We start with an introduction to singularity categories and equivalences between them (called „singular equivalences“ below). In particular, we recall known results about singular equivalences between commutative rings, which go back to works of Knörrer, Yang, Kawamata and a joint work with Karmazyn.
Then we explain a new singular equivalence between an affine surface and an affine threefold. This is the first (non-trivial) example of a singular equivalence involving rings of even and odd Krull dimension.
The only known non-trivial singular equivalences involving Gorenstein rings are due to Knörrer (using the equivalent notion of matrix factorizations). We offer a possible explanation for this by showing that Knörrer’s equivalences are the only singular equivalences between Gorenstein rings (of different Krull dimensions) that can be lifted to quasi-equivalences between the canonical dg enhancements of these singularity categories.