Geodesic
Gentle $m$-Calabi--Yau tilted algebras
Algebra and discrete mathematics, Tome 30 (2020) no. 1, pp. 44-62.

Voir la notice de l'article provenant de la source Math-Net.Ru

We prove that all gentle 2-Calabi–Yau tilted algebras are Jacobian, moreover their bound quiver can be obtained via block decomposition. For two related families, the $m$-cluster-tilted algebras of type $\mathbb{A}$ and $\tilde{\mathbb{A}}$, we prove that a module $M$ is stable Cohen-Macaulay if and only if $\Omega^{m+1} \tau M \simeq M$.
Keywords: 2-Calabi–Yau tilted algebras, Jacobian algebras, Gentle algebras, derived category, Cohen-Macaulay modules, cluster-tilted algebras. 
@article{ADM_2020_30_1_a4,
     author = {A. Garcia Elsener},
     title = {Gentle $m${-Calabi--Yau} tilted algebras},
     journal = {Algebra and discrete mathematics},
     pages = {44--62},
     publisher = {mathdoc},
     volume = {30},
     number = {1},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2020_30_1_a4/}
}
TY  - JOUR
AU  - A. Garcia Elsener
TI  - Gentle $m$-Calabi--Yau tilted algebras
JO  - Algebra and discrete mathematics
PY  - 2020
SP  - 44
EP  - 62
VL  - 30
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ADM_2020_30_1_a4/
LA  - en
ID  - ADM_2020_30_1_a4
ER  - 
%0 Journal Article
%A A. Garcia Elsener
%T Gentle $m$-Calabi--Yau tilted algebras
%J Algebra and discrete mathematics
%D 2020
%P 44-62
%V 30
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ADM_2020_30_1_a4/
%G en
%F ADM_2020_30_1_a4
A. Garcia Elsener. Gentle $m$-Calabi--Yau tilted algebras. Algebra and discrete mathematics, Tome 30 (2020) no. 1, pp. 44-62. http://geodesic.mathdoc.fr/item/ADM_2020_30_1_a4/

[1] C. Amiot, “Cluster categories for algebras of global dimension 2 and quivers with potential”, Ann. Inst. Fourier, 59:6 (2009), 2525–2590 | DOI | MR | Zbl

[2] C. Amiot, “On generalized cluster categories”, Representations of Algebras and Related Topics, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2011, 1–53 | MR | Zbl

[3] I. Assem, T. Brüstle, G. Charbonneau-Jodoin and P. G. Plamondon, “Gentle algebras arising from surface triangulations”, Algebr. Number Theory, 4:2 (2010), 201–229 | DOI | MR | Zbl

[4] I. Assem, D. Simson and A. Skowroński, Elements of the Representation Theory of Associative Algebras, London Math. Soc. Student Texts, 65, Cambridge University Press, 2006 | MR | Zbl

[5] K. Baur and R. Marsh, “A geometric description of $m$-cluster categories”, Trans. Amer. Math. Soc., 360:11 (2008), 5789–5803 | DOI | MR | Zbl

[6] A. Beligiannis, “The homological theory of contravariantly finite subcategories: Auslander-Buchweitz contexts Gorenstein categories and (co-)stabilization”, Comm. Algebra, 28:10 (2000), 4547–4596 | DOI | MR | Zbl

[7] A. Beligiannis, “Relative homology, higher cluster-tilting theory and categorified Auslander–Iyama correspondence”, J. Algebra, 444 (2015), 367–503 | DOI | MR | Zbl

[8] T. Brüstle, J. Zhang, “On the cluster category of a marked surface without punctures”, Algebr. Number Theory, 5:4 (2011), 529–566 | DOI | MR | Zbl

[9] A. Buan, R. Marsh, M. Reineke, I. Reiten and G. Todorov, “Tilting theory and cluster combinatorics”, Adv. Math., 204:2 (2006), 572–518 | DOI | MR

[10] A. Buan, R. Marsh and I. Reiten, “Cluster-tilted algebras”, Trans. Amer. Math. Soc., 359:1 (2007), 323–332 | DOI | MR | Zbl

[11] M. C. R. Butler and C. M. Ringel, “Auslander-reiten sequences with few middle terms and applications to string algebras”, Communications in Algebra, 15:1–2 (1987), 145–179 | DOI | MR | Zbl

[12] P. Caldero, F. Chapoton and R. Schiffler, “Quivers with relations arising from clusters ($A_n$ case)”, Trans. Amer. Math. Soc., 358:3 (2006), 1347–1364 | DOI | MR | Zbl

[13] L. David-Roesler, The AG-invariant for $(m+2)$-angulations, 2012, arXiv: 1210.6087 | Zbl

[14] H. Derksen, J. Weyman and A. Zelevinsky, “Quivers with potentials and their representations I: Mutations”, Sel. Math., 14:1 (2008), 59–119 | DOI | MR | Zbl

[15] S. Fomin, M. Shapiro, and D. Thurston, “Cluster algebras and triangulated surfaces. Part I: Cluster complexes”, Acta Mathematica, 201:1 (2008), 83–146 | DOI | MR | Zbl

[16] A. Garcia Elsener, Monomial Gorenstein algebras and the stably Calabi–Yau property, 2018, arXiv: 1807.07018

[17] A. Garcia Elsener and R. Schiffler, “On syzygies over 2-Calabi–Yau tilted algebras”, J. Algebra, 470 (2017), 91–121 | DOI | MR | Zbl

[18] Ch. Geiss and I. Reiten, “Gentle Algebras are Gorenstein”, Representations of algebras and related topics, Fields Institute Communications, 45, Amer. Math. Soc., Providence, RI, 2005, 129–133 | MR | Zbl

[19] V. Gubitosi, “$m$-cluster tilted algebras of type $\tilde{A}$”, Communications in Algebra, 46:8 (2018), 3563–3590 | DOI | MR | Zbl

[20] M. Kalck, “Singularity categories of gentle algebras”, Bulletin of the London Mathematical Society, 2014, bdu093 | MR

[21] B. Keller and I. Reiten, “Cluster-tilted algebras are Gorenstein and stably Calabi–Yau”, Adv. Math., 211:1 (2007), 123–151 | DOI | MR | Zbl

[22] B. Keller and I. Reiten, “Acyclic Calabi–Yau categories”, Compositio Mathematica, 144:5 (2008), 1332–1348 | DOI | MR | Zbl

[23] S. Ladkani, 2-CY-tilted algebras that are not Jacobian, 2014, arXiv: 1403.6814 | Zbl

[24] S. Ladkani, Finite-dimensional algebras are ($m>2$)-Calabi–Yau-tilted., 2016, arXiv: 1603.09709

[25] D. Labardini-Fragoso and D. Velasco, “On a family of Caldero–Chapoton algebras that have the Laurent phenomenon”, Journal of Algebra, 520 (2019), 90–135 | DOI | MR | Zbl

[26] G.J Murphy, “Derived equivalence classification of $m$-cluster tilted algebras of type $A_n$”, J. Algebra, 323:4 (2010), 920–965 | DOI | MR | Zbl

[27] H. Thomas, “Defining an $m$-cluster category”, J. Algebra, 318:1 (2007), 37–46 | DOI | MR | Zbl

[28] H. A. Torkildsen, “A Geometric Realization of the $m$-cluster category of affine type $A$”, Comm. Algebra, 43:6 (2015), 2541–2567 | DOI | MR | Zbl