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Dominant dimension and tilting modules

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We study which algebras have tilting modules that are both generated and cogenerated by projective–injective modules. Crawley–Boevey and Sauter have shown that Auslander algebras have such tilting modules; and for algebras of global dimension 2, Auslander algebras are classified by the existence of such tilting modules. In this paper, we show that the existence of such a tilting module is equivalent to the algebra having dominant dimension at least 2, independent of its global dimension. In general such a tilting module is not necessarily cotilting. Here, we show that the algebras which have a tilting–cotilting module generated–cogenerated by projective–injective modules are precisely 1-minimal Auslander–Gorenstein algebras. When considering such a tilting module, without the assumption that it is cotilting, we study the global dimension of its endomorphism algebra, and discuss a connection with the Finitistic Dimension Conjecture. Furthermore, as special cases, we show that triangular matrix algebras obtained from Auslander algebras and certain injective modules, have such a tilting module. We also give a description of which Nakayama algebras have such a tilting module.

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Acknowledgements

The authors thank the referees for pointing out some relevant references in the literature and comments to improve the clarity of the paper. The authors also thank Rene Marczinzik, Matthew Pressland, and Julia Sauter for helpful conversations and remarks, especially Matthew for pointing out an error in the original proof of Theorem 4.2.9. The third author would like to thank NTNU for their hospitality during her several visits while working on this project. This work was done when the first author was a Zelevinsky Research Instructor at Northeastern University; she thanks the Mathematics Department for their support.

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Authors and Affiliations

  1. Department of Mathematics, Hood College, Frederick, MD, 21701, USA

    Van C. Nguyen

  2. Institutt for Matematiske Fag, Norges Teknisk-Naturvitenskapelige Universitet, 7491, Trondheim, Norway

    Idun Reiten

  3. Department of Mathematics, Northeastern University, Boston, MA, 02115, USA

    Gordana Todorov & Shijie Zhu

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  1. Van C. Nguyen
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  2. Idun Reiten
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  3. Gordana Todorov
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  4. Shijie Zhu
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Correspondence to Gordana Todorov.

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Nguyen, V.C., Reiten, I., Todorov, G. et al. Dominant dimension and tilting modules. Math. Z. 292, 947–973 (2019). https://doi.org/10.1007/s00209-018-2111-4

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  • DOI: https://doi.org/10.1007/s00209-018-2111-4

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