Internally CalabiYau Algebras Matthew Pressland Universitt - PowerPoint PPT Presentation

Internally Calabi–Yau Algebras Matthew Pressland Universität Bielefeld Cluster Algebras and Geometry, Universität Münster 11 th March 2016 Matthew Pressland (Bielefeld) Internally Calabi–Yau Algebras Universität Münster

Main Definition Let A be a (not necessarily finite dimensional) C -algebra, and let e be an idempotent of A . Throughout, we will write A = A/AeA (the interior algebra) and B = eAe (the boundary algebra). Definition The algebra A is internally d -Calabi–Yau with respect to e if (i) gl . dim A ≤ d , and (ii) for any finite dimensional A -module M , and any A module N , there is a duality D Ext i A ( M, N ) = Ext d − i A ( N, M ) for all i , functorial in M and N . Matthew Pressland (Bielefeld) Internally Calabi–Yau Algebras Universität Münster

Voidology Definition The algebra A is internally d -Calabi–Yau with respect to e if (i) gl . dim A ≤ d , and (ii) for any finite dimensional A -module M , and any A module N , there is a duality D Ext i A ( M, N ) = Ext d − i A ( N, M ) for all i , functorial in M and N . Setting e = 0 recovers the (naïve) definition of a d -Calabi–Yau algebra. Setting e = 1 , (ii) becomes vacuous. If e � = 1 , (ii) = ⇒ gl . dim A ≥ d , and so gl . dim A = d in this case. Matthew Pressland (Bielefeld) Internally Calabi–Yau Algebras Universität Münster

Example 1 (finite dimensional, d = 3 ) β 1 2 α γ 3 βα = 0 = γβ e = e 1 + e 2 A = C . B = eAe is the preprojective algebra of type A 2 . Matthew Pressland (Bielefeld) Internally Calabi–Yau Algebras Universität Münster

Example 2 (infinite dimensional, d = 3 ) 6 1 7 5 2 8 9 4 3 The two paths back along any internal arrow are equal. 6 � e = e i i =1 Matthew Pressland (Bielefeld) Internally Calabi–Yau Algebras Universität Münster

Origins Let E be a Frobenius category: an exact category with enough projectives and enough injectives, and such that projective and injective objects coincide. Then E = E / proj E is triangulated. Assume that E is Krull–Schmidt, and E is d -Calabi–Yau. Let T ∈ E be d -cluster-tilting, i.e. add T = { X ∈ E : Ext i E ( X, T ) = 0 , 0 < i < d } . Theorem (Keller–Reiten) If gl . dim End E ( T ) op ≤ d + 1 , then it is internally ( d + 1) -Calabi–Yau with respect to projection onto a maximal projective summand. Matthew Pressland (Bielefeld) Internally Calabi–Yau Algebras Universität Münster

Bimodule version Write A ε = A ⊗ C A op , and Ω A = RHom A ε ( A, A ε ) . Let D A ( A ) be the full subcategory of the derived category of A consisting of objects whose total cohomology is a finite-dimensional A -module. Definition The algebra A is internally bimodule d -Calabi–Yau with respect to e if (i) p . dim A ε A ≤ d , and (ii) there is a triangle A → Ω A [ d ] → C → A [1] in D ( A ε ) , such that RHom A ( C, M ) = 0 = RHom A op ( C, N ) for all M ∈ D A ( A ) and N ∈ D A op ( A op ) . = Ω A [ d ] ∈ per A ε is bimodule If we can take C = 0 , then A ∼ d -Calabi–Yau. Matthew Pressland (Bielefeld) Internally Calabi–Yau Algebras Universität Münster

Consequences Definition The algebra A is internally bimodule d -Calabi–Yau with respect to e if (i) p . dim A ε A ≤ d , and (ii) there is a triangle A → Ω A [ d ] → C → A [1] in D ( A ε ) , such that RHom A ( C, M ) = 0 = RHom A op ( C, N ) for all M ∈ D A ( A ) and N ∈ D A ( A op ) . A is internally bimodule d -Calabi–Yau with respect to e if and only if the same is true for A op . If A is internally bimodule d -Calabi–Yau with respect to e then D Hom D ( A ) ( M, N ) = Hom D ( A ) ( N, M [ d ]) for any N ∈ D ( A ) and any M ∈ D A ( A ) . In particular, such an A is internally d -Calabi–Yau with respect to e . Matthew Pressland (Bielefeld) Internally Calabi–Yau Algebras Universität Münster

Main Theorem Theorem (P, cf. Amiot–Iyama–Reiten) Let A be a Noetherian algebra, and e an idempotent such that A is finite dimensional. If A and A op are internally ( d + 1) -Calabi–Yau with respect to e , then (i) B is Iwanaga–Gorenstein of Gorenstein dimension at most d + 1 , and so GP( B ) = { X ∈ mod B : Ext i B ( X, B ) = 0 , i > 0 } is Frobenius, (ii) eA ∈ GP( B ) is d -cluster-tilting, and = End B ( eA ) op and (iii) there are natural isomorphisms A ∼ A ∼ = End GP( B ) ( eA ) op . If A is internally bimodule ( d + 1) -Calabi–Yau with respect to e , then additonally (iv) GP( B ) is d -Calabi–Yau. Matthew Pressland (Bielefeld) Internally Calabi–Yau Algebras Universität Münster

Frozen Jacobian algebras Let Q be a quiver, and F a (not necessarily full) subquiver, called frozen. Let W be a linear combination of cycles of Q . For a cyclic path α n · · · α 1 of Q , define � ∂ α ( α n · · · α 1 ) = α i − 1 · · · α 1 α n · · · α i +1 α i = α and extend by linearity. The frozen Jacobian algebra J ( Q, F, W ) is J ( Q, F, W ) = C Q/ � ∂ α W : α ∈ Q 1 \ F 1 � , where C Q denotes the complete path algebra of Q over C . The frozen idempotent is e = � i ∈ F 0 e i . Matthew Pressland (Bielefeld) Internally Calabi–Yau Algebras Universität Münster

Example β 1 2 α γ 3 F is the full subquiver on vertices 1 and 2 . W = γβα e = e 1 + e 2 Matthew Pressland (Bielefeld) Internally Calabi–Yau Algebras Universität Münster

A bimodule resolution? Let A be a frozen Jacobian algebra, let S = A/ m ( A ) be the m semisimple part of A , and write ⊗ = ⊗ S . Write Q i for the dual S -bimodule to Q i \ F i . There is a natural complex m m 0 → A ⊗ Q 0 ⊗ A → A ⊗ Q 1 ⊗ A → A ⊗ Q 1 ⊗ A → A ⊗ Q 0 ⊗ A → A → 0 of A -bimodules (cf. Ginzburg and Broomhead for the case F = ∅ ). Theorem (P) If this complex is exact, then A is internally bimodule 3 -Calabi–Yau with respect to the frozen idempotent e . Matthew Pressland (Bielefeld) Internally Calabi–Yau Algebras Universität Münster

Dimer models Definition by example (in the disk): Matthew Pressland (Bielefeld) Internally Calabi–Yau Algebras Universität Münster

Associated frozen Jacobian algebra Definition by example (in the disk): 6 1 7 5 2 8 9 4 3 Matthew Pressland (Bielefeld) Internally Calabi–Yau Algebras Universität Münster