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Exotic triangulated categories Fernando Muro Universitat de Barcelona, Dept. lgebra i Geometria (joint work with S. Schwede and N. Strickland) Barcelona Topology Workshop 2007 Fernando Muro Exotic triangulated categories Goal Exhibiting

  1. Models for triangulated categories More generally we say that T has a model if there is an exact inclusion T ⊂ Ho M . Neeman defined a K -theory K ( T ) for triangulated categories. Theorem (Neeman’97) Let A be an abelian category and let T be a triangulated category with a bounded t-structure with heart A . If T admits a Waldhausen model then K ( A ) ≃ K ( T ) . Example T = D b ( A ) ⊂ D ( A ) = Ho Ch ( A ) . Neeman’s theorem can be used to obtain K ( A ) ≃ K ( B ) by embedding adequately two abelian categories A , B in T . Fernando Muro Exotic triangulated categories

  2. Models for triangulated categories More generally we say that T has a model if there is an exact inclusion T ⊂ Ho M . Neeman defined a K -theory K ( T ) for triangulated categories. Theorem (Neeman’97) Let A be an abelian category and let T be a triangulated category with a bounded t-structure with heart A . If T admits a Waldhausen model then K ( A ) ≃ K ( T ) . Example T = D b ( A ) ⊂ D ( A ) = Ho Ch ( A ) . Neeman’s theorem can be used to obtain K ( A ) ≃ K ( B ) by embedding adequately two abelian categories A , B in T . Fernando Muro Exotic triangulated categories

  3. Models for triangulated categories More generally we say that T has a model if there is an exact inclusion T ⊂ Ho M . Neeman defined a K -theory K ( T ) for triangulated categories. Theorem (Neeman’97) Let A be an abelian category and let T be a triangulated category with a bounded t-structure with heart A . If T admits a Waldhausen model then K ( A ) ≃ K ( T ) . Example T = D b ( A ) ⊂ D ( A ) = Ho Ch ( A ) . Neeman’s theorem can be used to obtain K ( A ) ≃ K ( B ) by embedding adequately two abelian categories A , B in T . Fernando Muro Exotic triangulated categories

  4. Models for triangulated categories More generally we say that T has a model if there is an exact inclusion T ⊂ Ho M . Neeman defined a K -theory K ( T ) for triangulated categories. Theorem (Neeman’97) Let A be an abelian category and let T be a triangulated category with a bounded t-structure with heart A . If T admits a Waldhausen model then K ( A ) ≃ K ( T ) . Example T = D b ( A ) ⊂ D ( A ) = Ho Ch ( A ) . Neeman’s theorem can be used to obtain K ( A ) ≃ K ( B ) by embedding adequately two abelian categories A , B in T . Fernando Muro Exotic triangulated categories

  5. A triangulated category without models Theorem A The category F ( Z / 4 ) of finitely generated free Z / 4 -modules has a unique triangulated structure with Σ = identity and exact triangle 2 2 2 Z / 4 → Z / 4 → Z / 4 → Z / 4 . − − − proof Theorem B There are not non-trivial exact functors F ( Z / 4 ) − → Ho M , Ho M − → F ( Z / 4 ) . proof Corollary F ( Z / 4 ) does not have models. remarks Fernando Muro Exotic triangulated categories

  6. A triangulated category without models Theorem A The category F ( Z / 4 ) of finitely generated free Z / 4 -modules has a unique triangulated structure with Σ = identity and exact triangle 2 2 2 Z / 4 → Z / 4 → Z / 4 → Z / 4 . − − − proof Theorem B There are not non-trivial exact functors F ( Z / 4 ) − → Ho M , Ho M − → F ( Z / 4 ) . proof Corollary F ( Z / 4 ) does not have models. remarks Fernando Muro Exotic triangulated categories

  7. A triangulated category without models Theorem A The category F ( Z / 4 ) of finitely generated free Z / 4 -modules has a unique triangulated structure with Σ = identity and exact triangle 2 2 2 Z / 4 → Z / 4 → Z / 4 → Z / 4 . − − − proof Theorem B There are not non-trivial exact functors F ( Z / 4 ) − → Ho M , Ho M − → F ( Z / 4 ) . proof Corollary F ( Z / 4 ) does not have models. remarks Fernando Muro Exotic triangulated categories

  8. F ( Z / 4 ) is triangulated Two candidate triangle morphisms ( α, β, γ ) and ( α ′ , β ′ , γ ′ ) are homotopic if there are morphisms (Θ , Φ , Ψ) q f i � B � C � Σ A A � B ′ � C ′ q ′ � Σ A ′ A ′ f ′ i ′ such that β ′ − β Φ i + f ′ Θ , = γ ′ − γ Ψ q + i ′ Φ , = Σ( α ′ − α ) Σ(Θ f ) + q ′ Ψ . = Fernando Muro Exotic triangulated categories

  9. F ( Z / 4 ) is triangulated � � � � � � � Two candidate triangle morphisms ( α, β, γ ) and ( α ′ , β ′ , γ ′ ) are homotopic if there are morphisms (Θ , Φ , Ψ) q f i A B C Σ A γ α β Σ α � B ′ � C ′ q ′ � Σ A ′ A ′ f ′ i ′ such that β ′ − β Φ i + f ′ Θ , = γ ′ − γ Ψ q + i ′ Φ , = Σ( α ′ − α ) Σ(Θ f ) + q ′ Ψ . = Fernando Muro Exotic triangulated categories

  10. F ( Z / 4 ) is triangulated � � � � � � � Two candidate triangle morphisms ( α, β, γ ) and ( α ′ , β ′ , γ ′ ) are homotopic if there are morphisms (Θ , Φ , Ψ) q f i A B C Σ A β ′ γ ′ α ′ Σ α ′ � B ′ � C ′ q ′ � Σ A ′ A ′ f ′ i ′ such that β ′ − β Φ i + f ′ Θ , = γ ′ − γ Ψ q + i ′ Φ , = Σ( α ′ − α ) Σ(Θ f ) + q ′ Ψ . = Fernando Muro Exotic triangulated categories

