Homology theories and Gorenstein dimensions for complexes
In this paper, the author establishes various results about Tate homology \(\widehat{\mathrm{Tor}}\), stable homology \(\widetilde{\mathrm{Tor}}\), and so-called unbounded homology \(\overline{\mathrm{Tor}}\) (which can all be seen as variants of the usual \(\mathrm{Tor}\) functor) for certain types of complexes of modules over a ring. For example, for suitably nice complexes \(M\) and \(N\) it is shown that the Tate homology \(\widehat{\mathrm{Tor}}_*(M,N)\) can be computed from Tate projective/flat resolutions and that the unbounded homology is balanced, i.e.~one has \(\overline{\mathrm{Tor}}_*(M,N) \cong \overline{\mathrm{Tor}}_*(N,M)\). The paper also contains some results that relate the Gorenstein projective/flat dimension of a complex to the Gorenstein projective/flat dimensions of the modules it consists of.
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