SpringerLink
Log in

Organising the module category

  • Special issue commemorating the Golden Jubilee of the Institute of Mathematics and Statistics of the University of São Paulo
  • Published:
São Paulo Journal of Mathematical Sciences Aims and scope Submit manuscript

A Correction to this article was published on 12 October 2021

This article has been updated

Abstract

We describe here the work developed with my students along the years in the IME-USP.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Change history

  • 10 October 2021

    In this article Reference 9 was incorrect, and was updated to read "Graduate Texts in Mathematics" instead of "Graduate Studies of Mathematics."

  • 12 October 2021

    A Correction to this paper has been published: https://doi.org/10.1007/s40863-021-00269-6

References

  1. Alvares, E.R., Assem, I., Castonguay, D., Vargas, R.R.S.: Right ADA algebras. J. Algebra Appl. 16(11), 1750210 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alvares, E., Assem, I., Coelho, F.U., Peña, M.I., Trepode, S.: From trisections in module categories to quasi-directed components. J. Algebra Appl 10, 409–433 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alvares, E., Coelho, F.U.: On translation quivers with weak sections, Representations of algebras and related topics, 1–8, Fields Inst. Commun., 45, Amer. Math. Soc., Providence, RI (2005)

  4. Alvares, E., Coelho, F.U.: Embeddings of non-semiregular translation quivers in quivers of type \(Z\Delta \). Algebra Represent Theory 10, 97–116 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Alvares, E., Coelho, F.U.: A note on the composite of two irreducible morphisms. Commun. Algebra 37(6), 2097–2099 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Angeleri-Hügel, L., Coelho, F.U.: Infinite generated tilting modules of finite projective dimensions. Forum Math. 13, 239–250 (2001)

    MathSciNet  MATH  Google Scholar 

  7. Angeleri-Hügel, L., Coelho, F.U.: Infinite generated complements to partial tilting modules. Math. Proc. Camb. Philos. Soc. 132, 88–95 (2002)

    Article  MATH  Google Scholar 

  8. Assem, I., Castonguay, D., Lanzilotta, M., Vargas, R.R.S.: Algebras determined by their supports. J. Pure Appl. Algebra 216(5), 1134–1145 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Assem, I., Coelho, F.U.: Basic Representation Theory of Algebras. Graduate Texts in Mathematics, p. x+311. Springer, Berlin (2020)

    Book  MATH  Google Scholar 

  10. Assem, I., Coelho, F.U.: Glueings of tilted algebras. J. Pure Appl. Algebra 96, 225–243 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Assem, I., Coelho, F.U.: Two sided gluings of tilted algebras. J. Algebra 269, 456–479 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Assem, I., Coelho, F.U., Lanzilotta, M., Smith, D., Trepode, S., Algebras determined by their left and right parts, XVto Colloquio Latinoamericano de Algebra, Cocoyoc (2003) Contemp. Math. 376, : 13–47. Am. Math, Soc (2005)

  13. Assem, I., Coelho, F.U., Trepode, S.: Simply connected tame quasi-tilted algebras. J. Pure Appl. Algebra 172, 139–160 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Assem, I., Coelho, F.U., Trepode, S.: The left and the right parts of a module category. J. Algebra 281, 518–534 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Assem, I., Coelho, F.U., Trepode, S.: The bound quiver of a split extension. J. Algebra Appl. 7, 405–423 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Assem, I., Coelho, F.U., Trepode, S.: Contravariantly finite subcategories closed under predecessors. J. Algebra 322(4), 1196–1213 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Assem, I., Coelho, F.U., Wagner, H.: On subcategory closed under predecessors and the representation dimension. J. Algebra 418, 174–196 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Assem, I., Happel, D., Trepode, S.: Tilting up iterated tilted algebras. Proc. Am. Math. Soc. 128(8), 2223–2232 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  19. Assem, I., Skowroński, A., Simson, D.: Elements of representation of algebras Vol 1. London Mathematical Society Student Texts, vol. 65. Cambridge University Press, Cambridge (2006)