  11. F ( Z / 4 ) is triangulated � � � � � � � � � � � Two candidate triangle morphisms ( α, β, γ ) and ( α ′ , β ′ , γ ′ ) are homotopic if there are morphisms (Θ , Φ , Ψ) q f i A B C Σ A γ α β β ′ γ ′ α ′ Σ α Σ α ′ � B ′ � C ′ q ′ � Σ A ′ A ′ f ′ i ′ such that β ′ − β Φ i + f ′ Θ , = γ ′ − γ Ψ q + i ′ Φ , = Σ( α ′ − α ) Σ(Θ f ) + q ′ Ψ . = Fernando Muro Exotic triangulated categories

  12. F ( Z / 4 ) is triangulated � � � � � � � � � � � Two candidate triangle morphisms ( α, β, γ ) and ( α ′ , β ′ , γ ′ ) are homotopic if there are morphisms (Θ , Φ , Ψ) q f i A B C Σ A γ α β β ′ γ ′ α ′ Σ α Σ α ′ � B ′ � C ′ q ′ � Σ A ′ A ′ f ′ i ′ such that β ′ − β Φ i + f ′ Θ , = γ ′ − γ Ψ q + i ′ Φ , = Σ( α ′ − α ) Σ(Θ f ) + q ′ Ψ . = Fernando Muro Exotic triangulated categories

  13. F ( Z / 4 ) is triangulated � � � � � � � � � � � Two candidate triangle morphisms ( α, β, γ ) and ( α ′ , β ′ , γ ′ ) are homotopic if there are morphisms (Θ , Φ , Ψ) q f i A B C Σ A � � � Θ Φ Ψ � � � � � � γ α β β ′ γ ′ α ′ � � � Σ α Σ α ′ � � � � � � � � � � � � � � � � B ′ � C ′ q ′ � Σ A ′ A ′ f ′ i ′ such that β ′ − β Φ i + f ′ Θ , = γ ′ − γ Ψ q + i ′ Φ , = Σ( α ′ − α ) Σ(Θ f ) + q ′ Ψ . = Fernando Muro Exotic triangulated categories

  14. F ( Z / 4 ) is triangulated � � � � � � � � � � � Two candidate triangle morphisms ( α, β, γ ) and ( α ′ , β ′ , γ ′ ) are homotopic if there are morphisms (Θ , Φ , Ψ) q f i A B C Σ A � � � Θ Φ Ψ � � � � � � γ α β β ′ γ ′ α ′ � � � Σ α Σ α ′ � � � � � � � � � � � � � � � � B ′ � C ′ q ′ � Σ A ′ A ′ f ′ i ′ such that β ′ − β Φ i + f ′ Θ , = γ ′ − γ Ψ q + i ′ Φ , = Σ( α ′ − α ) Σ(Θ f ) + q ′ Ψ . = Fernando Muro Exotic triangulated categories

  15. F ( Z / 4 ) is triangulated Homotopic morphisms have isomorphic mapping cones. A candidate triangle is contractible if the identity morphism is homotopic to the zero morphism. The exact triangles in F ( Z / 4 ) are the candidate triangles isomorphic to the direct sum of a contractible triangle and a triangle X 2 of the form 2 2 2 X − → X − → X − → X for some X ∈ F ( Z / 4 ) . Fernando Muro Exotic triangulated categories

  16. F ( Z / 4 ) is triangulated Homotopic morphisms have isomorphic mapping cones. A candidate triangle is contractible if the identity morphism is homotopic to the zero morphism. The exact triangles in F ( Z / 4 ) are the candidate triangles isomorphic to the direct sum of a contractible triangle and a triangle X 2 of the form 2 2 2 X − → X − → X − → X for some X ∈ F ( Z / 4 ) . Fernando Muro Exotic triangulated categories

  17. F ( Z / 4 ) is triangulated Homotopic morphisms have isomorphic mapping cones. A candidate triangle is contractible if the identity morphism is homotopic to the zero morphism. The exact triangles in F ( Z / 4 ) are the candidate triangles isomorphic to the direct sum of a contractible triangle and a triangle X 2 of the form 2 2 2 X − → X − → X − → X for some X ∈ F ( Z / 4 ) . Fernando Muro Exotic triangulated categories

  18. F ( Z / 4 ) is triangulated Let us check that F ( Z / 4 ) is triangulated. A → A → 0 → A is contractible. The translate of a contractible triangle is contractible. The translate of X 2 is X 2 . Any morphism in F ( Z / 4 ) is of the form   1 0 0 0 2 0  : W ⊕ X ⊕ Y − → W ⊕ X ⊕ Z .  0 0 0 It can be extended to an exact triangle which is the direct sum of X 2 and the contractible triangle „ 1 0 „ 0 0 „ 0 0 « « « 0 0 0 1 1 0 � W ⊕ Z � Y ⊕ Z � W ⊕ Y . W ⊕ Y Fernando Muro Exotic triangulated categories

  19. F ( Z / 4 ) is triangulated Let us check that F ( Z / 4 ) is triangulated. A → A → 0 → A is contractible. The translate of a contractible triangle is contractible. The translate of X 2 is X 2 . Any morphism in F ( Z / 4 ) is of the form   1 0 0 0 2 0  : W ⊕ X ⊕ Y − → W ⊕ X ⊕ Z .  0 0 0 It can be extended to an exact triangle which is the direct sum of X 2 and the contractible triangle „ 1 0 „ 0 0 „ 0 0 « « « 0 0 0 1 1 0 � W ⊕ Z � Y ⊕ Z � W ⊕ Y . W ⊕ Y Fernando Muro Exotic triangulated categories

  20. F ( Z / 4 ) is triangulated Let us check that F ( Z / 4 ) is triangulated. A → A → 0 → A is contractible. The translate of a contractible triangle is contractible. The translate of X 2 is X 2 . Any morphism in F ( Z / 4 ) is of the form   1 0 0 0 2 0  : W ⊕ X ⊕ Y − → W ⊕ X ⊕ Z .  0 0 0 It can be extended to an exact triangle which is the direct sum of X 2 and the contractible triangle „ 1 0 „ 0 0 „ 0 0 « « « 0 0 0 1 1 0 � W ⊕ Z � Y ⊕ Z � W ⊕ Y . W ⊕ Y Fernando Muro Exotic triangulated categories

