Before I had the habit of writing a blogpost discussing a new preprint, Shinnosuke Okawa, Andrea Ricolfi, and I (together with Wendy Lowen for the appendix) uploaded Moduli spaces of semiorthogonal decompositions in families. Shortly afterwards, the pandemic started, and we neglected to do anything with the paper. At some point we also discovered that there was an issue with the proof of the result in Subsection 8.3.3 (of v1), showing that the moduli space of non-trivial semiorthogonal decompositions is an open and closed algebraic subspace.

Over the years, we tried fixing this issue, but to no avail. So, now we have updated the paper, and I feel that it is appropriate to write a blogpost about it, because we

  • stated a (what we believe to be interesting) conjecture on smooth and proper dg algebras (see below)
  • rewritten the relevant part, making it clear which parts are conjectural
  • revised the rest of the paper not affected by this, improving the exposition wherever we could

We will also update the companion paper accordingly: whilst the geometric results are completely fine, the categorical results are conjectural.

A conjecture

As mentioned above, only the construction of the moduli space of non-trivial semiorthogonal decompositions is affected. We still hope that this can be fixed, but at the moment, we cannot do this, and have to turn this into a conjecture.

The following stronger conjecture on dg algebras seems to be out of reach of us (and some people we talked to) at the moment. Can you settle it?

Conjecture Let $R$ be a discrete valuation ring, and let $K$ be its fraction field. Let $\mathcal{A}$ be a dg algebra, smooth and proper over $R$. If $\mathcal{A}\otimes_RK$ is quasi-isomorphic to 0, then $\mathcal{A}$ is too.

Maybe it is too optimistic, and someone can find a counterexample. As explained in Section 8.4 of v2, for us it would suffice that a weaker conjecture holds. But for the purpose of this blogpost, I went for the most intriguging formulation.