Generalized tilting modules and Frobenius extensions

Dongxing Fu; Xiaowei Xu; Zhibing Zhao; Dongxing Fu; Xiaowei Xu; Zhibing Zhao

doi:10.3934/era.2022169

Electronic Research Archive

2022, Volume 30, Issue 9: 3337-3350. doi: 10.3934/era.2022169

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Research article

Generalized tilting modules and Frobenius extensions

1.
Center for Pure Mathematics, School of Mathematical Sciences, Anhui University, Hefei 230601, China
2.
School of Mathematical Sciences, Jilin University, Changchun 130012, China

Received: 12 April 2022 Revised: 12 June 2022 Accepted: 21 June 2022 Published: 13 July 2022

Let $ A/S $ be a ring extension with $ S $ commutative. We prove that $ \omega{\otimes}_SA_A $ is a generalized tilting module if $ \omega_S $ is a generalized tilting module. In this case, we obtain that $ ^\bot \omega $-resol.dim$ _S(M) $ and $ ^\bot (\omega\otimes_SA) $-resol.dim$ _A(M) $ are identical for any $ A $-module $ M $. As an application, we show that $ S $ satisfies gorenstein symmetric Conjecture if and only if so does $ A $. Furthermore, we introduce the concept of $ ^\bot\omega $-Gorenstein projective modules, and we obtain that the relative Gorenstein projectivity is invariant under Frobenius extensions.
- generalized tilting modules,
- Frobenius extensions,
- separable extensions,
- $ ^\bot\omega $-dimensions,
- $ ^\bot\omega $-Gorenstein projective modules
Citation: Dongxing Fu, Xiaowei Xu, Zhibing Zhao. Generalized tilting modules and Frobenius extensions[J]. Electronic Research Archive, 2022, 30(9): 3337-3350. doi: 10.3934/era.2022169

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Abstract

Let $ A/S $ be a ring extension with $ S $ commutative. We prove that $ \omega{\otimes}_SA_A $ is a generalized tilting module if $ \omega_S $ is a generalized tilting module. In this case, we obtain that $ ^\bot \omega $-resol.dim$ _S(M) $ and $ ^\bot (\omega\otimes_SA) $-resol.dim$ _A(M) $ are identical for any $ A $-module $ M $. As an application, we show that $ S $ satisfies gorenstein symmetric Conjecture if and only if so does $ A $. Furthermore, we introduce the concept of $ ^\bot\omega $-Gorenstein projective modules, and we obtain that the relative Gorenstein projectivity is invariant under Frobenius extensions.

