On syzygies over 2-Calabi-Yau tilted algebras

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• 作者：Ana Garcia Elsener^a ; ^{elsener@mdp.edu.ar" class="auth_mail" title="E-mail the corresponding author} ; Ralf Schiffler^b ; ^{schiffler@math.uconn.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Syzygy ; Cohen&ndash ; Macaulay module ; 2-Calabi&ndash ; Yau tilted algebra ; Cluster-tilted algebra ; Igusa&ndash ; Todorov function ; Triangulated surface ; Punctured polygon
• 刊名：Journal of Algebra
• 出版年：2017
• 出版时间：15 January 2017
• 年：2017
• 卷：470
• 期：Complete
• 页码：91-121
• 全文大小：1283 K

文摘

We characterize the syzygies and co-syzygies over 2-Calabi–Yau tilted algebras in terms of the Auslander–Reiten translation and the syzygy functor. We explore connections between the category of syzygies, the category of Cohen–Macaulay modules, the representation dimension of algebras and the Igusa–Todorov functions. In particular, we prove that the Igusa–Todorov dimensions of d-Gorenstein algebras are equal to d.

id="sp0190">For cluster-tilted algebras of Dynkin type id="mmlsi1" class="mathmlsrc">rmulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302939&_mathId=si1.gif&_user=111111111&_pii=S0021869316302939&_rdoc=1&_issn=00218693&md5=43e71405044821a87d7b3f67454ede18" title="Click to view the MathML source">Diner hidden">i1.gif" overflow="scroll">i mathvariant="double-struck">Di>, we give a geometric description of the stable Cohen–Macaulay category in terms of tagged arcs in the punctured disc. We also describe the action of the syzygy functor in a geometric way. This description allows us to compute the Auslander–Reiten quiver of the stable Cohen–Macaulay category using tagged arcs and geometric moves.