Vanishing of Tate homology -- an application of stable homology for complexes (Q327288)

Let \(R\) be an associative algebra over a commutative ring and let \(M\) be a complex over the opposite ring \(R^\circ\) of \(R\). In this paper, the authors find several sufficient conditions for the vanishing of Tate homology. For instance, they show that for any bounded above \(R\)-complex \(N\) and every \(i\in \mathbb{Z}\), one has \(\widehat{\mathrm{Tor}}^R_i (M,N) = 0\) if one of the following conditions holds: (1)\(R\) is a right noetherian ring, Gorenstein projective dimension of \(M\) is finite, \(M\) is homologically finite and homologically bounded, and injective dimension of \(N\) is finite. (2)\(R\) is a left coherent ring over which each flat \(R^\circ\)-module has finite projective dimension and Gorenstein projective dimension of \(M\) is finite, and injective dimension of \(N\) is finite.