  21. F ( Z / 4 ) is triangulated Let us check that F ( Z / 4 ) is triangulated. A → A → 0 → A is contractible. The translate of a contractible triangle is contractible. The translate of X 2 is X 2 . Any morphism in F ( Z / 4 ) is of the form   1 0 0 0 2 0  : W ⊕ X ⊕ Y − → W ⊕ X ⊕ Z .  0 0 0 It can be extended to an exact triangle which is the direct sum of X 2 and the contractible triangle „ 1 0 „ 0 0 „ 0 0 « « « 0 0 0 1 1 0 � W ⊕ Z � Y ⊕ Z � W ⊕ Y . W ⊕ Y Fernando Muro Exotic triangulated categories

  22. F ( Z / 4 ) is triangulated Let us check that F ( Z / 4 ) is triangulated. A → A → 0 → A is contractible. The translate of a contractible triangle is contractible. The translate of X 2 is X 2 . Any morphism in F ( Z / 4 ) is of the form   1 0 0 0 2 0  : W ⊕ X ⊕ Y − → W ⊕ X ⊕ Z .  0 0 0 It can be extended to an exact triangle which is the direct sum of X 2 and the contractible triangle „ 1 0 „ 0 0 „ 0 0 « « « 0 0 0 1 1 0 � W ⊕ Z � Y ⊕ Z � W ⊕ Y . W ⊕ Y Fernando Muro Exotic triangulated categories

  23. F ( Z / 4 ) is triangulated Let us check that F ( Z / 4 ) is triangulated. A → A → 0 → A is contractible. The translate of a contractible triangle is contractible. The translate of X 2 is X 2 . Any morphism in F ( Z / 4 ) is of the form   1 0 0 0 2 0  : W ⊕ X ⊕ Y − → W ⊕ X ⊕ Z .  0 0 0 It can be extended to an exact triangle which is the direct sum of X 2 and the contractible triangle „ 1 0 „ 0 0 „ 0 0 « « « 0 0 0 1 1 0 � W ⊕ Z � Y ⊕ Z � W ⊕ Y . W ⊕ Y Fernando Muro Exotic triangulated categories

  24. F ( Z / 4 ) is triangulated � � � � � Let us check that we can extend commutative squares 2 2 2 � X X X X α α β � Y � Y � Y Y 2 2 2 for any δ : X → Y . Suppose that   1 0 0 0 2 0  : X = L ⊕ M ⊕ N − α = → L ⊕ M ⊕ P = Y .  0 0 0 Fernando Muro Exotic triangulated categories

  25. F ( Z / 4 ) is triangulated � � � � � Let us check that we can extend commutative squares 2 2 2 � X X X X α α β � Y � Y � Y Y 2 2 2 for any δ : X → Y . Suppose that   1 0 0 0 2 0  : X = L ⊕ M ⊕ N − α = → L ⊕ M ⊕ P = Y .  0 0 0 Fernando Muro Exotic triangulated categories

  26. F ( Z / 4 ) is triangulated � � � � � � � Let us check that we can extend commutative squares 2 2 2 X X X X β + 2 · δ α α β � Y � Y � Y Y 2 2 2 for any δ : X → Y . Suppose that   1 0 0 0 2 0  : X = L ⊕ M ⊕ N − α = → L ⊕ M ⊕ P = Y .  0 0 0 Fernando Muro Exotic triangulated categories

  27. F ( Z / 4 ) is triangulated � � � � � � � Let us check that we can extend commutative squares 2 2 2 X X X X β + 2 · δ α α β � Y � Y � Y Y 2 2 2 for any δ : X → Y . Suppose that   1 0 0 0 2 0  : X = L ⊕ M ⊕ N − α = → L ⊕ M ⊕ P = Y .  0 0 0 Fernando Muro Exotic triangulated categories

  28. F ( Z / 4 ) is triangulated Let   0 0 0 0 1 0  : X = L ⊕ M ⊕ N − δ = → L ⊕ M ⊕ P = Y .  0 0 0 Since 2 · α = 2 · β then β = α + 2 · Φ for some Φ: X → Y , so ( δ, Φ , 0 ) is a homotopy from λ = ( α, β, β + 2 · δ ) to µ = ( α + 2 · δ, α + 2 · δ, α + 2 · δ ) ,   1 0 0 α + 2 · δ = 0 0 0  : X = L ⊕ M ⊕ N − → L ⊕ M ⊕ P = Y .  0 0 0 The mapping cone of µ (isomorphic to the mapping cone of λ ) is ( mapping cone of 1 L 2 ) exact . ⊕ M 2 ⊕ M 2 ⊕ N 2 ⊕ P 2 , � �� � contractible back remarks Fernando Muro Exotic triangulated categories

  29. F ( Z / 4 ) is triangulated Let   0 0 0 0 1 0  : X = L ⊕ M ⊕ N − δ = → L ⊕ M ⊕ P = Y .  0 0 0 Since 2 · α = 2 · β then β = α + 2 · Φ for some Φ: X → Y , so ( δ, Φ , 0 ) is a homotopy from λ = ( α, β, β + 2 · δ ) to µ = ( α + 2 · δ, α + 2 · δ, α + 2 · δ ) ,   1 0 0 α + 2 · δ = 0 0 0  : X = L ⊕ M ⊕ N − → L ⊕ M ⊕ P = Y .  0 0 0 The mapping cone of µ (isomorphic to the mapping cone of λ ) is ( mapping cone of 1 L 2 ) exact . ⊕ M 2 ⊕ M 2 ⊕ N 2 ⊕ P 2 , � �� � contractible back remarks Fernando Muro Exotic triangulated categories