    MATH  Google Scholar 

  20. Auslander, M.: Representation Dimension of Artin Algebras. Math. Notes, Queen Mary College, London (1971)

    MATH  Google Scholar 

  21. Auslander, M., Platzeck, M.I., Reiten, I.: Coxeter functors without diagrams. Trans. Am. Math. Soc. 250, 1–46 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  22. Auslander, M., Reiten, I., Smalø, S.O.: Representation Theory of Artin Algebras Cambridge Studies in Advanced Mathematics, vol. 36. Cambridge University Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  23. Auslander, M., Smalø, S.: Preprojective modules over Artin algebras. J. Algebra 66, 61–122 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  24. Bekkert, V., Coelho, F.U., Wagner, H.: Tree oriented pullback. Commun. Algebra 43, 4247–4257 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Bélanger, J., Tosar, C.: Shod string algebras. Commun. Algebra 33(8), 2465–2487 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  26. Braga, C.A., Coelho, F.U.: A note on tilting sequences. Central Eur. J. Math. 6(3), 364–371 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. Braga, C.A., Coelho, F.U.: Limits of tilting modules. Colloq. Math. 115(2), 207–217 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. Brenner, S., Butler, M.: Generalizations of the Bernstein–Gelfand–Ponomarev reflection functors. In: Proc. ICRA II (1979) SLN 832, pp. 103–170 (1980)

  29. Chaio, C., Coelho, F.U., Trepode, S.: On the composite of two irreducible morphisms in radical cube. J. Algebra 312, 650–667 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  30. Chaio, C., Coelho, F.U., Trepode, S.: On the composite of irreducible morphisms in almost sectional paths. J. Pure Appl. Algebra 212, 244–261 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  31. Chaio, C., Coelho, F.U., Trepode, S.: On the composite of three irreducible morphisms over string algebras São Paulo. J. Math. Sci. 4(2), 189–208 (2010)

    MathSciNet  MATH  Google Scholar 

  32. Chaio, C., Coelho, F.U., Trepode, S.: On the composite of three irreducible morphisms in the fourth power of the radical. Commun. Algebra 39(2), 555–559 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  33. Chaio, C., Le Meur, P., Trepode, S.: Degrees of irreducible morphisms and finite-representation type. J. Lond. Math. Soc. (2) 84(1), 35–57 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  34. Chust, V., Coelho, F.U.: On the correspondence between path algebras and generalized path algebras, preprint (2021)

  35. Chust, V., Coelho, F.U.: Representations and homological invariants of generalized bound path algebras, in preparation

  36. Coelho, F.U.: Components of Auslander–Reiten quivers containing only preprojective modules. J. Algebra 157, 472–488 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  37. Coelho, F.U.: Shod algebras. Resenhas 5(1), 25–61 (2001)

    MathSciNet  MATH  Google Scholar 

  38. Coelho, F.U.: The representation dimension of Artin algebras. São Paulo J. Math. Sci. 11(2), 348–360 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  39. Coelho, F.U., Aguiar, M.: A história da álgebra e o pensamento algébrico: correlações com o ensino, Revista Estudos Avançados 32 (94). IEA-USP 171–187 (2018)

  40. Coelho, F.U., Lanzilotta, M.: Algebras with small homological dimensions. Manus. Math. 100, 1–11 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  41. Coelho, F.U., Lanzilotta, M.: On non-semiregular components containing paths from injective to projective modules. Commun. Algebra 30, 4837–4849 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  42. Coelho, F.U., Lanzilotta, M.: Weakly shod algebras. J. Algebra 265, 379–403 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  43. Coelho, F.U., Lanzilotta, M., Savioli, A.M.P.D.: On the Hochschild cohomology of algebras with small homological dimensions. Ann. Sci. Math. Quèbec 26, 15–23 (2002)

    MathSciNet  MATH  Google Scholar 

  44. Coelho, F.U., Liu, S.X.: Generalized path algebras In: Interaction between ring theory and representations of algebras. In: Proceedings of the Conference Held in Murcia, Spain. pp. 53–66, Lecture Notes in Pure and Appl. Math., vol. 210, Dekker, New York (2000)

  45. Coelho, F.U., Marcos, E.N., Merklen, H.A., Skowroński, A.: Module categories with infinite radical square zero are of finite type. Commun. Algebra 22(11), 4511–4517 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  46. Coelho, F.U., Peña, J.A., Trepode, S.: Non-quasitilted tilt-critical algebras. In: Proceedings of the XVIth Latin American Algebra Colloquium (Spanish), 185–200, Bibl. Rev. Mat. Iberoamericana, Rev. Mat. Iberoamericana, Madrid (2007)

