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Relative Tor Functors for Level Modules with Respect to a Semidualizing Bimodule

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Abstract

Let R and S be rings and S C R a semidualizing bimodule. We investigate the relative Tor functors \(\text {Tor}_{i}^{\mathcal {M}\mathcal {L}_{C}}(-,-)\) defined via C-level resolutions, and these functors are exactly the relative Tor functors \(\text {Tor}_{i}^{\mathcal {M}\mathcal {F}_{C}}(-,-)\) defined by Salimi, Sather-Wagstaff, Tavasoli and Yassemi provided that S = R is a commutative Noetherian ring. Vanishing of these functors characterizes the finiteness of \(\mathcal {L}_{C}(S)\)-projective dimension. Applications go in two directions. The first is to characterize when every S-module has a monic (or epic) C-level precover (or preenvelope). The second is to give some criteria for the isomorphism \(\text {Tor}_{i}^{\mathcal {M}\mathcal {L}_{C}}(-,-)\cong \text {Tor}_{i}^{\mathcal {M}\mathcal {F}_{C}}(-,-)\) between the bifunctors.

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Authors and Affiliations

  1. School of Mathematics and Physics, Jiangsu University of Technology, Changzhou, 213001, China

    Jiangsheng Hu & Yuxian Geng

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  1. Jiangsheng Hu
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  2. Yuxian Geng
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Correspondence to Yuxian Geng.

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Presented by Jon F. Carlson.

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Hu, J., Geng, Y. Relative Tor Functors for Level Modules with Respect to a Semidualizing Bimodule. Algebr Represent Theor 19, 579–597 (2016). https://doi.org/10.1007/s10468-015-9589-9

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  • DOI: https://doi.org/10.1007/s10468-015-9589-9

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