Symmetric approximation sequences, Beilinson-Green algebras and derived equivalences


  • Shengyong Pan



In this paper, we will consider a class of locally $\Phi$-Beilinson-Green algebras, where $\Phi$ is an infinite admissible set of the integers, and show that symmetric approximation sequences in $n$-exangulated categories give rise to derived equivalences between quotient algebras of locally $\Phi$-Beilinson-Green algebras in the principal diagonals modulo some factorizable ghost and coghost ideals by the locally finite tilting family. Then we get a class of derived equivalent algebras that have not been obtained by using previous techniques. From higher exact sequences, we obtain derived equivalences between subalgebras of endomorphism algebras by constructing tilting complexes, which generalizes Chen and Xi's result for exact sequences. From a given derived equivalence, we get derived equivalences between locally $\Phi$-Beilinson-Green algebras of semi-Gorenstein modules. Finally, from given graded derived equivalences of group graded algebras, we get derived equivalences between associated Beilinson-Green algebras of group graded algebras.


Asashiba, H., Derived equivalences and smash products, Proceedings of the 49th Symposium on Ring Theory and Representation Theory, 12–17, Symp. Ring Theory Represent. Theory Organ. Comm., Shimane, 2017.

Aihara, T., and Iyama, O., Silting mutation in triangulated categories, J. Lond. Math. Soc. (2) 85 (2012), no. 3, 633–668.

Auslander, M., Reiten, I., and Smalø, S. O., Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics 36, Cambridge University Press, Cambridge, 1995.

bibitem Bei Beilinson, A. A., Coherent sheaves on $mathbb P^n$ and problems in linear algebra, Funct. Anal. Appl. 12 (1978), no. 3, 214–216.

Bennett-Tennenhaus, R., and Shah, A., Transport of structure in higher homological algebra, J. Algebra 574 (2021), 514–549.

Brundan, J., and Davidson, N., Categorical actions and crystals, Categorification and higher representation theory, 105–147, Contemp. Math., 683, Amer. Math. Soc., Providence, RI, 2017.

bibitem Buan2006 Buan, A. B., Marsh, R., Reineke, M., Reiten, I., and Todorov, G., Tilting theory and cluster combinatorics, Adv. Math. 204 (2006), no. 2, 572–618.

Chen, H. X. and Xi, C. C., Dominant dimensions, derived equivalenences and tilting modules, Israel J. Math. 215 (2016), no. 1, 349–395.

Chen, Q. H., Derived equivalences of repetitive algebras, Adv. Math. (China) 37 (2008), no. 2, 189–196.

bibitem C Chen, X. W., Graded self-injective algebras “are” trivial extensions, J. Algebra 322 (2009), no. 7, 2601–2606.

Chen, Y. P., Derived equivalences in $n$-angulated categories, Algebr. Represent. Theory 16 (2013), no. 6, 1661–1684.

Chen, Y. P., Derived equivalences between subrings, Comm. Algebra 42 (2014), no. 9, 4055–4065.

Chen, Y. P., and Hu, W., Approximations, ghosts and derived equivalences, Proc. Roy. Soc. Edinburgh Sect. A 150 (2020), no. 2, 813–840.

Dugas, A., A construction of derived equivalent pairs of symmetric algebras, Proc. Amer. Math Soc. 143 (2015), no. 6, 2281–2300.

Green, E. L., A criterion for relative global dimension zero with applications to graded rings, J. Algebra 34 (1975), 130–135.

Green, E. L., and Happel, D., Grading and derived categories, Algebr. Represent. Theory 14 (2011), no. 3, 497–513.

Happel, D., Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series 119, Cambridge University Press, Cambridge, 1988.

Happel, D., and Unger, L., Complements and the generalized Nakayama conjecture, Algebras and modules, II (Geiranger, 1996), 293–310, CMS Conf. Proc., 24, Amer. Math. Soc., Providence, RI, 1998.

Herschend, M., Liu, Y., and Nakaoka, H., $n$-exangulated categories (I): Definitions and fundamental properties, J. Algebra 570 (2021), 531–586.

