Posets, sheaves, and their derived equivalences Posets, sheaves, and their derived equivalences Sefi Ladkani Einstein Institute of Mathematics The Hebrew University of Jerusalem http://www.ma.huji.ac.il/~sefil/ 1
Posets, sheaves, and their derived equivalences Posets, diagrams and sheaves X – poset (finite partially ordered set) A – abelian category A X – the category of diagrams over X with values in A , or functors F : X → A consisting of: • An object F x of A for each x ∈ X . • A morphism r xx ′ ∈ Hom A ( F x , F x ′ ) for each x ≤ x ′ . such that r xx ′′ = r x ′ x ′′ r xx ′ for all x ≤ x ′ ≤ x ′′ (commutativity). U ⊆ X is open if x ∈ U , x ≤ x ′ ⇒ x ′ ∈ U Natural topology on X : Diagrams can be identified with sheaves over X with values in A . 2
Posets, sheaves, and their derived equivalences Universal derived equivalence Two posets X and Y are universally derived equivalent ( X u ∼ Y ) if D b ( A X ) ≃ D b ( A Y ) for any abelian category A . Fix a field k , and specialize: mod k – the category of finite dimensional vector spaces over k . (mod k ) X can be identified with the category of finitely generated right modules over the incidence algebra of X over k . X and Y are derived equivalent ( X ∼ Y ) if D b (mod kX ) ≃ D b (mod kY ) 3
Posets, sheaves, and their derived equivalences Constructions of derived equivalent posets Common theme: structured reversal of order relations. • Generalized reflections (universal derived equivalences) – Flip-Flops , with application to posets of tilting modules – Generalized BGP reflections – Hybrid construction • Mirroring with respect to a bipartite structure – Mates of triangular matrix algebras 4
Posets, sheaves, and their derived equivalences Flip-Flops Let ( X, ≤ X ), ( Y, ≤ Y ) be posets, f : X → Y order-preserving. Define two partial orders ≤ f + , ≤ f − on X ⊔ Y as follows: • Keep the original partial orders inside X and Y . • Add the relations x ≤ f + y ⇐ ⇒ f ( x ) ≤ Y y y ≤ f − x ⇐ ⇒ y ≤ Y f ( x ) for x ∈ X , y ∈ Y . + ) u Theorem. ( X ⊔ Y , ≤ f ∼ ( X ⊔ Y , ≤ f − ). 5
� � � � � � � � � � � � � � � � Posets, sheaves, and their derived equivalences Flip-Flop – Example 2 �→ 1 4 �→ 1 5 �→ 3 6 �→ 1 7 �→ 3 9 �→ 8 12 �→ 8 13 �→ 10 14 �→ 11 • 2 • 1 � � � � ������ � � � ���������� � ����������������� � � � � � • 4 • 2 • 3 � � � ���������� � � ������ � � ����������������� � ����������� � • 9 • 5 • 4 � � � � � � ���������� � � � � � � ������������ � � � • 6 � � � � � � � • 8 • 10 � • 5 � � � � � � � � � ���������� � � � � � � � ������ � ����� • 12 • 7 � ����������������� � ����������� • 6 • 9 • 11 � � � � � � � ���������� � � ���������� � � � � � � � � � � � • 1 • 13 � � � � ������������ � � ����������������� � � � � � � � � � � � � � � • 7 � � � � � � � � � � ���� � ���� � � ���������� � � � � • 8 • 3 � • 14 � ����� • 12 • 13 � � � � � � ���������� � � � � • 10 � � � � � � ����� � � � � � � � • 14 � � ���� � � • 11 ( X ⊔ Y , ≤ f ( X ⊔ Y , ≤ f + ) − ) 6
