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 affine1 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:

1. $\Psi_X$ 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.
2. $\mathbf{D}_{\mathrm{qc}}(\mathcal{O}_X)$ 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.