Amalgamation, unamalgamation and the phi-dimension conjecture - PowerPoint PPT Presentation

Amalgamation, unamalgamation and the phi-dimension conjecture Kiyoshi Igusa Brandeis University November 23, 2019 Kiyoshi IgusaBrandeis University Amalgamation and phi-dim November 23, 2019 1 / 15

Introduction This is joint work with Eric Hanson based on ongoing joint work with Gordana Todorov on “amalgamation” and “unamalgamation” which in turn originated in joint work with Dani ´ Alvarez-Gavela on Legendrian embeddings using plabic diagrams. (a) Plabic diagrams and amalgamation. (b) Counterexample to the φ -dimension conjecture. Kiyoshi IgusaBrandeis University Amalgamation and phi-dim November 23, 2019 2 / 15

phi-dimension conjecture The φ -dimension conjecture states that, for any artin algebra Λ, there is a uniform bound on the φ -dimension of the f.g. Λ-modules. For modules of finite projective dimension, the projective dimension is equal to the φ -dimension. Therefore, findim-Λ ≤ φ -dim-Λ. So, φ -dim Λ < ∞ implies findim-Λ < ∞ . φ -dim-Λ is defined to be the supremum of φ ( M ) for all Λ-modules M . To get a lower bound on φ ( M ) we use the following. Lemma Let X , Y be Λ-modules so that Ω k X �∼ = Ω k Y but Ω k +1 X ∼ = Ω k +1 Y . Then φ ( X ⊕ Y ) ≥ k . Kiyoshi IgusaBrandeis University Amalgamation and phi-dim November 23, 2019 3 / 15

Plabic diagrams Here is a plabic diagram (planar bicol- ored graph). Standard bipartite version (1) Coalescing vertices of same color (2) Add boundary vertices of opposite color. (We skip this step.) Plabic diagrams are assembled from the pieces on the left. Kiyoshi IgusaBrandeis University Amalgamation and phi-dim November 23, 2019 4 / 15

Jacobian algebra given by dual quiver We assemble the plabic diagram out of pieces: ⇒ The quiver is also assembled from pieces: ⇒ Kiyoshi IgusaBrandeis University Amalgamation and phi-dim November 23, 2019 5 / 15

Amalgamation (Fock-Goncharov) x 1 y 1 x 2 α y 2 ⇒ x 3 ∂ α W = 0 (using only solid arrows) gives x 1 x 2 x 3 = y 1 y 2 . Take triangles with dotted arrows. These are half-arrows. When you add two half-arrows you get either a solid arrow or no arrow. For the Jacobian algebra, only derivatives with respect to solid arrows are set equal to zero. Kiyoshi IgusaBrandeis University Amalgamation and phi-dim November 23, 2019 6 / 15

Amalgamation: an equivalent version x 2 x 3 x 1 x 2 x 1 x 3 x 2 x 3 x 1 Take triangles with solid arrows. Add “redundant” arrows (in red). y 2 x 3 y 2 y 3 y 1 y 1 y 2 x 2 x 3 ∗ y 3 y 2 x 1 x 3 x 3 y 1 y 1 y 2 y 3 y 1 x 2 x 2 x 3 x 1 x 2 x 1 x 3 x 3 x 1 x 2 Amalgamate by identifying. x 3 x 1 Red arrow ∗ = x 1 x 2 = y 1 y 2 Kiyoshi IgusaBrandeis University Amalgamation and phi-dim November 23, 2019 7 / 15

Result of amalgamation y 2 x 3 y 1 y 1 y 2 x 1 x 3 x 2 x 3 y 2 ∗ x 3 y 1 x 1 x 3 x 2 x 2 x 3 x 1 with relation: x 1 x 2 = y 1 y 2 ∗ = x 1 x 2 = y 1 y 2 Summary: Adding redundant arrows, identifying arrows, then removing redundant arrows gives the Jacobian algebra of the F-G amalgamation. We call this “amalgamation” (adding redundant arrow and identifying arrows). Kiyoshi IgusaBrandeis University Amalgamation and phi-dim November 23, 2019 8 / 15

