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GORENSTEIN PROJECTIVE OBJECTS IN FUNCTOR CATEGORIES

  • SONDRE KVAMME (a1)
Abstract

Let $k$ be a commutative ring, let ${\mathcal{C}}$ be a small, $k$ -linear, Hom-finite, locally bounded category, and let ${\mathcal{B}}$ be a $k$ -linear abelian category. We construct a Frobenius exact subcategory ${\mathcal{G}}{\mathcal{P}}({\mathcal{G}}{\mathcal{P}}_{P}({\mathcal{B}}^{{\mathcal{C}}}))$ of the functor category ${\mathcal{B}}^{{\mathcal{C}}}$ , and we show that it is a subcategory of the Gorenstein projective objects ${\mathcal{G}}{\mathcal{P}}({\mathcal{B}}^{{\mathcal{C}}})$ in ${\mathcal{B}}^{{\mathcal{C}}}$ . Furthermore, we obtain criteria for when ${\mathcal{G}}{\mathcal{P}}({\mathcal{G}}{\mathcal{P}}_{P}({\mathcal{B}}^{{\mathcal{C}}}))={\mathcal{G}}{\mathcal{P}}({\mathcal{B}}^{{\mathcal{C}}})$ . We show in examples that this can be used to compute ${\mathcal{G}}{\mathcal{P}}({\mathcal{B}}^{{\mathcal{C}}})$ explicitly.

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This is part of the authors PhD thesis. The author thanks Jan Geuenich and Julian Külshammer for helpful comments on a previous version of this paper, and the anonymous referee for useful suggestions and comments. The work was made possible by the funding provided by the Bonn International Graduate School in Mathematics.

