S TRING AND B AND C OMPLEXES OVER C ERTAIN A LGEBRA OF D IHEDRAL T YPE José A. Vélez-Marulanda V ALDOSTA S TATE U NIVERSITY Joint work with Hernán Giraldo U NIVERSIDAD DE A NTIOQUIA Maurice Auslander Distinguished Lectures and International Conference, Woods Hole, MA, May 2, 2016
S ET U P In this talk: k is an algebraically closed field of arbitrary characteristic. • The Λ ’s always denote finite-dimensional k -algebras. • Unless explicitly stated otherwise, all our modules are modules from the • right. We denote by mod Λ the abelian category of finitely generated right • Λ -modules, and P Λ denotes the full subcategory of mod Λ whose ob- jects are finitely generated projective Λ -modules. K b ( P Λ ) denotes the triangulated category of perfect complexes over • Λ and D b ( mod Λ ) denotes the bounded derived category of mod Λ . String and Band Complexes over Certain Algebra of Dihedral Type J.A. Vélez-Marulanda
M OTIVATION & B ACKGROUND Let V • be an object of D − ( mod Λ ) that has finitely many non-zero cohomology groups, all which have finite dimension over k . In 2015, F. M. B LEHER and V-M proved that V • has a well-defined versal deforma- • tion ring R ( Λ , V • ) , which is a complete local commutative Noetherian k -algebra with residue field k . Moreover, the isomorphism class of R ( Λ , V • ) is preserved un- der derived equivalences. They also proved that versal deformation rings of modules are preserved un- • der stable equivalences of Morita type (as introduced by M. B ROUÉ in 1994) between self-injective k -algebras. In 2016, in an ongoing research, V-M proved that versal deformation rings of • Cohen-Macaulay modules are preserved under singular equivalences of Morita type between Gorenstein k -algebras. These singular equivalences of Morita type where introduced in a preprint by X. • W. C HEN and L. G. S UN during 2012 and then formally discussed in a published article by G. Z HOU and A. Z IMMERMANN in 2013. The ultimate goal is to use “nice" descriptions of such complexes V • to explicitly describe R ( Λ , V • ) for when Λ is e.g. a Gorenstein k -algebra. String and Band Complexes over Certain Algebra of Dihedral Type J.A. Vélez-Marulanda
M OTIVATION & B ACKGROUND In general, it is a difficult problem to describe the indecomposable objects in • D b ( mod Λ ) . Assume that Λ is a gentle algebra as introduced by I. A SSEM and A. S KOWRO ´ • NSKI in 1987. In 2003, V. B EKKERT and H. A. M ERKLEN provided a combinatorial description • of the indecomposable objects in D b ( mod Λ ) . They used so-called string and band complexes, which are indecomposable objects in K b ( P Λ ) . They used the obtained results to prove that gentle algebras are derived tame • as introduced by C H . G EISS & H. K RAUSE in 2002. Then in 2011, G. B OBI ´ NSKI used these string and band complexes to describe the • almost split triangles in K b ( P Λ ) . He also showed the relation between the description provided by V. B EKKERT • and H. A. M ERKLEN with the Happel functor F : D b ( mod Λ ) → mod ˆ Λ , where mod ˆ Λ denotes the stable module category of the repetitive algebra ˆ Λ . Question: How about self-injective non-gentle algebras? String and Band Complexes over Certain Algebra of Dihedral Type J.A. Vélez-Marulanda
