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Derived invariance by syzygy complexes

Published online by Cambridge University Press:  13 February 2017

JIAQUN WEI
JIAQUN WEI*
Affiliation:
Institute of Mathematics, School of Mathematics Sciences, Nanjing Normal University, Nanjing 210023, P.R. China. e-mail: weijiaqun@njnu.edu.cn
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Abstract

We study derived invariance through syzygy complexes. In particular, we prove that syzygy-finite algebras and Igusa--Todorov algebras are invariant under derived equivalences. Consequently, we obtain that both classes of algebras are invariant under tilting equivalences. We also prove that derived equivalences preserve AC-algebras and the validity of the finitistic Auslander conjecture.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 164 , Issue 2 , March 2018 , pp. 325 - 343
Copyright
Copyright © Cambridge Philosophical Society 2017 

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References

REFERENCES

[1] Auslander, M. and Reiten, I. On a generalised version of the Nakayama Conjecture. Proc. Amer. Math. Soc. 52 (1975), 69–74.Google Scholar
[2] Avramov, L. and Iyengar, S. Constructing modules with prescribed cohomological support. Illinois. J. Math. 51 (2007), 1–20.Google Scholar
[3] Beligiannis, A. On algebras of finite Cohen–Macaulay type. Adv. Math. 226 (2011), 1973–2019.CrossRefGoogle Scholar
[4] Christensen, L. W. and Holm, H. Algebras that satisfy Auslander's condition on vanishing of cohomology. Math. Z. 265 (2010), 21–40.Google Scholar
[5] Diveris, K. and Purin, M. The generalised Auslander–Reiten condition for the bounded derived category. Arch. Math. (Basel) 98 (2012), 507–C511.Google Scholar
[6] Goodearl, K. R. and Zimmermann-Huisgen, B. Repetitive resolutions over classical orders and finite dimensional algebras. In Algebras and Modules II (I. Reiten, S.O. Smalø and O. Solberg, Eds.) Canad. Math. Soc. Conf. Proc. Series 24 (1998), 205–225.Google Scholar
[7] Happel, D. Homological conjectures in representation theory of finite dimensional algebras. Sherbrook Lecture Notes Series (1991).Google Scholar
[8] Happel, D. Triangulated categories in the representation theory of finite dimension algebras. London Math. Soc. Lect. Notes Ser. 119 (Cambridge, 1988).Google Scholar
[9] Hu, W. and Xi, C. Almost $\mathcal{D}$ -split sequences and derived equivalences. Adv. Math. 227 (2011), 292–318.CrossRefGoogle Scholar
[10] Huard, F., Lanzilotta, M. and Mendoza, O. Finitistic dimension through infinite projective dimension. Bull. London Math. Soc. 41 (2009), 367–376.Google Scholar
[11] Igusa, K. and Todorov, G. On the finitistic global dimension conjecture for artin algebras. In: Representations of algebras and related topics. Fields Inst. Commun. 45 (Amer. Math. Soc., Providence, RI, 2005), 201–204.Google Scholar
[12] Jørgensen, D. A. and Sega, L. M. Nonvanishing cohomology and classes of Gorenstein rings. Adv. Math. 188 (2004), 470–490.Google Scholar
[13] Pan, S. Generalised Auslander–Reiten conjecture and derived equivalences. Comm. Algebra 41 (2013), 3695–3704.Google Scholar
[14] Pan, S. and Xi, C. Finiteness of finitistic dimension is invariant under derived equivalences. J. Algebra 322 (2009), 21–24.Google Scholar
[15] Rickard, J. Morita theory for derived categories. J. London Math. Soc. 39 (1989), 436–456.Google Scholar
[16] Ringel, C. M. The torsionless modules of an artin algebra, preprint (2008). Avaible on https://www.math.uni-bielefeld.de/ringel/opus/torsionless.pdf.Google Scholar
[17] Schulz, R. A nonprojective module without self-extensions. Arch. Math. (Basel) 62 (1994), 497–500.CrossRefGoogle Scholar
[18] Smalø, S. O. and Zimmermann-Huisgen, B. The homology of string algebras I. J. Reine Angew. Math. 580 (2005), 1–37.Google Scholar
[19] Verdier, J. L. Catégories derivées. Lecture Notes in Math. 569 (Springer-Verlag, Berlin, New York, 1977), 262–311.Google Scholar
[20] Wei, J. Auslander bounds and homological conjectures. Rev. Mat. Iberoamericana 27 (2011), 871–884.Google Scholar
[21] Wei, J. Auslander's condition and tilting theory. Proc. Royal Soc. Edinburgh 140A (2010), 1111–1118.Google Scholar
[22] Wei, J. Finitistic dimension and Igusa–Todorov algebras. Adv. Math. 222 (2009), 2215–2226.Google Scholar
[23] Wei, J. Finitistic dimension conjecture and relative hereditary algebras. J. Algebra 322 (2009), 4198–4204.Google Scholar
[24] Wei, J. Finitistic dimension conjecture and conditions on ideals. Forum Math. 23 (2011), 549–564.Google Scholar
[25] Zimmermann-Huisgen, B. Homological domino efects and the first finistic demension conjecture. Invent. Math. 108 (1992), 369–383.Google Scholar
[26] Zimmermann-Huisgen, B. Syzygies and homological dimensions over left serial rings. In Methods in Module Theory (Abrams, G., Haefner, J., and Rangaswamy, K.M., Eds.) (Dekker, New York, 1992), 161–174.Google Scholar
[27] Zimmermann-Huisgen, B. The finitistic dimension conjectures - A tale of 3.5 decades. In Abelian Groups and Modules (Kluwer, Dordrecht, 1995), 501–517.Google Scholar