Gorenstein projective objects over cleft extensions

In this paper we introduce compatible cleft extensions of abelian categories, and we prove that if $(\mathcal{B},\mathcal{A},e,i,l)$ is a compatible cleft extension, then both the functor $l$ and the left adjoint of $i$ preserve Gorenstein projective objects. Moreover, we give some necessary conditions for an object of $\mathcal{A}$ to be Gorenstein projective, and we show that these necessary conditions are also sufficient in some special case. As applications, we unify some known results on the description of Gorenstein projective modules over triangular matrix rings, Morita context rings with zero homomorphisms and $\theta$-extensions.