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Homological aspects of the adjoint cotranspose

Volume 150 / 2017

Xi Tang, Zhaoyong Huang Colloquium Mathematicum 150 (2017), 293-311 MSC: 18G25, 16E05, 16E10. DOI: 10.4064/cm7121-12-2016 Published online: 29 September 2017

Abstract

Let $R$ and $S$ be rings and $_R\omega_S$ a semidualizing bimodule. We introduce and study the adjoint cotransposes of modules and adjoint $n$-$\omega$-cotorsionfree modules. We show that the Auslander class with respect to $_R\omega_S$ is the intersection of the class of adjoint $\infty$-$\omega$-cotorsionfree modules and the right $\operatorname{Tor}$-orthogonal class of $\omega_S$. As a consequence, the classes of adjoint $\infty$-$\omega$-cotorsionfree modules and of $\infty$-$\omega$-cotorsionfree modules are equivalent under Foxby equivalence if and only if they coincide with the Auslander and Bass classes with respect to $\omega$ respectively. Moreover, we give some equivalent characterizations when the left and right projective dimensions of $_R\omega_S$ are finite in terms of the properties of (adjoint) $\infty$-$\omega$-cotorsionfree modules.

Authors

  • Xi TangCollege of Science
    Guilin University of Technology
    541004 Guilin, Guangxi, P.R. China
    e-mail
  • Zhaoyong HuangDepartment of Mathematics
    Nanjing University
    210093 Nanjing, Jiangsu, P.R. China
    e-mail

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