## Rocky Mountain Journal of Mathematics

### Three results for $\tau$-rigid modules

#### Abstract

$\tau$-rigid modules are essential in the $\tau$-tilting theory introduced by Adachi, Iyama and Reiten. In this paper, we give equivalent conditions for Iwanaga-Gorenstein algebras with self-injective dimension at most one in terms of $\tau$-rigid modules. We show that every indecomposable module over iterated tilted algebras of Dynkin type is $\tau$-rigid. Finally, we give a $\tau$-tilting theorem on homological dimension which is an analog to that of classical tilting modules.

#### Article information

Source
Rocky Mountain J. Math., Volume 49, Number 8 (2019), 2791-2808.

Dates
First available in Project Euclid: 31 January 2020

https://projecteuclid.org/euclid.rmjm/1580461794

Digital Object Identifier
doi:10.1216/RMJ-2019-49-8-2791

Mathematical Reviews number (MathSciNet)
MR4058350

Zentralblatt MATH identifier
07163199

Subjects
Primary: 16G10: Representations of Artinian rings 16E10.

#### Citation

Xie, Zongzhen; Zan, Libo; Zhang, Xiaojin. Three results for $\tau$-rigid modules. Rocky Mountain J. Math. 49 (2019), no. 8, 2791--2808. doi:10.1216/RMJ-2019-49-8-2791. https://projecteuclid.org/euclid.rmjm/1580461794

#### References

• T. Adachi, The classification of $\tau$-tilting modules over Nakayama algebras, J. Algebra 452 (2016), 227–262.
• T. Adachi, Characterizing $\tau$-tilting modules over algebras with radical square zero, Proc. Amer. Math. Soc. 144 (2016), no. 11, 4673–4685.
• T. Aihara and O. Iyama, Silting mutation in triangulated categories, J. London. Math. Soc. 85 (2012), no. 3, 633–668.
• T. Adachi, O. Iyama and I. Reiten, $\tau$-tilting theory, Compos. Math. 150 (2014), no. 3, 415–452.
• M. Auslander and S.O. Smalø, Addendum to “Almost split sequences in subcategories”, J. Algebra 69 (1981), 426–454. Addendum to J. Algebra 71 (1981), 592–594.
• I. Assem and D. Happel, Generalized tilted algebra of type An, Comm. Algebra 20 (1981), 2101–2125.
• I. Assem, D. Simson and A. Skowroński, Elements of the representation theory of associative algebras I: techniques of reperesentation theory, London Math. Soc. Student Texts 65, Cambridge Univ. Press, Cambridge (2006).
• S. Brenner and M.C.R. Bulter, Generalization of the Bernstein-Gelfand-Ponomarev reflection functors, 103–169 in Representation theory II: Proc. Second Internat. Conf. (Carleton University, Ottawa, Ontario, 1979), Lecture Notes in Math. 832 Springer-Verlag, 103–169 (1980).
• A.B. Buan, R. Marsh, M. Reineke, I. Reiten and G. Todorov, Tilting theory and cluster combinatorics, Adv. Math. 204 (2006), no. 2, 572–618.
• A.B. Buan and Y. Zhou, Endomorphism algebras of 2-term silting complexes Algebr. Represent. Th. 21 (2018), no. 1, 181–194.
• L. Demonet, O. Iyama and G. Jasso, $\tau$-tilting finite algebras and g-vectors, Int. Math. Res. Not. IMRN 3 (2019), 852–892.
• L. Demonet, O. Iyama and Y. Palu, $\tau$-tilting theory for Schurian algebras, in preparation.
• L. Demonet, O. Iyama, N. Reading, I. Reiten and H. Thomas, Lattice theory of torsion class, arXiv:1711.01785.
• F. Eisele, G. Janssens and T. Raedschelders, A reduction theorem for $\tau$-rigid modules, Math. Zeit. 290 (2018), no. 3-4, 1377–1413.
• D. Happel and C.M. Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), no. 2, 399–443.
• D. Happel and L. Unger, Modules of finite projective dimension and cocovers, Math. Ann. 306 (1996), no. 1, 445–457.
• Z. Huang and Y. Zhang, G-stable support $\tau$-tilting modules, Front. Math. China 11 (2016), no. 4, 1057–1077.
• O. Iyama, P. Jorgensen and D. Yang, Intermediate co-$t$-structures, two-term silting objects, $\tau$-tilting modules and torsion classes, Algebra Number Theory 8 (2014), no. 10, 2413–2431.
• O. Iyama and Y. Yoshino, Mutations in triangulated categories and rigid Cohen-Macaulay modules, Invent. Math. 172 (2008), no. 1, 117–168.
• O. Iyama and X. Zhang, Classifying $\tau$-tilting modules over the Auslander algebra of $K[X]/(x^{n})$, to appear in J. Math. Soc. Japan, arXiv: 1602.05037.
• O. Iyama and X. Zhang, Tilting modules over Auslander-Gorenstein algebra, Pacific J. Math. 298 (2019), no. 2, 399–416.
• G. Jasso, Reduction of $\tau$-tilting modules and torsion pairs, Int. Math. Res. Not. IMRN 16 (2015), 7190–7237.
• R. Kase, Weak orders on symmetric groups and posets of support $\tau$-tilting modules, Int. J. Algebr. Comput. 27 (2017), no. 5, 501–546.
• B. Keller and I. Reiten, Cluster-tilted algebras are Gorenstein and stably Calabi-Yau, Adv. Math. 211 (2007), no. 1, 123–151.
• Y. Miyashita, Tilting modules of finite projective dimension, Math. Zeit. 193 (1986), no. 1, 113–146.
• Y. Mizuno, Classifying $\tau$-tilting modules over preprojective algebras of Dynkin type, Math. Zeit. 277 (2014), no. 3, 665–690.
• J. Wei, $\tau$-tilting theory and $*$-modules, J. Algebra 414 (2014), 1–5.
• X. Zhang, $\tau$-rigid modules for algebras with radical square zero, to appear in Algebr. Colloq., arXiv:1211.5622.
• X. Zhang, $\tau$-rigid modules over Auslander algebras, Taiwanese J. Math. 21 (2017), no. 4, 327–338.
• S. Zito, $\tau$-rigid modules from tilted to cluster-tilted algebras, Comm. Algebra. 47 (2019), no. 9, 3716–3734.
• S. Zito, $\tau$-tilting finite tilted and cluster-tilted algebras, arXiv:1902.05866v1.