This blogpost is concerned with Deformation theory for a morphism in the derived category with fixed lift of the codomain, written together with Wendy Lowen, Shinnosuke Okawa, and Andrea Ricolfi. If you want to go straight to the details, you can directly start reading the paper, this blogpost is just to give a bit of background and history.

Origin story

Following a referee report and subsequent revision for Moduli spaces of semiorthogonal decompositions in families, we have performed an appendectomy on what was formerly Appendix A in that paper (whose v3 will be online on Monday, I'll describe those significant changes in a separate blogpost), turning it into a standalone paper. After all, the main result proved in this former appendix is of independent interest. However, it being buried in an appendix was not an ideal situation.

So now it is a standalone paper which also includes our original proof that semiorthogonal decompositions deform uniquely in smooth and proper families. The referee has suggested a more self-contained proof for this uniqueness, which will be featured in v3 of the original paper, but more on that in a later blogpost.

What's in the paper?

The paper establishes a deformation theory of morphisms, explaining when they lift to derived categories of deformations of abelian categories. The case of objects was worked out by Lowen in 2005. One important conclusion from the results for objects is a precise theorem that explains the heuristic that exceptional objects should lift uniquely along deformations.

To do the same lifting for morphisms, you have to say first what you precisely want to do with the objects that are the domain and codomain of the morphism. As the title suggests, we want to fix a lift of the codomain, and then we try to understand how the domain and morphism itself can be lifted.

The reason for this choice of setup is that it is exactly what is needed for our application, namely to understand the deformation theory of semiorthogonal decompositions. After all, a semiorthogonal decomposition can be described using a decomposition triangle of projection functors, with the middle term being the identity functor, encoded using the structure sheaf of the diagonal. We therefore know which lift of the codomain we want to take: the structure sheaf of the diagonal.

And thus, that is what we have done:

  • Theorem A gives the technical result for lifting morphisms, identifying the obstruction class and torsor structure for the set of lifts;
  • Corollary B gives the algebro-geometric version, which requires a restriction argument, because in algebraic geometry one uses tensor products for pullbacks, but those are a priori not compatible with using Grothendieck categories and injective objects;
  • Theorem C shows that semiorthogonal decompositions deform uniquely in smooth and proper families.

I hope that this abstract and general machinery can be useful for you too!