Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-25T14:06:02.741Z Has data issue: false hasContentIssue false
  • English
  • Français

SPECIAL TILTING MODULES FOR ALGEBRAS WITH POSITIVE DOMINANT DIMENSION

Part of: Representation theory of rings and algebras Homological algebra Abelian categories

Published online by Cambridge University Press:  09 December 2020

MATTHEW PRESSLAND  and
JULIA SAUTER
MATTHEW PRESSLAND
Affiliation:
Institut für Algebra und Zahlentheorie, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart Germany e-mail: presslmw@mathematik.uni-stuttgart.de
JULIA SAUTER
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, Postfach 100 131, D-33501 Bielefeld, Germany e-mail: jsauter@math.uni-bielefeld.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study certain special tilting and cotilting modules for an algebra with positive dominant dimension, each of which is generated or cogenerated (and usually both) by projective-injectives. These modules have various interesting properties, for example, that their endomorphism algebras always have global dimension less than or equal to that of the original algebra. We characterise minimal d-Auslander–Gorenstein algebras and d-Auslander algebras via the property that these special tilting and cotilting modules coincide. By the Morita–Tachikawa correspondence, any algebra of dominant dimension at least 2 may be expressed (essentially uniquely) as the endomorphism algebra of a generator-cogenerator for another algebra, and we also study our special tilting and cotilting modules from this point of view, via the theory of recollements and intermediate extension functors.

MSC classification

Primary: 16G10: Representations of Artinian rings
Secondary: 18E30: Derived categories, triangulated categories 18G05: Projectives and injectives 18G20: Homological dimension
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 64 , Issue 1 , January 2022 , pp. 79 - 105
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

References

REFERENCES

Assem, I., Simson, D. and Skowroński, A., Elements of the representation theory of associative algebras. Vol. 1: Techniques of representation theory, London Mathematical Society Student Texts, vol. 65 (Cambridge University Press, Cambridge, 2006).Google Scholar
Auslander, M., Representation dimension of artin algebras (1971). Reprinted in Selected works of Maurice Auslander. Part 1 (American Mathematical Society, Providence, RI, 1999), 505–575.Google Scholar
Auslander, M., Platzeck, M. I and Todorov, G., Homological theory of idempotent ideals, Trans. Amer. Math. Soc. 332(2) (1992), 667–692.CrossRefGoogle Scholar
Auslander, M. and Reiten, I., Cohen-Macaulay and Gorenstein Artin algebras, Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991), 221–245.CrossRefGoogle Scholar
Brenner, S. and Butler, M. C. R., Generalizations of the Bernstein–Gel’fand–Ponomarev reflection functors, Representation theory, II (Proceedings of Second International Conference, Carleton University, Ottawa, Ontario, 1979), 1980, 103–169.CrossRefGoogle Scholar
Chen, H. and Xi, C., Dominant dimensions, derived equivalences and tilting modules, Israel J. Math. 215 (2016)(1), 349–395. (arXiv:1503.02385 [math.RT]).CrossRefGoogle Scholar
Cline, E., Parshall, B. and Scott, L., Derived categories and Morita theory, J. Algebra 104(2) (1986), 397–409.CrossRefGoogle Scholar
Crawley-Boevey, W. and Sauter, J., On quiver Grassmannians and orbit closures for representation-finite algebras, Math. Z. 285(1–2) (2017), 367–395. (arXiv:1509.03460 [math.RT]).CrossRefGoogle Scholar
Franjou, V. and Pirashvili, T., Comparison of abelian categories recollements, Doc. Math. 9 (2004), 41–56. (arXiv:math/0403081 [math.CT]).Google Scholar
Gao, N. and Koenig, S., Grade, dominant dimension and Gorenstein algebras, J. Algebra 427 (2015), 118–141.CrossRefGoogle Scholar
Gastaminza, S., Happel, D., Platzeck, M. I., Redondo, M. J. and Unger, L., Global dimensions for endomorphism algebras of tilting modules, Arch. Math. (Basel) 75(4) (2000), 247–255.CrossRefGoogle Scholar
Happel, D., Triangulated categories in the representation theory of finite-dimensional algebras , London Mathematical Society Lecture Note Series, vol. 119 (Cambridge University Press, Cambridge, 1988).Google Scholar
Iyama, O., Auslander correspondence, Adv. Math. 210(1) (2007), 51–82. (arXiv:math/0411631 [math.RT]).CrossRefGoogle Scholar
Iyama, O., Higher-dimensional Auslander–Reiten theory on maximal orthogonal subcategories, Adv. Math. 210(1) (2007), 22–50. (arXiv:math/0407052 [math.RT]).CrossRefGoogle Scholar
Iyama, O., Auslander–Reiten theory revisited, Trends in representation theory of algebras and related topics, 2008, 349–397. (arXiv:0803.2841 [math.RT]).CrossRefGoogle Scholar
Iyama, O. and Solberg, Ø., Auslander–Gorenstein algebras and precluster tilting, Adv. Math. 326 (2018), 200–240. (arXiv:1608.04179 [math.RT]).CrossRefGoogle Scholar
Kuhn, N. J., Generic representations of the finite general linear groups and the Steenrod algebra II. K-Theory 8(4) (1994), 395–428.CrossRefGoogle Scholar
Miyashita, Y., Tilting modules of finite projective dimension, Math. Z. 193(1) (1986), 113–146.CrossRefGoogle Scholar
Morita, K., Flat modules, injective modules and quotient rings, Math. Z. 120 (1971), 25–40.CrossRefGoogle Scholar
Müller, B. J., The classification of algebras by dominant dimension, Canad. J. Math. 20 (1968), 398–409.CrossRefGoogle Scholar
Nguyen, V. C., Reiten, I., Todorov, G., and Zhu, S., Dominant dimension and tilting modules, Math. Z. 292(3–4) (2019), 947–973. (arXiv:1706.00475 [math.RT]).CrossRefGoogle Scholar
Pressland, M. and Sauter, J., On quiver Grassmannians and orbit closures for gen-finite modules (2018). To appear in Algebr. Represent. Theory. (arXiv:1802.01848 [math.RA]).Google Scholar
Psaroudakis, C. and Vitória, J., Recollements of module categories, Appl. Categ. Struct. 22(4) (2014), 579–593. (arXiv:1304.2692 [math.RT]).CrossRefGoogle Scholar
Rickard, J., Morita theory for derived categories, J. London Math. Soc. (2) 39(3) (1989), 436–456.CrossRefGoogle Scholar
Ringel, C. M., Artin algebras of dominant dimension at least 2, 2007. Seminar notes, Bielefeld. Available at www.math.uni-bielefeld.de/~ringel/opus/domdim.pdf.Google Scholar
Tachikawa, H., On left QF – 3 rings, Pacific J. Math. 32 (1970), 255–268.CrossRefGoogle Scholar
Tachikawa, H., Quasi-Frobenius rings and generalizations. QF-3 and QF-1 rings, Lecture Notes in Mathematics, Vol. 351 (Springer-Verlag, BerlinNew York, 1973). Notes by C.M. Ringel.Google Scholar