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

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

Feb 06, 2024 377 likes •777 views

Higher-dimensional Auslander algebras of type A and the higher-dimensional Waldhausen S -constructions Gustavo Jasso 1 (joint with Tobias Dyckerho 2 ) 1 Universitt Bonn 2 Universitt Hamburg th Internaonal Conference on

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Higher-dimensional Auslander algebras of type A and the higher-dimensional Waldhausen S-constructions

Gustavo Jasso1 (joint with Tobias Dyckerho2)

1Universität Bonn 2Universität Hamburg th Internaonal Conference on Representaons of Algebras Prague, Czech Republic, August ,

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Aims for today Relate Iyama’s higher-dimensional Auslander–Reiten theory to construcons in ▶ algebraic topology / homotopy theory ▶ algebraic K-theory Important perspecve Abstract representaon theory in the sense of Groth and Šťovíček

  • G. Jasso (jt. with T. Dyckerho)

th Internaonal Conference on Representaons of Algebras /

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Aims for today Relate Iyama’s higher-dimensional Auslander–Reiten theory to construcons in ▶ algebraic topology / homotopy theory ▶ algebraic K-theory Important perspecve Abstract representaon theory in the sense of Groth and Šťovíček

  • G. Jasso (jt. with T. Dyckerho)

th Internaonal Conference on Representaons of Algebras /

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The Dold-Kan nerve N(A[1])

B B B B B B B B B @

a00 a01 a02 a0;n1 a0n a11 a12 a1;n1 a1n ... . . . . . . ... an2;n1 an2;n an1;n1 an1;n ann

1 C C C C C C C C C A

. For each 0 i n aii = 0 . For all 0 i < j < k n aij aik + ajk = 0

“Euler relaon”

  • G. Jasso (jt. with T. Dyckerho)

th Internaonal Conference on Representaons of Algebras /

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The Dold-Kan nerve N(A[1])

B B B B B B B B B @

a00 a01 a02 a0;n1 a0n a11 a12 a1;n1 a1n ... . . . . . . ... an2;n1 an2;n an1;n1 an1;n ann

1 C C C C C C C C C A

. For each 0 i n aii = 0 . For all 0 i < j < k n aij aik + ajk = 0

“Euler relaon”

  • G. Jasso (jt. with T. Dyckerho)

th Internaonal Conference on Representaons of Algebras /

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The Dold-Kan nerve N(A[1])

B B B B B B B B B @

a00 a01 a02 a0;n1 a0n a11 a12 a1;n1 a1n ... . . . . . . ... an2;n1 an2;n an1;n1 an1;n ann

1 C C C C C C C C C A

. For each 0 i n aii = 0 . For all 0 i < j < k n aij aik + ajk = 0

“Euler relaon”

  • G. Jasso (jt. with T. Dyckerho)

th Internaonal Conference on Representaons of Algebras /

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The Waldhausen S-construction

X00 X01 X02 X0;n1 X0n X11 X12 X1;n1 X1n ... . . . . . . ... Xn2;n1 Xn2;n Xn1;n1 Xn1;n Xnn

. For all i 2 [n] Xii = 0 . For all 0 i < j < k n Xij Xik Xii Xjk is an

  • G. Jasso (jt. with T. Dyckerho)

th Internaonal Conference on Representaons of Algebras /

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The Waldhausen S-construction

X00 X01 X02 X0;n1 X0n X11 X12 X1;n1 X1n ... . . . . . . ... Xn2;n1 Xn2;n Xn1;n1 Xn1;n Xnn

. For all i 2 [n] Xii = 0 . For all 0 i < j < k n Xij Xik Xii Xjk is an

  • G. Jasso (jt. with T. Dyckerho)

th Internaonal Conference on Representaons of Algebras /

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The Waldhausen S-construction

X00 X01 X02 X0;n1 X0n X11 X12 X1;n1 X1n ... . . . . . . ... Xn2;n1 Xn2;n Xn1;n1 Xn1;n Xnn

. For all i 2 [n] Xii = 0 . For all 0 i < j < k n Xij Xik Xii Xjk is an

  • G. Jasso (jt. with T. Dyckerho)

th Internaonal Conference on Representaons of Algebras /

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The Waldhausen S-construction

