Special homological dimensions and Intersection Theorem

Let $(R,\text{fm})$ be commutative Noetherian local ring. It is shown that $R$ is Cohen–Macaulay ring if there exists a Cohen–Macaulay finite (i.e. finitely generated) $R$–module with finite upper Gorenstein dimension. In addition, we show that, in the Intersection Theorem, projective dimension can be replaced by quasi–projective dimension.