  30. F ( Z / 4 ) is triangulated Let   0 0 0 0 1 0  : X = L ⊕ M ⊕ N − δ = → L ⊕ M ⊕ P = Y .  0 0 0 Since 2 · α = 2 · β then β = α + 2 · Φ for some Φ: X → Y , so ( δ, Φ , 0 ) is a homotopy from λ = ( α, β, β + 2 · δ ) to µ = ( α + 2 · δ, α + 2 · δ, α + 2 · δ ) ,   1 0 0 α + 2 · δ = 0 0 0  : X = L ⊕ M ⊕ N − → L ⊕ M ⊕ P = Y .  0 0 0 The mapping cone of µ (isomorphic to the mapping cone of λ ) is ( mapping cone of 1 L 2 ) exact . ⊕ M 2 ⊕ M 2 ⊕ N 2 ⊕ P 2 , � �� � contractible back remarks Fernando Muro Exotic triangulated categories

  31. F ( Z / 4 ) is triangulated Let   0 0 0 0 1 0  : X = L ⊕ M ⊕ N − δ = → L ⊕ M ⊕ P = Y .  0 0 0 Since 2 · α = 2 · β then β = α + 2 · Φ for some Φ: X → Y , so ( δ, Φ , 0 ) is a homotopy from λ = ( α, β, β + 2 · δ ) to µ = ( α + 2 · δ, α + 2 · δ, α + 2 · δ ) ,   1 0 0 α + 2 · δ = 0 0 0  : X = L ⊕ M ⊕ N − → L ⊕ M ⊕ P = Y .  0 0 0 The mapping cone of µ (isomorphic to the mapping cone of λ ) is ( mapping cone of 1 L 2 ) exact . ⊕ M 2 ⊕ M 2 ⊕ N 2 ⊕ P 2 , � �� � contractible back remarks Fernando Muro Exotic triangulated categories

  32. F ( Z / 4 ) is triangulated Let   0 0 0 0 1 0  : X = L ⊕ M ⊕ N − δ = → L ⊕ M ⊕ P = Y .  0 0 0 Since 2 · α = 2 · β then β = α + 2 · Φ for some Φ: X → Y , so ( δ, Φ , 0 ) is a homotopy from λ = ( α, β, β + 2 · δ ) to µ = ( α + 2 · δ, α + 2 · δ, α + 2 · δ ) ,   1 0 0 α + 2 · δ = 0 0 0  : X = L ⊕ M ⊕ N − → L ⊕ M ⊕ P = Y .  0 0 0 The mapping cone of µ (isomorphic to the mapping cone of λ ) is ( mapping cone of 1 L 2 ) exact . ⊕ M 2 ⊕ M 2 ⊕ N 2 ⊕ P 2 , � �� � contractible back remarks Fernando Muro Exotic triangulated categories

  33. F ( Z / 4 ) is triangulated A candidate triangle in F ( Z / 4 ) f i q A − → B − → C − → A is quasi-exact if q f i f A − → B − → C − → A → B is an exact sequence of Z / 4-modules. Example X 2 is quasi-exact. Contractible triangles are quasi-exact. Fernando Muro Exotic triangulated categories

  34. F ( Z / 4 ) is triangulated A candidate triangle in F ( Z / 4 ) f i q A − → B − → C − → A is quasi-exact if q f i f A − → B − → C − → A → B is an exact sequence of Z / 4-modules. Example X 2 is quasi-exact. Contractible triangles are quasi-exact. Fernando Muro Exotic triangulated categories

  35. F ( Z / 4 ) is triangulated A candidate triangle in F ( Z / 4 ) f i q A − → B − → C − → A is quasi-exact if q f i f A − → B − → C − → A → B is an exact sequence of Z / 4-modules. Example X 2 is quasi-exact. Contractible triangles are quasi-exact. Fernando Muro Exotic triangulated categories

  36. F ( Z / 4 ) is triangulated � � � � � q f i � A contractible A B C α β α � B ′ � C ′ � A ′ quasi-exact A ′ f ′ i ′ q ′ C is free. Let (Θ , Φ , Ψ) be a contracting homotopy for the upper row, γ = γ ′ + ( i ′ β − γ ′ i )Φ . Similarly if the first row is quasi-exact and the second row is contractible since Z / 4 is a Frobenius ring, so the duality functor Hom Z / 4 ( − , Z / 4 ) preserves contractible triangles and quasi-exact triangles. Fernando Muro Exotic triangulated categories

  37. F ( Z / 4 ) is triangulated � � � � � q f i � A contractible A B C α β α � B ′ � C ′ � A ′ quasi-exact A ′ f ′ i ′ q ′ C is free. Let (Θ , Φ , Ψ) be a contracting homotopy for the upper row, γ = γ ′ + ( i ′ β − γ ′ i )Φ . Similarly if the first row is quasi-exact and the second row is contractible since Z / 4 is a Frobenius ring, so the duality functor Hom Z / 4 ( − , Z / 4 ) preserves contractible triangles and quasi-exact triangles. Fernando Muro Exotic triangulated categories

  38. F ( Z / 4 ) is triangulated � � � � � � � q f i contractible A B C A α β / γ ′ α � � � B ′ � C ′ � A ′ quasi-exact A ′ f ′ i ′ q ′ C is free. Let (Θ , Φ , Ψ) be a contracting homotopy for the upper row, γ = γ ′ + ( i ′ β − γ ′ i )Φ . Similarly if the first row is quasi-exact and the second row is contractible since Z / 4 is a Frobenius ring, so the duality functor Hom Z / 4 ( − , Z / 4 ) preserves contractible triangles and quasi-exact triangles. Fernando Muro Exotic triangulated categories

  39. F ( Z / 4 ) is triangulated � � � � � � � q f i contractible A B C A α β / γ ′ α � � � B ′ � C ′ � A ′ quasi-exact A ′ f ′ i ′ q ′ C is free. Let (Θ , Φ , Ψ) be a contracting homotopy for the upper row, γ = γ ′ + ( i ′ β − γ ′ i )Φ . Similarly if the first row is quasi-exact and the second row is contractible since Z / 4 is a Frobenius ring, so the duality functor Hom Z / 4 ( − , Z / 4 ) preserves contractible triangles and quasi-exact triangles. Fernando Muro Exotic triangulated categories