� � � � � � � � � � � � � � � A LGEBRAS OF D IHEDRAL T YPE Consider the following quivers. τ 0 � • τ 1 � • τ 0 � • τ 1 � • 3 A = • 3 B = • ζ 0 0 1 2 0 1 2 γ 1 γ 2 γ 1 γ 2 τ 0 � • τ 1 � • τ 0 3 D = • 3 L = • � • ζ 0 ζ 2 ζ 0 � ✽✽✽✽✽✽✽ 0 1 2 0 1 � ✝✝✝✝✝✝✝ γ 1 γ 2 τ 2 τ 1 • 2 τ 0 τ 0 3 Q = • � • 3 R = • � • ζ 0 ζ 1 ζ 0 ζ 1 � ✽✽✽✽✽✽✽ � ✽✽✽✽✽✽✽ 0 0 1 1 � ✝✝✝✝✝✝✝ � ✝✝✝✝✝✝✝ τ 2 τ 1 τ 2 τ 1 • • 2 2 ζ 2 String and Band Complexes over Certain Algebra of Dihedral Type J.A. Vélez-Marulanda
A LGEBRAS OF D IHEDRAL T YPE Let Λ be one of the following bounded path algebras. = k [ 3 A ] / � τ 0 τ 1 , γ 2 γ 1 , ( γ 1 τ 0 ) 2 − ( τ 1 γ 2 ) 2 � D ( 3 A ) 2,2 2 = k [ 3 B ] / � γ 1 ζ 0 , ζ 0 τ 0 , τ 0 τ 1 , γ 2 γ 1 , ( γ 1 τ 0 ) 2 − ( τ 1 γ 2 ) 2 , ( τ 0 γ 1 ) 2 − ζ 2 D ( 3 B ) 2,2,2 0 � 2 0 , ( γ 2 τ 1 ) 2 − ζ 2 D ( 3 D ) 1,2,2,2 = k [ 3 D ] / � γ 1 ζ 0 , ζ 0 τ 0 , τ 0 τ 1 , γ 2 γ , τ 1 ζ 2 , ζ 2 γ 2 , τ 0 γ 1 − ζ 2 2 , γ 1 τ 0 − ( τ 1 γ 2 ) 2 � 2 D ( 3 L ) 2,2 = k [ 3 L ] / � ζ 0 τ 0 , τ 2 ζ 0 , ( τ 0 τ 1 τ 2 ) 2 − ζ 2 0 , ( τ 1 τ 2 τ 0 ) 2 τ 1 � D ( 3 Q ) 1,2,2 = k [ 3 Q ] / � ζ 0 τ 0 , τ 2 ζ 0 , τ 0 ζ 1 , ζ 1 τ 1 , τ 0 τ 1 τ 2 − ζ 2 0 , τ 1 τ 2 τ 0 − ζ 2 1 � D ( 3 Q ) 2,2,2 = k [ 3 Q ] / � ζ 0 τ 0 , τ 2 ζ 0 , τ 0 ζ 1 , ζ 1 τ 1 , ( τ 0 τ 1 τ 2 ) 2 − ζ 2 0 , ( τ 1 τ 2 τ 0 ) 2 − ζ 2 1 � D ( 3 R ) 1,2,2,2 = k [ 3 R ] / � ζ 0 τ 0 , τ 0 ζ 1 , ζ 1 τ 1 , τ 1 ζ 2 , ζ 2 τ 2 , τ 2 ζ 0 , τ 0 τ 1 τ 2 − ζ 2 0 , τ 1 τ 2 τ 0 − ζ 2 1 , τ 2 τ 0 τ 1 − ζ 2 2 � String and Band Complexes over Certain Algebra of Dihedral Type J.A. Vélez-Marulanda
A LGEBRAS OF D IHEDRAL T YPE Theorem 1. (T. H OLM , 1999) The algebras D ( 3 B ) 2,2,2 , D ( 3 D ) 1,2,2,2 , D ( 3 Q ) 2,2,2 2 2 and D ( 3 R ) 1,2,2,2 (resp. D ( A ) 2,2 2 , D ( L ) 2,2 and D ( 3 Q ) 1,2,2 ) are derived equiv- alent. Thus, we can restrict ourselves to the algebras Λ 0 = D ( 3 R ) 1,2,2,2 and Λ 1 = D ( 3 Q ) 1,2,2 . Remark 2. The results obtained for Λ 0 can be adjusted for Λ 1 by letting ζ 2 = 1 2 in the graph of 3 Q , where 1 2 denotes the path of length zero that starts and ends at the vertex 2 . String and Band Complexes over Certain Algebra of Dihedral Type J.A. Vélez-Marulanda
I NDECOMPOSABLE P ROJECTIVE Λ 0 - MODULES M [ 1 0 ] M [ 1 1 ] τ 0 τ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ζ 0 . . . . . . ζ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M [ 1 1 ] . . M [ 1 2 ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P 0 = . P 1 = . . . . . . . . . τ 1 . M [ 1 0 ] τ 2 . M [ 1 1 ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M [ 1 2 ] . . M [ 1 0 ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ζ 0 . . ζ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . τ 2 . . . . τ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M [ 1 0 ] M [ 1 1 ] M [ 1 2 ] τ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ζ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M [ 1 0 ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P 2 = . . . . . τ 0 . M [ 1 2 ] . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M [ 1 1 ] . . . . . . . . . . . . . . . . . . . . . ζ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . τ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M [ 1 2 ] String and Band Complexes over Certain Algebra of Dihedral Type J.A. Vélez-Marulanda
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