Hoshino, M., and Kato, Y., An elementary construction of tilting complexes, J. Pure Appl. Algebra 177 (2003), no. 2, 159–175.

bibitem Hu2013a Hu, W., Koenig, S., and Xi, C. C., Derived equivalences from cohomological approximations and mutations of Φ-Yoneda algebras, Proc. Roy. Soc. Edinburgh Sect. A 143 (2013), 589–629.

bibitem Hu2017 Hu, W., and Pan, S. Y., Stable functors of derived equivalences and Gorenstein projective modules, Math. Narchr. 290 (2017), no. 10, 1512–1530.

Hu, W., and Xi, C. C., $mathcal D$-split sequences and derived equivalences, Adv. Math. 227 (2011), no. 1, 292–318.

Hu, W., and Xi, C. C., Derived equivalences for Φ-Auslander-Yoneda algebras, Trans. Amer. Math. Soc. 365 (2013), n0. 11, 5681–5711.

Hughes, D., and Waschbüsch, J., Trivial extensions of tilted algebras, Proc. London Math. Soc. (3) 46 (1983), no. 2, 347–364.

bibitem Iyama2014a Iyama, O. and Wemyss, M., Maximal modifications and Auslander-Reiten duality for non-isolated singularities, Invent. Math. 197 (2014), no. 3, 521–586.

Karoubi, M., Algèbres de Clifford et $K$-théorie, Ann. Sci. École Norm. Sup. (4) 1 (1968), 161–270.

bibitem Keller1994 Keller, B., Deriving DG categories, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 1, 63–102.

bibitem Keller2006Keller, B., On differential graded categories, International Congress of Mathematicians. Vol. II, 151–190, Eur. Math. Soc., Zürich, 2006.

Marcus, A., Tilting complexes for group graded algebras, J. Group Theory 6 (2003), no. 2, 175–193.

Marcus, A., and Pan, S. Y., Tilting complexes for group graded self-injective algebras, Tsukuba J. Math. 43 (2019), no. 2, 211–222.

Mori, I., B-construction and C-construction, Comm. Algebra 41 (2013), no. 6, 2071–2091.

Nastasescu, C., and Van Oystaeyen, F., Methods of graded rings, Lecture Notes in Mathematics, 1836. Springer-Verlag, Berlin, 2004.

Neeman, A., Triangulated categories. Annals of Mathematics Studies, 148. Princeton University Press, Princeton, NJ, 2001.

Pan, S. Y., Derived equivalences for Cohen–Macaulay Auslander algebras, J. Pure Appl. Algebra 216 (2012), no. 2, 355–363.

Pan, S. Y., Stable equivalences of Morita type for Φ-Beilinson-Green algebras, Math. Nachr. 294 (2021), no. 5, 977–996.

Pan, S. Y., and Peng, Z., A note on derived equivalences for Φ-Green algebras, Algebr. Represent. Theory 17 (2014), no. 6, 1707–1720.

Peng, Z., Derived equivalences between Φ-Green algebras and $n$-homological ring epimorphisms, Ph.D. thesis, 2013.

Rickard, J., Morita theory for derived categories, J. London Math. Soc. (2) 39 (1989), no. 3, 436–456.

Ringel, C. M., and Zhang, P., Gorenstein-projective and semi-Gorenstein-projective modules, Algebra Number Theory 14 (2020), no. 1, 1–36.

Rudakov, A. N., Exceptional collections, mutations and helices, Helices and vector bundles, 1–6, London Math. Soc. Lecture Note Ser., 148, Cambridge Univ. Press, Cambridge, 1990.

Van den Bergh, M., Non-commutative crepant resolutions, The legacy of Niels Henrik Abel, 749–770, Springer, Berlin, 2004.

Wisbauer, R., Foundations of module and ring theory, Algebra, Logic and Applications, 3. Gordon and Breach Science Publishers, Philadelphia, PA, 1991.

Xi, C. C., On the finitistic dimension conjecture, I: related to representation-finite algebras, J. Pure Appl. Algebra 193 (2004), no. 1–3, 287–305.



How to Cite

Pan, S. (2022). Symmetric approximation sequences, Beilinson-Green algebras and derived equivalences. MATHEMATICA SCANDINAVICA, 128(3).