  40. F ( Z / 4 ) is triangulated � � � � � � � q f i contractible A B C A γ α β α � � � B ′ � C ′ � A ′ quasi-exact A ′ f ′ i ′ q ′ C is free. Let (Θ , Φ , Ψ) be a contracting homotopy for the upper row, γ = γ ′ + ( i ′ β − γ ′ i )Φ . Similarly if the first row is quasi-exact and the second row is contractible since Z / 4 is a Frobenius ring, so the duality functor Hom Z / 4 ( − , Z / 4 ) preserves contractible triangles and quasi-exact triangles. Fernando Muro Exotic triangulated categories

  41. F ( Z / 4 ) is triangulated � � � � � � � q f i contractible A B C A γ α β α � � � B ′ � C ′ � A ′ quasi-exact A ′ f ′ i ′ q ′ C is free. Let (Θ , Φ , Ψ) be a contracting homotopy for the upper row, γ = γ ′ + ( i ′ β − γ ′ i )Φ . Similarly if the first row is quasi-exact and the second row is contractible since Z / 4 is a Frobenius ring, so the duality functor Hom Z / 4 ( − , Z / 4 ) preserves contractible triangles and quasi-exact triangles. Fernando Muro Exotic triangulated categories

  42. F ( Z / 4 ) is triangulated Let T and T ′ be contractible triangles in F ( Z / 4 ) . Any commutative square between the first arrows of X 2 ⊕ T and Y 2 ⊕ T ′ can be extended to a morphism � ϕ 11 � ϕ 12 → Y 2 ⊕ T ′ , : X 2 ⊕ T − ϕ 21 ϕ 22 such that the mapping cone of ϕ 11 : X 2 → Y 2 is exact. Morphisms from or to contractible triangles are null-homotopic, so � ϕ 11 � ϕ 11 � � 0 ϕ 12 ≃ , 0 0 ϕ 21 ϕ 22 whose mapping cone is ( mapping cone of ϕ 11 ) ⊕ ( translate of T ) exact . ⊕ T ′ , � �� � � �� � exact contractible back remarks Fernando Muro Exotic triangulated categories

  43. F ( Z / 4 ) is triangulated Let T and T ′ be contractible triangles in F ( Z / 4 ) . Any commutative square between the first arrows of X 2 ⊕ T and Y 2 ⊕ T ′ can be extended to a morphism � ϕ 11 � ϕ 12 → Y 2 ⊕ T ′ , : X 2 ⊕ T − ϕ 21 ϕ 22 such that the mapping cone of ϕ 11 : X 2 → Y 2 is exact. Morphisms from or to contractible triangles are null-homotopic, so � ϕ 11 � ϕ 11 � � 0 ϕ 12 ≃ , 0 0 ϕ 21 ϕ 22 whose mapping cone is ( mapping cone of ϕ 11 ) ⊕ ( translate of T ) exact . ⊕ T ′ , � �� � � �� � exact contractible back remarks Fernando Muro Exotic triangulated categories

  44. F ( Z / 4 ) is triangulated Let T and T ′ be contractible triangles in F ( Z / 4 ) . Any commutative square between the first arrows of X 2 ⊕ T and Y 2 ⊕ T ′ can be extended to a morphism � ϕ 11 � ϕ 12 → Y 2 ⊕ T ′ , : X 2 ⊕ T − ϕ 21 ϕ 22 such that the mapping cone of ϕ 11 : X 2 → Y 2 is exact. Morphisms from or to contractible triangles are null-homotopic, so � ϕ 11 � ϕ 11 � � 0 ϕ 12 ≃ , 0 0 ϕ 21 ϕ 22 whose mapping cone is ( mapping cone of ϕ 11 ) ⊕ ( translate of T ) exact . ⊕ T ′ , � �� � � �� � exact contractible back remarks Fernando Muro Exotic triangulated categories

  45. F ( Z / 4 ) is triangulated Let T and T ′ be contractible triangles in F ( Z / 4 ) . Any commutative square between the first arrows of X 2 ⊕ T and Y 2 ⊕ T ′ can be extended to a morphism � ϕ 11 � ϕ 12 → Y 2 ⊕ T ′ , : X 2 ⊕ T − ϕ 21 ϕ 22 such that the mapping cone of ϕ 11 : X 2 → Y 2 is exact. Morphisms from or to contractible triangles are null-homotopic, so � ϕ 11 � ϕ 11 � � 0 ϕ 12 ≃ , 0 0 ϕ 21 ϕ 22 whose mapping cone is ( mapping cone of ϕ 11 ) ⊕ ( translate of T ) exact . ⊕ T ′ , � �� � � �� � exact contractible back remarks Fernando Muro Exotic triangulated categories

  46. F ( Z / 4 ) is orthogonal to Ho M We are going to define two kinds of objects in a triangulated category T according to the cofiber of 2 · 1 X : X → X . Example If S is the sphere spectrum there is an exact triangle in Ho Sp 2 · 1 S q i → S / 2 S − → S − − → Σ S , where S / 2 is the mod 2 Moore spectrum. The map 2 · 1 S / 2 : S / 2 → S / 2 is the composite q η i S / 2 → S / 2 , − → Σ S − → S − where η is the stable Hopf map, which satisfies 2 · η = 0 . Fernando Muro Exotic triangulated categories

  47. F ( Z / 4 ) is orthogonal to Ho M We are going to define two kinds of objects in a triangulated category T according to the cofiber of 2 · 1 X : X → X . Example If S is the sphere spectrum there is an exact triangle in Ho Sp 2 · 1 S q i → S / 2 S − → S − − → Σ S , where S / 2 is the mod 2 Moore spectrum. The map 2 · 1 S / 2 : S / 2 → S / 2 is the composite q η i S / 2 → S / 2 , − → Σ S − → S − where η is the stable Hopf map, which satisfies 2 · η = 0 . Fernando Muro Exotic triangulated categories

