Gorenstein projective modules and Frobenius extensions
Abstract
We prove that for a Frobenius extension, if a module over the extension ring is Gorenstein projective, then its underlying module over the 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 $\mathbb{Z}\subset \mathbb{Z}G$). Moreover, for the Frobenius extension $R\subset A=R[x]/(x^2)$, we show that: a graded $A$-module is Gorenstein projective in $\mathrm{GrMod}(A)$, 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- July 2017
- DOI:
- 10.48550/arXiv.1707.05885
- arXiv:
- arXiv:1707.05885
- Bibcode:
- 2017arXiv170705885R
- Keywords:
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- Mathematics - K-Theory and Homology;
- 16G50;
- 13B02;
- 16W50
- E-Print:
- 15 pages. Comments are welcome. It has been accepted for publication in SCIENCE CHINA Mathematics