Higher-dimensional Auslander algebras of type A and the - PowerPoint PPT Presentation

Higher-dimensional Auslander algebras of type A and the higher-dimensional Waldhausen S -constructions Gustavo Jasso 1 (joint with Tobias Dyckerho� 2 ) 1 Universität Bonn 2 Universität Hamburg ��th Interna�onal Conference on Representa�ons of Algebras Prague, Czech Republic, August ��, ����

Important perspec�ve Abstract representa�on theory in the sense of Groth and Šťovíček Aims for today Relate Iyama’s higher-dimensional Auslander–Reiten theory to construc�ons in ▶ algebraic topology / homotopy theory ▶ algebraic K -theory G. Jasso (jt. with T. Dyckerho�) ��th Interna�onal Conference on Representa�ons of Algebras �/�

Aims for today Relate Iyama’s higher-dimensional Auslander–Reiten theory to construc�ons in ▶ algebraic topology / homotopy theory ▶ algebraic K -theory Important perspec�ve Abstract representa�on theory in the sense of Groth and Šťovíček G. Jasso (jt. with T. Dyckerho�) ��th Interna�onal Conference on Representa�ons of Algebras �/�

�. For each 0 � i � n a ii = 0 �. For all 0 � i < j < k � n a ij � a ik + a jk = 0 “Euler rela�on” The Dold-Kan nerve N ( A [1]) 0 1 a 00 a 01 a 02 � � � a 0 ;n � 1 a 0 n a 11 a 12 a 1 ;n � 1 a 1 n � � � B C B C . . ... . . B C . . B C B C ... B C a n � 2 ;n � 1 a n � 2 ;n B C B C B a n � 1 ;n � 1 a n � 1 ;n C @ A a nn G. Jasso (jt. with T. Dyckerho�) ��th Interna�onal Conference on Representa�ons of Algebras �/�

�. For all 0 � i < j < k � n a ij � a ik + a jk = 0 “Euler rela�on” The Dold-Kan nerve N ( A [1]) �. For each 0 � i � n 0 1 a 00 a 01 a 02 � � � a 0 ;n � 1 a 0 n a 11 a 12 a 1 ;n � 1 a 1 n � � � a ii = 0 B C B C . . ... . . B C . . B C B C ... B C a n � 2 ;n � 1 a n � 2 ;n B C B C B a n � 1 ;n � 1 a n � 1 ;n C @ A a nn G. Jasso (jt. with T. Dyckerho�) ��th Interna�onal Conference on Representa�ons of Algebras �/�

The Dold-Kan nerve N ( A [1]) �. For each 0 � i � n 0 1 a 00 a 01 a 02 � � � a 0 ;n � 1 a 0 n a 11 a 12 a 1 ;n � 1 a 1 n � � � a ii = 0 B C B C . . ... . . B C . . B C �. For all 0 � i < j < k � n B C ... B C a n � 2 ;n � 1 a n � 2 ;n B C B C a ij � a ik + a jk = 0 B a n � 1 ;n � 1 a n � 1 ;n C @ A a nn “Euler rela�on” G. Jasso (jt. with T. Dyckerho�) ��th Interna�onal Conference on Representa�ons of Algebras �/�

�. For all i 2 [ n ] X ii = 0 �. For all 0 � i < j < k � n X ij X ik X ii X jk is an The Waldhausen S -construction X 00 X 01 X 02 X 0 ;n � 1 X 0 n � � � X 11 X 12 X 1 ;n � 1 X 1 n � � � . . ... . . . . ... X n � 2 ;n � 1 X n � 2 ;n X n � 1 ;n � 1 X n � 1 ;n X nn G. Jasso (jt. with T. Dyckerho�) ��th Interna�onal Conference on Representa�ons of Algebras �/�

�. For all 0 � i < j < k � n X ij X ik X ii X jk is an The Waldhausen S -construction �. For all i 2 [ n ] X 00 X 01 X 02 X 0 ;n � 1 X 0 n � � � X ii = 0 X 11 X 12 X 1 ;n � 1 X 1 n � � � . . ... . . . . ... X n � 2 ;n � 1 X n � 2 ;n X n � 1 ;n � 1 X n � 1 ;n X nn G. Jasso (jt. with T. Dyckerho�) ��th Interna�onal Conference on Representa�ons of Algebras �/�

�. For all 0 � i < j < k � n X ij X ik X ii X jk is an The Waldhausen S -construction �. For all i 2 [ n ] X 00 X 01 X 02 X 0 ;n � 1 X 0 n � � � X ii = 0 X 11 X 12 X 1 ;n � 1 X 1 n � � � . . ... . . . . ... X n � 2 ;n � 1 X n � 2 ;n X n � 1 ;n � 1 X n � 1 ;n X nn G. Jasso (jt. with T. Dyckerho�) ��th Interna�onal Conference on Representa�ons of Algebras �/�

The Waldhausen S -construction �. For all i 2 [ n ] X 00 X 01 X 02 � � � X 0 ;n � 1 X 0 n X ii = 0 X 11 X 12 � � � X 1 ;n � 1 X 1 n �. For all 0 � i < j < k � n . . ... . . . . X ij X ik ... X n � 2 ;n � 1 X n � 2 ;n X ii X jk X n � 1 ;n � 1 X n � 1 ;n X nn is an exact triangle G. Jasso (jt. with T. Dyckerho�) ��th Interna�onal Conference on Representa�ons of Algebras �/�

The Waldhausen S -construction �. For all i 2 [ n ] X 00 X 01 X 02 � � � X 0 ;n � 1 X 0 n X ii = 0 X 11 X 12 � � � X 1 ;n � 1 X 1 n �. For all 0 � i < j < k � n . . ... . . . . X ij X ik ... X n � 2 ;n � 1 X n � 2 ;n X ii X jk X n � 1 ;n � 1 X n � 1 ;n X nn is an exact triangle G. Jasso (jt. with T. Dyckerho�) ��th Interna�onal Conference on Representa�ons of Algebras �/�

The Waldhausen S -construction �. For all i 2 [ n ] X 00 X 01 X 02 � � � X 0 ;n � 1 X 0 n X ii = 0 X 11 X 12 � � � X 1 ;n � 1 X 1 n �. For all 0 � i < j < k � n . . ... . . . . X ij X ik ... X n � 2 ;n � 1 X n � 2 ;n □ X ii X jk X n � 1 ;n � 1 X n � 1 ;n X nn is an exact triangle co�bre sequence G. Jasso (jt. with T. Dyckerho�) ��th Interna�onal Conference on Representa�ons of Algebras �/�

f f w x w x g 0 0 g 0 u y \ ( ) " 0 y h h 0 z 0 z ( f ) � = u � = ( h ) (homotopy) biCartesian biCartesian cubes in stable 1 -categories X : I m +1 ! A I = f 0 ! 1 g ( m + 1) -cube G. Jasso (jt. with T. Dyckerho�) ��th Interna�onal Conference on Representa�ons of Algebras �/�

f w x g 0 u y h 0 z ( f ) � = u � = ( h ) biCartesian cubes in stable 1 -categories X : I m +1 ! A I = f 0 ! 1 g ( m + 1) -cube f w x 0 0 g \ ( ) " 0 y h 0 z (homotopy) biCartesian G. Jasso (jt. with T. Dyckerho�) ��th Interna�onal Conference on Representa�ons of Algebras �/�

biCartesian cubes in stable 1 -categories X : I m +1 ! A I = f 0 ! 1 g ( m + 1) -cube f f w x w x g □ 0 0 g 0 u y \ ( ) " 0 y □ h h 0 z 0 z cofib ( f ) � = u � = fib ( h ) (homotopy) biCartesian G. Jasso (jt. with T. Dyckerho�) ��th Interna�onal Conference on Representa�ons of Algebras �/�

�. For all 0 � i < j < n X iij = X ijj = 0 �. For all 0 � i < j < k < l � n X ijk X ijl X jjk X jjl X ikk X ikl X jkk X jkl is (homotopy) biCartesian. The Waldhausen S h 2 i -construction X 012 X 013 X 014 X 112 X 113 X 114 X 022 X 023 X 024 X 122 X 123 X 124 X 223 X 224 X 033 X 034 X 133 X 134 X 233 X 234 G. Jasso (jt. with T. Dyckerho�) ��th Interna�onal Conference on Representa�ons of Algebras �/�

�. For all 0 � i < j < k < l � n X ijk X ijl X jjk X jjl X ikk X ikl X jkk X jkl is (homotopy) biCartesian. The Waldhausen S h 2 i -construction �. For all 0 � i < j < n X 012 X 013 X 014 X iij = X ijj = 0 X 112 X 113 X 114 X 022 X 023 X 024 X 122 X 123 X 124 X 223 X 224 X 033 X 034 X 133 X 134 X 233 X 234 G. Jasso (jt. with T. Dyckerho�) ��th Interna�onal Conference on Representa�ons of Algebras �/�

The Waldhausen S h 2 i -construction �. For all 0 � i < j < n X 012 X 013 X 014 X iij = X ijj = 0 X 112 X 113 X 114 �. For all 0 � i < j < k < l � n X 022 X 023 X 024 X ijk X ijl X 122 X 123 X 124 X jjk X jjl X 223 X 224 X 033 X 034 X ikk X ikl X jkk X jkl X 133 X 134 X 233 X 234 is (homotopy) biCartesian. G. Jasso (jt. with T. Dyckerho�) ��th Interna�onal Conference on Representa�ons of Algebras �/�

� S h m i ( A ) n � ! Fun � ( P ( m; n ) ; A ) X 012 X 013 X 014 0 X 023 X 024 0 X 034 G. Jasso (jt. with T. Dyckerho�) ��th Interna�onal Conference on Representa�ons of Algebras �/�

� S h m i ( A ) n � ! Fun � ( P ( m; n ) ; A ) X 012 X 013 X 014 0 0 0 X 023 X 024 0 0 X 034 G. Jasso (jt. with T. Dyckerho�) ��th Interna�onal Conference on Representa�ons of Algebras �/�

� S h m i ( A ) n � ! Fun � ( P ( m; n ) ; A ) X 012 X 013 X 014 0 0 0 X 023 X 024 0 X 123 0 X 034 G. Jasso (jt. with T. Dyckerho�) ��th Interna�onal Conference on Representa�ons of Algebras �/�

� S h m i ( A ) n � ! Fun � ( P ( m; n ) ; A ) X 012 X 013 X 014 0 0 0 0 X 023 X 024 0 X 123 0 X 034 G. Jasso (jt. with T. Dyckerho�) ��th Interna�onal Conference on Representa�ons of Algebras �/�

� S h m i ( A ) n � ! Fun � ( P ( m; n ) ; A ) X 012 X 013 X 014 0 0 0 0 X 023 X 024 0 X 123 X 124 0 X 034 G. Jasso (jt. with T. Dyckerho�) ��th Interna�onal Conference on Representa�ons of Algebras �/�

� S h m i ( A ) n � ! Fun � ( P ( m; n ) ; A ) X 012 X 013 X 014 0 0 0 0 X 023 X 024 0 X 123 X 124 0 X 034 0 G. Jasso (jt. with T. Dyckerho�) ��th Interna�onal Conference on Representa�ons of Algebras �/�

� S h m i ( A ) n � ! Fun � ( P ( m; n ) ; A ) X 012 X 013 X 014 0 0 0 0 X 023 X 024 0 X 123 X 124 0 X 034 0 X 134 G. Jasso (jt. with T. Dyckerho�) ��th Interna�onal Conference on Representa�ons of Algebras �/�

� S h m i ( A ) n � ! Fun � ( P ( m; n ) ; A ) X 012 X 013 X 014 0 0 0 0 X 023 X 024 0 X 123 X 124 0 0 0 X 034 0 X 134 0 G. Jasso (jt. with T. Dyckerho�) ��th Interna�onal Conference on Representa�ons of Algebras �/