  48. F ( Z / 4 ) is orthogonal to Ho M We are going to define two kinds of objects in a triangulated category T according to the cofiber of 2 · 1 X : X → X . Example If S is the sphere spectrum there is an exact triangle in Ho Sp 2 · 1 S q i → S / 2 S − → S − − → Σ S , where S / 2 is the mod 2 Moore spectrum. The map 2 · 1 S / 2 : S / 2 → S / 2 is the composite q η i S / 2 → S / 2 , − → Σ S − → S − where η is the stable Hopf map, which satisfies 2 · η = 0 . Fernando Muro Exotic triangulated categories

  49. F ( Z / 4 ) is orthogonal to Ho M Definition Let A ∈ T and let 2 · 1 A i q A − → A − → C − → Σ A be an exact triangle. A Hopf map for A is a map η : Σ A → A such that 2 · 1 C = i η q , 2 · η 0 . = If A admits a Hopf map we say that A is hopfian. Exact functors preserve Hopf maps and hopfian objects. Fernando Muro Exotic triangulated categories

  50. F ( Z / 4 ) is orthogonal to Ho M Definition Let A ∈ T and let 2 · 1 A i q A − → A − → C − → Σ A be an exact triangle. A Hopf map for A is a map η : Σ A → A such that 2 · 1 C = i η q , 2 · η 0 . = If A admits a Hopf map we say that A is hopfian. Exact functors preserve Hopf maps and hopfian objects. Fernando Muro Exotic triangulated categories

  51. F ( Z / 4 ) is orthogonal to Ho M Definition Let A ∈ T and let 2 · 1 A i q A − → A − → C − → Σ A be an exact triangle. A Hopf map for A is a map η : Σ A → A such that 2 · 1 C = i η q , 2 · η 0 . = If A admits a Hopf map we say that A is hopfian. Exact functors preserve Hopf maps and hopfian objects. Fernando Muro Exotic triangulated categories

  52. F ( Z / 4 ) is orthogonal to Ho M Proposition If T admits a model then all objects are hopfian. Proof. Sp is “the free stable model category on one generator” S [Schwede-Shipley’02]. In particular for any object A ∈ Ho M there is an exact functor F A : Ho Sp − → Ho M with F A ( S ) = A , so A is hopfian as S . Fernando Muro Exotic triangulated categories

  53. F ( Z / 4 ) is orthogonal to Ho M Proposition If T admits a model then all objects are hopfian. Proof. Sp is “the free stable model category on one generator” S [Schwede-Shipley’02]. In particular for any object A ∈ Ho M there is an exact functor F A : Ho Sp − → Ho M with F A ( S ) = A , so A is hopfian as S . Fernando Muro Exotic triangulated categories

  54. F ( Z / 4 ) is orthogonal to Ho M Definition An object E ∈ T is exotic if there is an exact triangle 2 · 1 E 2 · 1 E q E − → E − → E − → Σ E . Example Z / 4 is exotic in F ( Z / 4 ) . Indeed all objects in F ( Z / 4 ) are exotic. Exact functors preserve exotic objects. Fernando Muro Exotic triangulated categories

  55. F ( Z / 4 ) is orthogonal to Ho M Definition An object E ∈ T is exotic if there is an exact triangle 2 · 1 E 2 · 1 E q E − → E − → E − → Σ E . Example Z / 4 is exotic in F ( Z / 4 ) . Indeed all objects in F ( Z / 4 ) are exotic. Exact functors preserve exotic objects. Fernando Muro Exotic triangulated categories

  56. F ( Z / 4 ) is orthogonal to Ho M Proposition If X ∈ T is both hopfian and exotic then X = 0 . Proof. If η : Σ X → X is a Hopf map and 2 · 1 X 2 · 1 X q X − → X − → X − → Σ X is exact then 2 · 1 X = ( 2 · 1 X ) η q = 0 , therefore 0 0 q X − → X − → X − → Σ X is exact, so X = 0. Fernando Muro Exotic triangulated categories

  57. F ( Z / 4 ) is orthogonal to Ho M Proposition If X ∈ T is both hopfian and exotic then X = 0 . Proof. If η : Σ X → X is a Hopf map and 2 · 1 X 2 · 1 X q X − → X − → X − → Σ X is exact then 2 · 1 X = ( 2 · 1 X ) η q = 0 , therefore 0 0 q X − → X − → X − → Σ X is exact, so X = 0. Fernando Muro Exotic triangulated categories

  58. F ( Z / 4 ) is orthogonal to Ho M Proposition If X ∈ T is both hopfian and exotic then X = 0 . Proof. If η : Σ X → X is a Hopf map and 2 · 1 X 2 · 1 X q X − → X − → X − → Σ X is exact then 2 · 1 X = ( 2 · 1 X ) η q = 0 , therefore 0 0 q X − → X − → X − → Σ X is exact, so X = 0. Fernando Muro Exotic triangulated categories

  59. F ( Z / 4 ) is orthogonal to Ho M Proposition If X ∈ T is both hopfian and exotic then X = 0 . Proof. If η : Σ X → X is a Hopf map and 2 · 1 X 2 · 1 X q X − → X − → X − → Σ X is exact then 2 · 1 X = ( 2 · 1 X ) η q = 0 , therefore 0 0 q X − → X − → X − → Σ X is exact, so X = 0. Fernando Muro Exotic triangulated categories

  60. F ( Z / 4 ) is orthogonal to Ho M Proof of Theorem B. All objects in Ho M are hopfian and all objects in F ( Z / 4 ) are exotic. Therefore F : F ( Z / 4 ) − → Ho M is an exact functor the image of F consists of objects which are both hopfian and exotic, so F = 0. Similarly for F : Ho M − → F ( Z / 4 ) . back remarks Fernando Muro Exotic triangulated categories

  61. F ( Z / 4 ) is orthogonal to Ho M Proof of Theorem B. All objects in Ho M are hopfian and all objects in F ( Z / 4 ) are exotic. Therefore F : F ( Z / 4 ) − → Ho M is an exact functor the image of F consists of objects which are both hopfian and exotic, so F = 0. Similarly for F : Ho M − → F ( Z / 4 ) . back remarks Fernando Muro Exotic triangulated categories