  47. Coelho, F.U., Peña, J.A., Trepode, S.: On minimal non-tilted algebras. Colloq. Math. 111, 71–84 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  48. Coelho, F.U., Peña, J.A., Trepode, S.: Tilt-critical algebras of tame type. Commun. Algebra 36(9), 3574–3584 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  49. Coelho, F.U., Savioli, A.M.P.D.: A note on non-supportive elements in monomial algebras. Bol. Soc. Parana. Mat 19, 27–32 (1999)

    MathSciNet  MATH  Google Scholar 

  50. Coelho, F.U., Savioli, A.M.P.D.: On shod extensions of algebras. Commun. Algebra 32, 1307–1317 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  51. Coelho, F.U., Silva, Danilo D.: Relative degrees of irreducible morphisms. J. Algebra. 428, 471–489 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  52. Coelho, F.U., Silva, Danilo D.: On the postprojective partitions and components of the Auslander–Reiten quivers. Commun. Algebra 44, 4165–4176 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  53. Coelho, F.U., Vargas, R.R.S.: Mesh algebras. Commun. Algebra 30, 3703–3722 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  54. Coelho, F.U., Tosar, C.: On the derived categories and Quasitilted algebras. Algebra Represent Theory 12(1), 77–92 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  55. Coelho, F.U., Wagner, H.: On linearly oriented pullback and classes of algebras. Algebra Represent Theory 23(3), 739–758 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  56. Happel, D.: Triangulated Categories in the Representation Theory of Finite-Dimensional Algebras. London Mathematical Society Lecture Note Series, vol. 119, p. x+208. Cambridge University Press, Cambridge (1988)

    Book  MATH  Google Scholar 

  57. Happel, D., Reiten, I., Smalø, S.O.: Tilting in abelian categories and Quasitilted algebras, Mem. Am. Math. Soc. 120(575) (1996)

  58. Happel, D., Ringel, C.: Tilted algebras. Trans. Am. Math. Soc. 274, 399–443 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  59. Igusa, K., Todorov, G.: On the finitistic global dimension conjecture for Artin algebras. In: Representations of algebras and related topics, vol. 45 of Fields Inst. Comm. AMS, pp 201–204 (2005)

  60. Igusa, K., Platzeck, M.I., Todorov, G., Zacharia, D.: Auslander algebras of finite representation type. Commun. Algebra 15(1–2), 377–424 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  61. Levesque, J.: Nakayama oriented pullbacks and stably hereditary algebras. J. Pure Appl. Algebra 212, 1149–1161 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  62. Li, S.: The embedding of the components of the Auslander–Reiten quiver of an Artin algebra containing no oriented cycles. Commun. Algebra 28(10), 4635–4645 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  63. Liu, S.: The degree of irreducible morphisms and the shapes of Auslander–Reiten quivers. J. Lond. Math. Soc. 45, 32–54 (1992)

    Article  MATH  Google Scholar 

  64. Reiten, I., Skowroński, A.: A characterization of algebras with small homological dimensions. Adv. Math. 179, 122–154 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  65. Reiten, I., Skowroński, A.: Generalized double tilted algebras. J. Math. Soc. Jpn. 56(1), 269–288 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  66. Silva, Danilo D.: Artin algebras of finite type and finite categories of \(\Delta \)-good modules. Commun. Algebra 44(5), 2234–2242 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  67. Trepode, S. E., Roldán’s conjecture in the case \(A_n\) (Spanish) Proceedings of the Third ”Dr. Antonio A. R. Monteiro” Congress on Mathematics (Spanish) (Bahía Blanca: 51–68, p. 1996. Univ. Nac. del Sur, Bahía Blanca (1995)

Download references

Funding

Funding was provided by CNPq (Grant Number: Pq 312590/2020-2).

Author information

Authors and Affiliations

  1. Institute of Mathematics and Statistics of the University of São Paulo, São Paulo, Brazil

    Flávio U. Coelho

Authors
  1. Flávio U. Coelho
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Flávio U. Coelho.

Additional information

Communicated by Vyacheslav Futorny.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Coelho, F.U. Organising the module category. São Paulo J. Math. Sci. 16, 62–82 (2022). https://doi.org/10.1007/s40863-021-00241-4

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40863-021-00241-4

Keywords

Mathematics Subject Classification

Search

Navigation