References

[1]	T. Wakamatsu, Tilting modules and Auslander's Gorenstein property, J. Algebra, 275 (2004), 3–39. https://doi.org/10.1016/j.jalgebra.2003.12.008 doi: 10.1016/j.jalgebra.2003.12.008
[2]	F. Mantese, I. Reiten, Wakamatsu tilting modules, J. Algebra, 278 (2004), 532–552. https://doi.org/10.1016/j.jalgebra.2004.03.023 doi: 10.1016/j.jalgebra.2004.03.023
[3]	A. Beligiannis, I. Reiten, Homological and Homotopical Aspects of Torsion Theories, Memoirs of the American Mathematical Society, 2007. https://doi.org/10.1090/memo/0883
[4]	Z. Y. Huang, Generalized tilting modules with finite injective dimension, J. Algebra, 31 (2007), 619–634. https://doi.org/10.1016/j.jalgebra.2006.11.025 doi: 10.1016/j.jalgebra.2006.11.025
[5]	Z. Y. Huang, Wakamatsu tilting modules, U-dominant dimension and $k$-Gorenstein modules, in Abelian Groups, Rings, Modules, and Homological Algebra, Chapman Hall/CRC, (2006), 183–202.
[6]	Z. B. Zhao, X. N. Du, Generalized tilting modules with finite injective dimension, Adv. Math. (China), 43 (2014), 844–850.
[7]	F. Kasch, Grundlagen einer theorie der Frobenius-Erweiterungen, Math. Ann., 127 (1954), 453–474. https://doi.org/10.1007/BF01361137 doi: 10.1007/BF01361137
[8]	J. Kock, Frobenius Algebras and 2D Topological Quantum Field Theories, Cambridge University Press, 2004. https://doi.org/10.1017/CBO9780511615443
[9]	L. Kadison, The jones polynomial and certain separable Frobenius extensions, J. Algebra, 186 (1996), 461–475. https://doi.org/10.1006/jabr.1996.0383 doi: 10.1006/jabr.1996.0383
[10]	L. Kadison, New Example of Frobenius Extension, University Lecture Series, 1999. https://doi.org/10.1090/ulect/014
[11]	C. C. Xi, S. J. Yin, Cellularity of centrosymmetric martrix algebras and Frobenius extensions, Linear Algebra Appl., 590 (2020), 317–329. https://doi.org/10.1016/j.laa.2020.01.002 doi: 10.1016/j.laa.2020.01.002
[12]	D. Fischman, S. Montgomery, H. J. Schneider, Frobenius extensions of subalgebras of Hopf algebras, Trans. Amer. Math. Soc., 349 (1997), 4857–4895. https://doi.org/10.1090/S0002-9947-97-01814-X doi: 10.1090/S0002-9947-97-01814-X
[13]	H. J. Schneider, Normal basis and transitivity of crossed products for Hopf algebras, J. Algebra, 151 (1992), 289–312. https://doi.org/10.1016/0021-8693(92)90034-J doi: 10.1016/0021-8693(92)90034-J
[14]	K. Hirata, K. Sugano, On semisimple extensions and separable extensions over noncommutative rings, J. Math. Soc. Japan., 18 (1966), 360–373. https://doi.org/10.2969/jmsj/01840360 doi: 10.2969/jmsj/01840360
[15]	K. Sugano, Separable extensions and Frobenius extensions, Osaka J. Math., 7, (1970), 291–299.
[16]	X. W. Chen, Totally reflexive extensions and modules, J. Algebra, 379 (2013), 322–332. https://doi.org/10.1016/j.jalgebra.2013.01.014 doi: 10.1016/j.jalgebra.2013.01.014
[17]	Z. Y. Huang, J. X. Sun, Invariant properties of represenations under excellent extensions, J. Algebra, 358 (2012), 87–101. https://doi.org/10.1016/j.jalgebra.2012.03.004 doi: 10.1016/j.jalgebra.2012.03.004
[18]	W. Ren, Gorenstein projective and injective dimensions over Frobenius extensions, Commun. Algebra, 46 (2018), 1–7. https://doi.org/10.1080/00927872.2017.1355714 doi: 10.1080/00927872.2017.1355714
[19]	C. C. Xi, Frobenius bimodules and flat-dominant dimensions, Sci. China Math., 64 (2020), 33–44. https://doi.org/10.1007/s11425-018-9519-2 doi: 10.1007/s11425-018-9519-2
[20]	Z. B. Zhao, Gorenstein homological invariant properties under Frobenius extensions, Sci. China Math., 62 (2019), 2487–2496. https://doi.org/10.1007/s11425-018-9432-2 doi: 10.1007/s11425-018-9432-2
[21]	R. S. Pierce, Projective modules over Artinian algebras, in Graduate Texts in Mathematics, Springer, (1982), 88–107. https://doi.org/10.1007/978-1-4757-0163-0_6
[22]	W. Ren, Gorenstein projective mdoules and Frobenius extensions, Sci. China Math., 61 (2018), 1175–1186. https://doi.org/10.1007/s11425-017-9138-y doi: 10.1007/s11425-017-9138-y
[23]	M. Auslander, M. Bridger, Stable Module Theory, Memoirs of the American Mathematical Society, 1969. https://doi.org/10.1090/memo/0094
[24]	K. Morita, Adojint pairs of functors and Frobenius extension, Sci. Rep. T. K. D. Sect. A., 9 (1965), 40–71.
[25]	T. Wakamatsu, On modules with trivial self-extensions, J. Algebra, 114 (1988), 106–114. https://doi.org/10.1016/0021-8693(88)90215-3 doi: 10.1016/0021-8693(88)90215-3
[26]	J. Wang, X. W. Xu, Z. B. Zhao, $\mathfrak{X}$-Gorenstein projective dimensions, preprint, arXiv: 1801.09127.
[27]	T. Nakamaya, T. Tsuzuku, On frobenius extension I, Nagoya. Math. J., (1960), 89–110. https://doi.org/10.1017/S0027763000002075 doi: 10.1017/S0027763000002075

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