Gorenstein dimensions of unbounded complexes under base change (Q2812474)

In the paper under review, the author studies transfer of homological properties along ring homomorphisms. More precisely, let \(R\) and \(S\) be commutative Noetherian rings and let \(\varphi:R\rightarrow S\) be a ring homomorphism. If \(X\) is an \(R\)-complex and \(U\) is an \(S\)-complex with finite projective (or injective) dimension, then the relations of Gorenstein projective (or Gorenstein injective) dimensions of unbounded complexes between \(U\bigotimes^L_R X\) (or \(\mathbf{R}\mathrm{Hom}_R(X,U)\)) and \(X\) are studied (Here \(-\bigotimes^L_R -\) and \(\mathbf{R}\mathrm{Hom}_R(-,-)\) denote the derived \(\mathrm{Hom}\) and derived tensor product of complexes). Also, it is shown that if \(\varphi\) is a faithfully flat and module-finite ring homomorphism and \(\mathrm{dim} S\) is finite then for every \(R\)-complex \(X\), the Gorenstein projective dimension of \(S\bigotimes^L_R X\) is equal to Gorenstein projective dimension of \(X\).