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[1] Auslander, M. and Bridger, M., Stable Module Theory, Memoirs of the American Mathematical Society 94 , American Mathematical Society, Providence, RI, 1969.
[2] Auslander, M. and Buchweitz, R.-O., The homological theory of maximal Cohen-Macaulay approximations , Mém. Soc. Math. Fr. (N.S.) 38 (1989), 537; Colloque en l’honneur de Pierre Samuel (Orsay, 1987).
[3] Avramov, L. L. and Martsinkovsky, A., Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension , Proc. Lond. Math. Soc. (3) 85(2) (2002), 393440.
[4] Barr, M. and Beck, J., “ Homology and standard constructions ”, in Sem. on Triples and Categorical Homology Theory (ETH, Zürich, 1966/67), Springer, Berlin, 1969, 245335.
[5] Beligiannis, A., The homological theory of contravariantly finite subcategories: Auslander-Buchweitz contexts, Gorenstein categories and (co-)stabilization , Comm. Algebra 28(10) (2000), 45474596.
[6] Beligiannis, A., Cohen-Macaulay modules, (co)torsion pairs and virtually Gorenstein algebras , J. Algebra 288(1) (2005), 137211.
[7] Beligiannis, A., On algebras of finite Cohen-Macaulay type , Adv. Math. 226(2) (2011), 19732019.
[8] Beligiannis, A. and Krause, H., Thick subcategories and virtually Gorenstein algebras , Illinois J. Math. 52(2) (2008), 551562.
[9] Beligiannis, A. and Reiten, I., Homological and homotopical aspects of torsion theories , Mem. Amer. Math. Soc. 188(883) (2007), viii+207.
[10] Bennis, D. and Mahdou, N., Strongly Gorenstein projective, injective, and flat modules , J. Pure Appl. Algebra 210(2) (2007), 437445.
[11] Bühler, T., Exact categories , Expo. Math. 28(1) (2010), 169.
[12] Dell’Ambrogio, I., Stevenson, G. and Stovicek, J., Gorenstein homological algebra and universal coefficient theorems , Math. Z. 287(3–4) (2017), 11091155.
[13] Enochs, E., Estrada, S. and Garcia-Rozas, J. R., Gorenstein categories and Tate cohomology on projective schemes , Math. Nachr. 281(4) (2008), 525540.
[14] Enochs, E., Estrada, S. and Garcia-Rozas, J. R., Injective representations of infinite quivers. Applications , Canad. J. Math. 61(2) (2009), 315335.
[15] Enochs, E. E. and Jenda, O. M. G., Gorenstein injective and projective modules , Math. Z. 220(4) (1995), 611633.
[16] Enochs, E. E. and Jenda, O. M. G., Relative Homological Algebra. Volume 1, extended edn, De Gruyter Expositions in Mathematics 30 , Walter de Gruyter GmbH & Co., KG, Berlin, 2011.
[17] Enochs, E. E. and Jenda, O. M. G., Relative Homological Algebra. Volume 2, De Gruyter Expositions in Mathematics 54 , Walter de Gruyter GmbH & Co., KG, Berlin, 2011.
[18] Eshraghi, H., Hafezi, R. and Salarian, S., Total acyclicity for complexes of representations of quivers , Comm. Algebra 41(12) (2013), 44254441.
[19] Holm, H., Gorenstein homological dimensions , J. Pure Appl. Algebra 189(1–3) (2004), 167193.
[20] Henrik, H. and Jørgensen, P., Cotorsion pairs in categories of quiver representations, Kyoto J. Math., to appear, preprint, 2016, arXiv:1604.01517.
[21] Hu, W., Luo, X.-H., Xiong, B.-L. and Zhou, G., Gorenstein projective bimodules via monomorphism categories and filtration categories , J. Pure Appl. Algebra 223(3) (2019), 10141039.
[22] Huang, Z., Proper resolutions and Gorenstein categories , J. Algebra 393 (2013), 142169.
[23] Jensen, B. T., King, A. D. and Su, X., A categorification of Grassmannian cluster algebras , Proc. Lond. Math. Soc. (3) 113(2) (2016), 185212.
[24] Jørgensen, P., Existence of Gorenstein projective resolutions and Tate cohomology , J. Eur. Math. Soc. (JEMS) 9(1) (2007), 5976.
[25] Jørgensen, P. and Kato, K., Symmetric Auslander and bass categories , Math. Proc. Cambridge Philos. Soc. 150(2) (2011), 227240.
[26] Jørgensen, P. and Zhang, J. J., Gourmet’s guide to Gorensteinness , Adv. Math. 151(2) (2000), 313345.
[27] Kelly, G. M., Basic concepts of enriched category theory , Repr. Theory Appl. Categ. 10 (2005), vi+137, Reprint of the 1982 original [Cambridge University Press, Cambridge; MR0651714].
[28] Kvamme, S., A generalization of the Nakayama functor, preprint, 2017, arXiv:1611.01654v2, pp. 1–38.
[29] Luo, X.-H. and Zhang, P., Monic representations and Gorenstein-projective modules , Pacific J. Math. 264(1) (2013), 163194.
[30] Luo, X.-H. and Zhang, P., Separated monic representations I: Gorenstein-projective modules , J. Algebra 479 (2017), 134.
[31] Mac Lane, S., Categories for the Working Mathematician, 2nd edn, Graduate Texts in Mathematics 5 , Springer, New York, 1998.
[32] Nájera Chávez, A., On Frobenius (completed) orbit categories , Algebr. Represent. Theory 20(4) (2017), 10071027.
[33] Oberst, U. and Röhrl, H., Flat and coherent functors , J. Algebra 14 (1970), 91105.
[34] Popescu, N., Abelian Categories with Applications to Rings and Modules, London Mathematical Society Monographs 3 , Academic Press, London–New York, 1973.
[35] Pressland, M., Internally Calabi–Yau algebras and cluster-tilting objects , Math. Z. 287(1–2) (2017), 555585.
[36] Shen, D., A description of Gorenstein projective modules over the tensor products of algebras, preprint, 2016, arXiv:1602.00116.
[37] Stovicek, J., Derived equivalences induced by big cotilting modules , Adv. Math. 263 (2014), 4587.
[38] Yang, X. and Liu, Z., Gorenstein projective, injective, and flat complexes , Comm. Algebra 39(5) (2011), 17051721.
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Nagoya Mathematical Journal
  • ISSN: 0027-7630
  • EISSN: 2152-6842
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