X00 X01 X02 X0;n1 X0n X11 X12 X1;n1 X1n ... . . . . . . ... Xn2;n1 Xn2;n Xn1;n1 Xn1;n Xnn

. For all i 2 [n] Xii = 0 . For all 0 i < j < k n Xij Xik Xii Xjk is an exact triangle

  • G. Jasso (jt. with T. Dyckerho)

th Internaonal Conference on Representaons of Algebras /

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The Waldhausen S-construction

X00 X01 X02 X0;n1 X0n X11 X12 X1;n1 X1n ... . . . . . . ... Xn2;n1 Xn2;n Xn1;n1 Xn1;n Xnn

. For all i 2 [n] Xii = 0 . For all 0 i < j < k n Xij Xik Xii Xjk is an exact triangle

  • G. Jasso (jt. with T. Dyckerho)

th Internaonal Conference on Representaons of Algebras /

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SLIDE 12

The Waldhausen S-construction

X00 X01 X02 X0;n1 X0n X11 X12 X1;n1 X1n ... . . . . . . ... Xn2;n1 Xn2;n Xn1;n1 Xn1;n Xnn

. For all i 2 [n] Xii = 0 . For all 0 i < j < k n Xij Xik Xii Xjk □ is an exact triangle cobre sequence

  • G. Jasso (jt. with T. Dyckerho)

th Internaonal Conference on Representaons of Algebras /

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biCartesian cubes in stable 1-categories I = f0 ! 1g X : Im+1 ! A (m + 1)-cube

w x y z

f g h

(homotopy) biCartesian

\( )"

w x u y z

f g h

(f) = u = (h)

  • G. Jasso (jt. with T. Dyckerho)

th Internaonal Conference on Representaons of Algebras /

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biCartesian cubes in stable 1-categories I = f0 ! 1g X : Im+1 ! A (m + 1)-cube

w x y z

f g h

(homotopy) biCartesian

\( )"

w x u y z

f g h

(f) = u = (h)

  • G. Jasso (jt. with T. Dyckerho)

th Internaonal Conference on Representaons of Algebras /

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biCartesian cubes in stable 1-categories I = f0 ! 1g X : Im+1 ! A (m + 1)-cube

w x y z

f g h

(homotopy) biCartesian

\( )"

w x u y z

f

□

g

□

h

cofib(f) = u = fib(h)

  • G. Jasso (jt. with T. Dyckerho)

th Internaonal Conference on Representaons of Algebras /

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The Waldhausen Sh2i-construction

X012 X013 X014 X022 X023 X024 X033 X034 X112 X113 X114 X122 X123 X124 X133 X134 X223 X224 X233 X234 . For all 0 i < j < n Xiij = Xijj = 0 . For all 0 i < j < k < l n

Xijk Xijl Xjjk Xjjl Xikk Xikl Xjkk Xjkl

is (homotopy) biCartesian.

  • G. Jasso (jt. with T. Dyckerho)

th Internaonal Conference on Representaons of Algebras /

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The Waldhausen Sh2i-construction

X012 X013 X014 X022 X023 X024 X033 X034 X112 X113 X114 X122 X123 X124 X133 X134 X223 X224 X233 X234 . For all 0 i < j < n Xiij = Xijj = 0 . For all 0 i < j < k < l n

Xijk Xijl Xjjk Xjjl Xikk Xikl Xjkk Xjkl

is (homotopy) biCartesian.

  • G. Jasso (jt. with T. Dyckerho)

th Internaonal Conference on Representaons of Algebras /

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The Waldhausen Sh2i-construction

X012 X013 X014 X022 X023 X024 X033 X034 X112 X113 X114 X122 X123 X124 X133 X134 X223 X224 X233 X234 . For all 0 i < j < n Xiij = Xijj = 0 . For all 0 i < j < k < l n

Xijk Xijl Xjjk Xjjl Xikk Xikl Xjkk Xjkl

is (homotopy) biCartesian.