  62. F ( Z / 4 ) is orthogonal to Ho M Proof of Theorem B. All objects in Ho M are hopfian and all objects in F ( Z / 4 ) are exotic. Therefore F : F ( Z / 4 ) − → Ho M is an exact functor the image of F consists of objects which are both hopfian and exotic, so F = 0. Similarly for F : Ho M − → F ( Z / 4 ) . back remarks Fernando Muro Exotic triangulated categories

  63. Remarks Theorems A and B are not only true for R = Z / 4 but for any commutative local ring R with maximal ideal m = ( 2 ) � = 0 such that m 2 = 0. For instance R = W 2 ( k ) , k a perfect field of char k = 2. Let k be a field of char k = 2. The category F ( k [ ε ] /ε 2 ) of finitely generated free modules over the ring of dual numbers k [ ε ] /ε 2 has a unique triangulated structure with Σ = identity and exact triangle k [ ε ] /ε 2 → k [ ε ] /ε 2 ε ε → k [ ε ] /ε 2 ε → k [ ε ] /ε 2 . − − − However F ( k [ ε ] /ε 2 ) does have a model. skip model Fernando Muro Exotic triangulated categories

  64. Remarks Theorems A and B are not only true for R = Z / 4 but for any commutative local ring R with maximal ideal m = ( 2 ) � = 0 such that m 2 = 0. For instance R = W 2 ( k ) , k a perfect field of char k = 2. Let k be a field of char k = 2. The category F ( k [ ε ] /ε 2 ) of finitely generated free modules over the ring of dual numbers k [ ε ] /ε 2 has a unique triangulated structure with Σ = identity and exact triangle k [ ε ] /ε 2 → k [ ε ] /ε 2 ε ε → k [ ε ] /ε 2 ε → k [ ε ] /ε 2 . − − − However F ( k [ ε ] /ε 2 ) does have a model. skip model Fernando Muro Exotic triangulated categories

  65. Remarks Theorems A and B are not only true for R = Z / 4 but for any commutative local ring R with maximal ideal m = ( 2 ) � = 0 such that m 2 = 0. For instance R = W 2 ( k ) , k a perfect field of char k = 2. Let k be a field of char k = 2. The category F ( k [ ε ] /ε 2 ) of finitely generated free modules over the ring of dual numbers k [ ε ] /ε 2 has a unique triangulated structure with Σ = identity and exact triangle k [ ε ] /ε 2 → k [ ε ] /ε 2 ε ε → k [ ε ] /ε 2 ε → k [ ε ] /ε 2 . − − − However F ( k [ ε ] /ε 2 ) does have a model. skip model Fernando Muro Exotic triangulated categories

  66. Remarks Proposition The triangulated category F ( k [ ε ] /ε 2 ) is exact equivalent to D c ( A ) , so it has a model given by differential graded right modules over a differential graded algebra A. A is a DGA such that H 0 ( A ) = k [ ε ] /ε 2 , any right DG A -module M has H 0 ( M ) free as a k [ ε ] /ε 2 -module, and the equivalence is given by → F ( k [ ε ] /ε 2 ) . H 0 : D c ( A ) − skip algebra Fernando Muro Exotic triangulated categories

  67. Remarks Proposition The triangulated category F ( k [ ε ] /ε 2 ) is exact equivalent to D c ( A ) , so it has a model given by differential graded right modules over a differential graded algebra A. A is a DGA such that H 0 ( A ) = k [ ε ] /ε 2 , any right DG A -module M has H 0 ( M ) free as a k [ ε ] /ε 2 -module, and the equivalence is given by → F ( k [ ε ] /ε 2 ) . H 0 : D c ( A ) − skip algebra Fernando Muro Exotic triangulated categories

  68. Remarks A = k � a , u , v , v − 1 � / I with | a | = | u | = 0 , | v | = − 1 . The two-sided ideal I is generated by a 2 , au + ua + 1 , av + va , uv + vu . The differential is defined by d ( a ) = u 2 v , d ( u ) = 0 , d ( v ) = 0 . H ∗ ( A ) = k [ x , x − 1 ] ⊗ k k [ ε ] /ε 2 , with ε = { u } , x = { v } . We have a non-trivial Massey product � ε, ε · x , ε � = 1 mod ε. Given y ∈ H 0 ( M ) with y · ε = 0 then � y , ε, ε · x � · ε = y · � ε, ε · x , ε � = y ⇒ H 0 ( M ) is free . Fernando Muro Exotic triangulated categories

  69. Remarks A = k � a , u , v , v − 1 � / I with | a | = | u | = 0 , | v | = − 1 . The two-sided ideal I is generated by a 2 , au + ua + 1 , av + va , uv + vu . The differential is defined by d ( a ) = u 2 v , d ( u ) = 0 , d ( v ) = 0 . H ∗ ( A ) = k [ x , x − 1 ] ⊗ k k [ ε ] /ε 2 , with ε = { u } , x = { v } . We have a non-trivial Massey product � ε, ε · x , ε � = 1 mod ε. Given y ∈ H 0 ( M ) with y · ε = 0 then � y , ε, ε · x � · ε = y · � ε, ε · x , ε � = y ⇒ H 0 ( M ) is free . Fernando Muro Exotic triangulated categories

  70. Remarks A = k � a , u , v , v − 1 � / I with | a | = | u | = 0 , | v | = − 1 . The two-sided ideal I is generated by a 2 , au + ua + 1 , av + va , uv + vu . The differential is defined by d ( a ) = u 2 v , d ( u ) = 0 , d ( v ) = 0 . H ∗ ( A ) = k [ x , x − 1 ] ⊗ k k [ ε ] /ε 2 , with ε = { u } , x = { v } . We have a non-trivial Massey product � ε, ε · x , ε � = 1 mod ε. Given y ∈ H 0 ( M ) with y · ε = 0 then � y , ε, ε · x � · ε = y · � ε, ε · x , ε � = y ⇒ H 0 ( M ) is free . Fernando Muro Exotic triangulated categories

