The singularity category $D_{sg}(R)$ of a commutative Noetherian ring $R$ is a triangulated category which measures singularity of $R$.
Two commutative Noetherian rings $R$ and $S$ are said to be singularly equivalent if their singularity categories are equivalent as triangulated categories.
Singular equivalence have deeply been studied in non-commutative setting and various examples are known.
However, in commutative setting, only one non trivial example is known, which is so-called Knorrer's periodicity.
The aim of this talk is to give a necessary condition for singular equivalence by using singular loci and as an application, I will show that Knorrer's periodicity fails for non-regular local rings.
The key tool to prove our main result is the support theory for triangulated categories without tensor structure.