Abstract
We consider the smallest triangulated subcategory of the unbounded derived module category of a ring that contains the injective modules and is closed under set indexed coproducts. If this subcategory is the entire derived category, then we say that injectives generate for the ring. In particular, we ask whether, if injectives generate for a collection of rings, do injectives generate for related ring constructions, and vice versa. We provide sufficient conditions for this statement to hold for various constructions including recollements, ring extensions and module category equivalences.
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References
Angeleri Hügel, L., Koenig, S., Liu, Q.: Recollements and tilting objects. J. Pure Appl. Algebra 215(4), 420–438 (2011)
Angeleri Hügel, L., Koenig, S., Liu, Q., Yang, D.: Ladders and simplicity of derived module categories. J. Algebra 472, 15–66 (2017)
Angeleri Hügel, L., Koenig, S., Liu, Q., Yang, D.: Recollements and stratifying ideals. J. Algebra 484, 47–65 (2017)
Bell, A.D., Farnsteiner, R.: On the theory of Frobenius extensions and its application to Lie superalgebras. Trans. Am. Math. Soc. 335(1), 407–424 (1993)
Bergh, P.A., Erdmann, K.: The representation dimension of Hecke algebras and symmetric groups. Adv. Math. 228(4), 2503–2521 (2011)
Beı̆linson, A.A., Bernstein, J., Deligne, P.: Faisceaux pervers. In: Analysis and Topology on Singular Spaces, I (Luminy, 1981), volume 100 of Astérisque, pp 5–171. Soc. Math., France, Paris (1982)
Bökstedt, M., Neeman, A.: Homotopy limits in triangulated categories. Compos. Math. 86(2), 209–234 (1993)
Bravo, D., Paquette, C.: Idempotent reduction for the finitistic dimension conjecture. Proc. Am. Math. Soc. 148(5), 1891–1900 (2020)
Cartan, H., Samuel, E.: Homological Algebra. Princeton University Press, Princeton (1956)
Chen, H.X., Xi, C.: Recollements of derived categories III: finitistic dimensions. J. Lond. Math. Soc. Second Ser. 95(2), 633–658 (2017)
Cline, E., Parshall, B., Scott, L.: Stratifying endomorphism algebras. Mem. Am. Math. Soc. 124(591), viii+ 119 (1996)
Fossum, R.M., Griffith, P.A., Reiten, I.: Trivial Extensions of Abelian Categories. Lecture Notes in Mathematics, vol. 456. Springer-Verlag, Berlin-New York (1975)
Fuller, K.R., Saorín, M.: On the finitistic dimension conjecture for Artinian rings. Manuscripta Math. 74(2), 117–132 (1992)
Green, E.L., Psaroudakis, C., Solberg, Ø.: Reduction techniques for the finitistic dimension. Trans. Amer. Math. Soc. 374(10), 6839–6879 (2021)
Happel, D.: Homological conjectures in representation theory of finite dimensional algebras. Sherbrook Lecture Notes Series, Université de Sherbrooke (1991)
Happel, D.: Reduction techniques for homological conjectures. Tsukuba J. Math. 17 (1), 115–130 (1993)
Huang, Z., Sun, J.: Invariant properties of representations under excellent extensions. J. Algebra 358, 87–101 (2012)
Huisgen-Zimmerman, B.: The finitistic dimension conjectures - a tale of 3.5 decades. Abelian Groups Modules (Padova) 1994(343), 501–517 (1995)
Kadison, L.: New examples of Frobenius extensions, volume 14 of University Lecture Series. American Mathematical Society, Providence RI (1999)
Kalck, M., Yang. D.: Relative singularity categories I: Auslander resolutions. Adv. Math. 301, 973–1021 (2016)
Kasch, F.: Grundlagen einer Theorie der Frobeniuserweiterungen. Math. Ann. 127, 453–474 (1954)
Keller, B.: Unbounded derived categories and homological conjectures. Talk at summer school on “Homological conjectures for finite dimensional algebras”. Nordfjordeid (2001)
Koenig, S.: Tilting complexes, perpendicular categories and recollements of derived module categories of rings. J. Pure Appl. Algebra 73(3), 211–232 (1991)
Linckelmann, M.: Finite generation of Hochschild cohomology of Hecke algebras of finite classical type in characteristic zero. Bull. Lond. Math. Soc. 43(5), 871–885 (2011)
Nakayama, T., Tsuzuku, T.: On Frobenius extensions I. Nagoya Math. J. 17, 89–110 (1960)
Passman, D, S.: The Algebraic Structure of Group Rings. Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney (1977)
Psaroudakis, C.: Homological theory of recollements of abelian categories. J. Algebra 398, 63–110 (2014)
Psaroudakis, C.: A representation-theoretic approach to recollements of abelian categories. In: Surveys in Representation Theory of Algebras, vol. 716 of Contemp. Math. Amer. Math. Soc., Providence, RI, pp 67–154 (2018)
Psaroudakis, C., Vitória, J.: Recollements of module categories. Appl. Categ. Struct. 22(4), 579–593 (2014)
Qin, Y.: Reduction techniques of singular equivalences. arXiv:2103.10393 (2021)
Rickard, J.: Morita theory for derived categories. J. Lond. Math. Soc. Second Ser. 39(3), 436–456 (1989)
Rickard, J.: Unbounded derived categories and the finitistic dimension conjecture. Adv. Math. 106735(21), 354 (2019)
Shamsuddin, A.: Finite normalizing extensions. J. Algebra 151(1), 218–220 (1992)
Soueif, L.: Normalizing extensions and injective modules, essentially bounded normalizing extensions. Commun. Algebra 15(8), 1607–1619 (1987)
Wang, C., Xi, C.: Finitistic dimension conjecture and radical-power extensions. J. Pure Appl. Algebra 221(4), 832–846 (2017)
Xi, C.: On the representation dimension of finite dimensional algebras. J. Algebra 226(1), 332–346 (2000)
Xi, C.: On the finitistic dimension conjecture. I. Related to representation-finite algebras. J. Pure Appl. Algebra 193(1-3), 287–305 (2004)
Xi, C.: On the finitistic dimension conjecture. II. Related to finite global dimension. Adv. Math. 201(1), 116–142 (2006)
Xi, C.: On the finitistic dimension conjecture. III. Related to the pair \(eAe\subseteq A\). J. Algebra 319(9), 3666–3688 (2008)
Xue, W.: On almost excellent extensions. Algebra Colloq. 3(2), 125–134 (1996)
Acknowledgements
I would like to thank my Ph.D. supervisor Jeremy Rickard for his guidance and many useful discussions. I am grateful for the financial support provided by the Engineering and Physical Sciences Research Council Doctoral Training Partnership award EP/N509619/1.
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This work was supported by Engineering and Physical Sciences Research Council Doctoral Training Partnership award EP/N509619/1.
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Cummings, C. Ring Constructions and Generation of the Unbounded Derived Module Category. Algebr Represent Theor 26, 281–315 (2023). https://doi.org/10.1007/s10468-021-10094-2
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DOI: https://doi.org/10.1007/s10468-021-10094-2