A model structure approach to the Tate–Vogel cohomology

We study Tate–Vogel cohomology of complexes by applying the model structure induced by a complete hereditary cotorsion pair (A,B)(A,B) of modules. Vanishing of Tate–Vogel cohomology characterizes the finiteness of AA dimension and BB dimension of complexes defined by Yang and Ding [57]. Applications go in three directions. The first is to characterize when a left and right Noetherian ring is Gorenstein. The second is to obtain some criteria for the validity of the Finitistic Dimension Conjecture. The third is to investigate the relationships between flat dimension and Gorenstein flat dimension for complexes.