Equivariant Cleft Extensions and Singular Equivalences

Abstract: We study the equivariant lifting of cleft extensions of abelian categories and its impact on singularity categories. Specifically, we establish the necessary framework for lifting a cleft extension to a G-equivariant cleft extension. Furthermore, we prove that a restriction functor associated to a cleft extension induces a singular equivalence if and only if its equivariant counterpart does. As a concrete application, we demonstrate that the skew group ring of a $G$-equivariant $θ$-extension is isomorphic to a $\widehatθ$-extension of the base skew group ring, allowing us to lift singular equivalences for these structures.