  • G. Jasso (jt. with T. Dyckerho)

th Internaonal Conference on Representaons of Algebras /

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Shmi(A)n

  • ! Fun(P(m; n); A)

X012 X013 X014 X023 X024 X034

  • G. Jasso (jt. with T. Dyckerho)

th Internaonal Conference on Representaons of Algebras /

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Shmi(A)n

  • ! Fun(P(m; n); A)

X012 X013 X014 X023 X024 X034

  • G. Jasso (jt. with T. Dyckerho)

th Internaonal Conference on Representaons of Algebras /

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Shmi(A)n

  • ! Fun(P(m; n); A)

X012 X013 X014 X023 X024 X034 X123

  • G. Jasso (jt. with T. Dyckerho)

th Internaonal Conference on Representaons of Algebras /

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Shmi(A)n

  • ! Fun(P(m; n); A)

X012 X013 X014 X023 X024 X034 X123

  • G. Jasso (jt. with T. Dyckerho)

th Internaonal Conference on Representaons of Algebras /

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Shmi(A)n

  • ! Fun(P(m; n); A)

X012 X013 X014 X023 X024 X034 X123 X124

  • G. Jasso (jt. with T. Dyckerho)

th Internaonal Conference on Representaons of Algebras /

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Shmi(A)n

  • ! Fun(P(m; n); A)

X012 X013 X014 X023 X024 X034 X123 X124

  • G. Jasso (jt. with T. Dyckerho)

th Internaonal Conference on Representaons of Algebras /

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Shmi(A)n

  • ! Fun(P(m; n); A)

X012 X013 X014 X023 X024 X034 X123 X124 X134

  • G. Jasso (jt. with T. Dyckerho)

th Internaonal Conference on Representaons of Algebras /

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Shmi(A)n

  • ! Fun(P(m; n); A)

X012 X013 X014 X023 X024 X034 X123 X124 X134

  • G. Jasso (jt. with T. Dyckerho)

th Internaonal Conference on Representaons of Algebras /

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Shmi(A)n

  • ! Fun(P(m; n); A)

X012 X013 X014 X023 X024 X034 X123 X124 X134 X234

  • G. Jasso (jt. with T. Dyckerho)

th Internaonal Conference on Representaons of Algebras /

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Shmi(A)n

  • ! Fun(S; A)

X013 X014 X023 X024 X034 X123

  • G. Jasso (jt. with T. Dyckerho)

th Internaonal Conference on Representaons of Algebras /

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Shmi(A)n

  • ! Fun(S; A)

X013 X014 X023 X024 X034 X123

  • G. Jasso (jt. with T. Dyckerho)

th Internaonal Conference on Representaons of Algebras /

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Shmi(A)n

  • ! Fun(S; A)

X013 X014 X023 X024 X034 X123 X012

  • G. Jasso (jt. with T. Dyckerho)

th Internaonal Conference on Representaons of Algebras /

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Shmi(A)n

  • ! Fun(S; A)

X013 X014 X023 X024 X034 X123 X012

  • G. Jasso (jt. with T. Dyckerho)

th Internaonal Conference on Representaons of Algebras /

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Shmi(A)n

  • ! Fun(S; A)

X013 X014 X023 X024 X034 X123 X012 X124

  • G. Jasso (jt. with T. Dyckerho)

th Internaonal Conference on Representaons of Algebras /

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Shmi(A)n

  • ! Fun(S; A)

X013 X014 X023 X024 X034 X123 X012 X124

  • G. Jasso (jt. with T. Dyckerho)

th Internaonal Conference on Representaons of Algebras /

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Shmi(A)n

  • ! Fun(S; A)

X013 X014 X023 X024 X034 X123 X012 X124 X134

  • G. Jasso (jt. with T. Dyckerho)

th Internaonal Conference on Representaons of Algebras /

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Shmi(A)n

  • ! Fun(S; A)

X013 X014 X023 X024 X034 X123 X012 X124 X134

  • G. Jasso (jt. with T. Dyckerho)

th Internaonal Conference on Representaons of Algebras /

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Shmi(A)n

  • ! Fun(S; A)

X013 X014 X023 X024 X034 X123 X012 X124 X134 X234

  • G. Jasso (jt. with T. Dyckerho)

th Internaonal Conference on Representaons of Algebras /

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Recollements Shmi

n (A)

Shmi

n+1(A)

Shm1i

n

(A)

si di di+1

a d0 a s0 a d1 a s1 a a dn a sn a dn+1 a

  • G. Jasso (jt. with T. Dyckerho)

th Internaonal Conference on Representaons of Algebras /

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Thank you for your aenon!

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