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Gorenstein projective modules and Frobenius extensions

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Abstract

We prove that for a Frobenius extension, if a module over the extension ring is Gorenstein projective, then its underlying module over the base ring is Gorenstein projective; the converse holds if the Frobenius extension is either left-Gorenstein or separable (e.g., the integral group ring extension ℤ ⊂ ℤG). Moreover, for the Frobenius extension R ⊂ A = R[x]=(x2), we show that: a graded A-module is Gorenstein projective in GrMod(A), if and only if its ungraded A-module is Gorenstein projective, if and only if its underlying R-module is Gorenstein projective. It immediately follows that an R-complex is Gorenstein projective if and only if all its items are Gorenstein projective R-modules.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 11401476) and China Postdoctoral Science Foundation (Grant No. 2016M591592). The author thanks Professor Xiao-Wu Chen for sharing his thoughts on this topic. This research was completed when the author was a postdoctor at Fudan University supervised by Professor Quan-Shui Wu. The author thanks the referees for helpful comments and suggestions.

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Authors and Affiliations

  1. School of Mathematical Sciences, Chongqing Normal University, Chongqing, 401331, China

    Wei Ren

  2. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Wei Ren

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  1. Wei Ren
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Correspondence to Wei Ren.

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Ren, W. Gorenstein projective modules and Frobenius extensions. Sci. China Math. 61, 1175–1186 (2018). https://doi.org/10.1007/s11425-017-9138-y

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