex that acts as a compact generator one has to refer to Bondal--Van den Bergh (2003).

In the Stacks project it is given in tag 09IS. Observe that the statement is for $\mathbf{D}_{\mathrm{qc}}(\mathcal{O}_X)$, but assuming quasicompact and having affine diagonal these categories turn out to be equivalent, which was proven in Bökstedt--Neeman (1993) for the quasicompact separated case, and in tag 08DB in the Stacks project assuming affine diagonal.

## When $X$ is an algebraic space

In this case the result holds if $X$ is **quasicompact** and **has affine diagonal**.

This result is not published, but is due to Van den Bergh, from 2005.

In the Stacks project it is given in tag 09IY. Observe that the statement is for $\mathbf{D}_{\mathrm{qc}}(\mathcal{O}_X)$, but assuming quasicompact and affine diagonal these turn out to be equivalent, which is given in tag 08H1.

## When $X$ is an algebraic stack

In this case the result holds if $X$ is **quasicompact** and **has affine ^{1} and quasi-finite diagonal**.

In particular, if I'm not mistaken the result applies to quasicompact Deligne--Mumford stacks having affine diagonal: by definition these have an unramified diagonal, and unramified implies quasi-finite (everything being quasicompact).

The result in this case is again a combination of knowing when the natural functor is an equivalence, and when the codomain is compactly generated by a single perfect complex. The results in the literature are:

*$\Psi_X$ is an equivalence*if $X$ is quasicompact and has affine diagonal, provided (!) $\mathbf{D}_{\mathrm{qc}}(\mathcal{O}_X)$ is compactly generated. So compact generation of this category becomes now important for two reasons. This is theorem 1.2 in Hall--Neeman--Rydh.*$\mathbf{D}_{\mathrm{qc}}(\mathcal{O}_X)$ is compactly generated by a single perfect complex*if $X$ is quasicompact and has quasifinite separated diagonal. This is theorem B in Hall--Rydh.

There are more general situations in which $\Psi_X$ is an equivalence, the harder question is to have a single compact generator, and one cannot expect such a thing in general for an algebraic stack not having suitable finiteness properties, as one can conclude by considering $\mathrm{B}\mathbb{G}_{\mathrm{m}}$ which needs a countable set of compact generators.

The main conclusion of this post is probably that I should dedicate a blogpost (or even several) to the wonderful properties of diagonals.

^{1} I assume here that affine implies separated for algebraic stacks, a statement I couldn't find in the Stacks project.