  71. Remarks A = k � a , u , v , v − 1 � / I with | a | = | u | = 0 , | v | = − 1 . The two-sided ideal I is generated by a 2 , au + ua + 1 , av + va , uv + vu . The differential is defined by d ( a ) = u 2 v , d ( u ) = 0 , d ( v ) = 0 . H ∗ ( A ) = k [ x , x − 1 ] ⊗ k k [ ε ] /ε 2 , with ε = { u } , x = { v } . We have a non-trivial Massey product � ε, ε · x , ε � = 1 mod ε. Given y ∈ H 0 ( M ) with y · ε = 0 then � y , ε, ε · x � · ε = y · � ε, ε · x , ε � = y ⇒ H 0 ( M ) is free . Fernando Muro Exotic triangulated categories

  72. Remarks A = k � a , u , v , v − 1 � / I with | a | = | u | = 0 , | v | = − 1 . The two-sided ideal I is generated by a 2 , au + ua + 1 , av + va , uv + vu . The differential is defined by d ( a ) = u 2 v , d ( u ) = 0 , d ( v ) = 0 . H ∗ ( A ) = k [ x , x − 1 ] ⊗ k k [ ε ] /ε 2 , with ε = { u } , x = { v } . We have a non-trivial Massey product � ε, ε · x , ε � = 1 mod ε. Given y ∈ H 0 ( M ) with y · ε = 0 then � y , ε, ε · x � · ε = y · � ε, ε · x , ε � = y ⇒ H 0 ( M ) is free . Fernando Muro Exotic triangulated categories

  73. Remarks A = k � a , u , v , v − 1 � / I with | a | = | u | = 0 , | v | = − 1 . The two-sided ideal I is generated by a 2 , au + ua + 1 , av + va , uv + vu . The differential is defined by d ( a ) = u 2 v , d ( u ) = 0 , d ( v ) = 0 . H ∗ ( A ) = k [ x , x − 1 ] ⊗ k k [ ε ] /ε 2 , with ε = { u } , x = { v } . We have a non-trivial Massey product � ε, ε · x , ε � = 1 mod ε. Given y ∈ H 0 ( M ) with y · ε = 0 then � y , ε, ε · x � · ε = y · � ε, ε · x , ε � = y ⇒ H 0 ( M ) is free . Fernando Muro Exotic triangulated categories

  74. Remarks A = k � a , u , v , v − 1 � / I with | a | = | u | = 0 , | v | = − 1 . The two-sided ideal I is generated by a 2 , au + ua + 1 , av + va , uv + vu . The differential is defined by d ( a ) = u 2 v , d ( u ) = 0 , d ( v ) = 0 . H ∗ ( A ) = k [ x , x − 1 ] ⊗ k k [ ε ] /ε 2 , with ε = { u } , x = { v } . We have a non-trivial Massey product � ε, ε · x , ε � = 1 mod ε. Given y ∈ H 0 ( M ) with y · ε = 0 then � y , ε, ε · x � · ε = y · � ε, ε · x , ε � = y ⇒ H 0 ( M ) is free . Fernando Muro Exotic triangulated categories

  75. Remarks A = k � a , u , v , v − 1 � / I with | a | = | u | = 0 , | v | = − 1 . The two-sided ideal I is generated by a 2 , au + ua + 1 , av + va , uv + vu . The differential is defined by d ( a ) = u 2 v , d ( u ) = 0 , d ( v ) = 0 . H ∗ ( A ) = k [ x , x − 1 ] ⊗ k k [ ε ] /ε 2 , with ε = { u } , x = { v } . We have a non-trivial Massey product � ε, ε · x , ε � = 1 mod ε. Given y ∈ H 0 ( M ) with y · ε = 0 then � y , ε, ε · x � · ε = y · � ε, ε · x , ε � = y ⇒ H 0 ( M ) is free . Fernando Muro Exotic triangulated categories

  76. Remarks A = k � a , u , v , v − 1 � / I with | a | = | u | = 0 , | v | = − 1 . The two-sided ideal I is generated by a 2 , au + ua + 1 , av + va , uv + vu . The differential is defined by d ( a ) = u 2 v , d ( u ) = 0 , d ( v ) = 0 . H ∗ ( A ) = k [ x , x − 1 ] ⊗ k k [ ε ] /ε 2 , with ε = { u } , x = { v } . We have a non-trivial Massey product � ε, ε · x , ε � = 1 mod ε. Given y ∈ H 0 ( M ) with y · ε = 0 then � y , ε, ε · x � · ε = y · � ε, ε · x , ε � = y ⇒ H 0 ( M ) is free . Fernando Muro Exotic triangulated categories

  77. Remarks A = k � a , u , v , v − 1 � / I with | a | = | u | = 0 , | v | = − 1 . The two-sided ideal I is generated by a 2 , au + ua + 1 , av + va , uv + vu . The differential is defined by d ( a ) = u 2 v , d ( u ) = 0 , d ( v ) = 0 . H ∗ ( A ) = k [ x , x − 1 ] ⊗ k k [ ε ] /ε 2 , with ε = { u } , x = { v } . We have a non-trivial Massey product � ε, ε · x , ε � = 1 mod ε. Given y ∈ H 0 ( M ) with y · ε = 0 then � y , ε, ε · x � · ε = y · � ε, ε · x , ε � = y ⇒ H 0 ( M ) is free . Fernando Muro Exotic triangulated categories

  78. Remarks Theorem (Hovey-Lockridge’07) Let R be a commutative ring. The category F ( R ) is triangulated with Σ = identity if and only if R is a finite product of fields, rings of dual numbers over fields of characteristic 2 , and local rings with m = ( 2 ) � = 0 and m 2 = 0 . Corollary The triangulated category F ( R ) admits a model if and only if R is a finite product of fields and rings of dual numbers over fields of characteristic 2 . Fernando Muro Exotic triangulated categories

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