Gorenstein homological invariant properties under Frobenius extensions
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We prove that for a Frobenius extension, a module over the extension ring is Gorenstein projective if and only if its underlying module over the base ring is Gorenstein projective. For a separable Frobenius extension between Artin algebras, we obtain that the extension algebra is CM (Cohen-Macaulay)-finite (resp. CM-free) if and only if so is the base algebra. Furthermore, we prove that the reprensentation dimension of Artin algebras is invariant under separable Frobenius extensions.
KeywordsFrobenius extensions separable extensions Gorenstein projective modules reprensenation dimension
MSC(2010)13B02 16E10 16G10 16G50
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This work was supported by National Natural Science Foundation of China (Grant No. 11571329) and the Natural Science Foundation of Anhui Province (Grant No. 1708085MA01). The research was completed during the author’s visit at the University of Washington. He thanks Professor James Zhang for his hospitality. The author thanks Dr. Jie Li for pointing out an error in the manuscript. The author thanks the referees for the helpful comments and valuable suggestions.