Abstract
Let C be a semidualizing module over a commutative noetherian ring R. We exhibit an isomorphism \(\operatorname{Tor}^{{\mathcal{F}_C}\mathcal{M}}_{i}(-,-) \cong \operatorname{Tor}^{{\mathcal{P}_C}\mathcal{M}}_{i}(-,-)\) between the bifunctors defined via C-flat and C-projective resolutions. We show how the vanishing of these functors characterizes the finiteness of \({{\mathcal{F}_C}\text{-}\operatorname{pd}}\), and use this to give a relation between the \({{\mathcal{F}_C}\text{-}\operatorname{pd}}\) of a module and of a pure submodule. On the other hand, we show that other isomorphisms force C to be trivial.
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This material is based on work supported by North Dakota EPSCoR and National Science Foundation Grant EPS-0814442. The research of Siamak Yassemi was in part supported by a grant from IPM (No. 91130214).
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Salimi, M., Sather-Wagstaff, S., Tavasoli, E. et al. Relative Tor Functors with Respect to a Semidualizing Module. Algebr Represent Theor 17, 103–120 (2014). https://doi.org/10.1007/s10468-012-9389-4
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DOI: https://doi.org/10.1007/s10468-012-9389-4