The counterexample Let C be the algebra given by the triangular quiver on the left modulo rad 2 = 0. Let A be the algebra given by the y 1 quiver on the right modulo rad 2 = 0. y 2 x 1 x 3 (This is the amalgamation of two copies of C .) x 2 Theorem Λ = A ⊗ C has infinite φ -dimension. 1 1 Two days after us, Barrios and Mata also posted an example (arXiv:1911.02325). Kiyoshi IgusaBrandeis University Amalgamation and phi-dim November 23, 2019 9 / 15

Outline of proof To prove this we construct a sequence of pair of Λ-modules WX k , WY k so that Ω 3 k WX k �∼ = Ω 3 k WY k but Ω 3 k +1 WX k ∼ = Ω 3 k +1 WY k . This implies that φ ( WX k ⊕ WY k ) ≥ 3 k . So, φ -dim Λ ≥ 3 k for all k . So, φ -dim Λ = ∞ . The modules WX k , WY k are constructed out of chain complexes of A -modules X k , Y k . Kiyoshi IgusaBrandeis University Amalgamation and phi-dim November 23, 2019 10 / 15

Wrapping chain complexes A Λ = A ⊗ C module M is a triple of modules M 1 , M 2 , M 3 and maps d : M i → M i − 1 so that d 2 = 0. Conversely, given any chain complex of A -modules V ∗ : 0 ← V 0 ← V 1 ← V 2 ← V 3 ← · · · define WV ∗ to be the triple of A -modules M 1 = V 1 ⊕ V 4 ⊕ V 7 ⊕ · · · , M 2 = V 2 ⊕ V 5 ⊕ · · · , M 3 = V 0 ⊕ V 3 ⊕ V 6 ⊕ · · · with boundary maps M i → M i − 1 given by the boundary maps of V ∗ . Lemma W is an exact functor which commutes with Ω and takes exact sequences to exact sequences. Kiyoshi IgusaBrandeis University Amalgamation and phi-dim November 23, 2019 11 / 15

� � � � � � � � � � � The chain complexes X k , Y k The A -chain complexes X k , Y k are truncated projective resolutions of different simple A -module S 3 , S 4 of length 3 k . � 4 2 � 1 3 The branch point moves to the left under syzygy: P K M P 0 P 1 P 2 P 4 P 4 L Kiyoshi IgusaBrandeis University Amalgamation and phi-dim November 23, 2019 12 / 15

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Branch point moves to the left under syzygy P K M P 0 P 1 P 2 P 3 P 4 L P ′ P Ω K Ω M Ω L P 1 P 2 P 3 P 4 P 5 Ω 2 K P P ′ P ′′ Ω 2 M Ω 2 L P 2 P 3 P 4 P 5 P 6 Ω 3 K P ′ P ′′ P ′′′ P Ω 3 M Ω 3 L P 3 P 4 P 5 P 6 P 7 Once more and we loose the information of how the resolution was truncated. Kiyoshi IgusaBrandeis University Amalgamation and phi-dim November 23, 2019 13 / 15

Branches fall off The chain complexes Ω X k , Ω Y k are both truncated projective resolution of the same module S 1 . They are truncated in different ways in degree 3 k . So, after taking 3 k syzygies, when the “branch” “falls off” we cannot tell the difference and they become isomorphic: Ω 3 k +1 X k ∼ = Ω 3 k +1 Y k Since the wrapping functor is exact and takes projectives to projectives we get Ω 3 k +1 WX k ∼ = Ω 3 k +1 WY k . Since Ω 3 k X k and Ω 3 k Y k are truncated differently we can show that Ω 3 k WX k �∼ = Ω 3 k WY k . So, φ -dim Λ is unbounded. Kiyoshi IgusaBrandeis University Amalgamation and phi-dim November 23, 2019 14 / 15

Conclusion THANK YOU! Kiyoshi IgusaBrandeis University Amalgamation and phi-dim November 23, 2019 15 / 15

Figure from ribbon Legendrians paper (with D. ´ Alvarez-Gavela) (example used to illustrate the proof of the main theorem) Kiyoshi IgusaBrandeis University Amalgamation and phi-dim November 23, 2019 16 / 15