Posets, sheaves, and their derived equivalences Application – Posets of tilting modules Q – quiver without oriented cycles, k – field T Q – poset of tilting modules of kQ [Riedtmann-Schofield, Happel-Unger] x – a source in Q Q ′ – the BGP reflection with respect to x . T x Q – tilting modules containing the simple at x as summand Theorem. T Q and T Q ′ are related via a flip-flop. Q ′ , ≤ f ′ Q , ≤ f T Q ≃ ( T Q \ T x Q ⊔ T x T Q ′ ≃ ( T Q ′ \ T x Q ′ ⊔ T x + ) − ) u Corollary. If Q 1 ∼ Q 2 then T Q 1 ∼ T Q 2 . 7
Posets, sheaves, and their derived equivalences Generalized BGP reflections Let ( Y, ≤ ) be poset, Y 0 ⊆ Y a subset with the property for all y � = y ′ in Y 0 [ y, · ] ∩ [ y ′ , · ] = φ = [ · , y ] ∩ [ · , y ′ ] Define two partial orders ≤ Y 0 + , ≤ Y 0 − on {∗} ∪ Y as follows: • Keep the original partial order inside Y . • Add the relations ∗ < Y 0 + y ⇐ ⇒ ∃ y 0 ∈ Y 0 with y 0 ≤ y y < Y 0 − ∗ ⇐ ⇒ ∃ y 0 ∈ Y 0 with y ≤ y 0 for y ∈ Y . 8
� � � � � � Posets, sheaves, and their derived equivalences Generalized BGP reflections – continued The vertex ∗ is a source in the Hasse diagram of ≤ Y 0 + , with arrows ending at the vertices of Y 0 . The Hasse diagram of ≤ Y 0 − is obtained by reverting the orientations of the arrows from ∗ , making it into a sink . + ) u Theorem. ( {∗} ∪ Y , ≤ Y 0 ∼ ( {∗} ∪ Y , ≤ Y 0 − ). Example. • • � � � ������� � � ������� � � � � � � � � � � � � • � ∗ • • • • • ∗ � � � � ������� � � ������� � � � � � � � � � � • • 9
Posets, sheaves, and their derived equivalences Hybrid construction – setup ( X, ≤ X ), ( Y, ≤ Y ) – posets, { Y x } x ∈ X – collection of subsets Y x ⊆ Y , with the properties: • For all x ∈ X , for all y � = y ′ in Y x [ y, · ] ∩ [ y ′ , · ] = φ = [ · , y ] ∩ [ · , y ′ ] ∼ • For all x ≤ x ′ , there exists an isomorphism ϕ x,x ′ : Y x − → Y x ′ with y ≤ Y ϕ x,x ′ ( y ) for all y ∈ Y x It follows that { Y x } x ∈ X is a local system of subsets of Y : for all x ≤ x ′ ≤ x ′′ . ϕ x,x ′′ = ϕ x ′ ,x ′′ ϕ x,x ′ 10
Posets, sheaves, and their derived equivalences Hybrid construction – result Define two partial orders on ≤ + , ≤ − on X ⊔ Y as follows: • Keep the original partial orders inside X and Y . • Add the relations x ≤ + y ⇐ ⇒ ∃ y x ∈ Y x with y x ≤ Y y y ≤ − x ⇐ ⇒ ∃ y x ∈ Y x with y ≤ Y y x for x ∈ X , y ∈ Y . Theorem. ( X ⊔ Y , ≤ + ) u ∼ ( X ⊔ Y , ≤ − ). Remarks. • When X = {∗} , we recover the generalized BGP reflection. • When Y x = {∗} for all x ∈ X , we recover the flip-flop. 11
Posets, sheaves, and their derived equivalences Mirroring with respect to a bipartite structure Let S be bipartite . ( S = S 0 ⊔ S 1 with s < s ′ ⇒ s ∈ S 0 and s ′ ∈ S 1 ) Let X = { X s } s ∈ S be a collection of posets indexed by S . Define two partial orders ≤ + and ≤ − on � s ∈ S X s as follows: • Keep the original partial order inside each X s . • Add the relations x s < + x t ⇐ ⇒ s < t x t < − x s ⇐ ⇒ t < s for x s ∈ X s , x t ∈ X t . Theorem. ( � s ∈ S X s , ≤ + ) ∼ ( � s ∈ S X s , ≤ − ). 12
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Posets, sheaves, and their derived equivalences Bipartite structure – example S = • • � � � � � � � � � � � � � � • • • X = • • • • • • • � � � ������� � � ������� � � � � � � � � � � � • • • • • • ( � ( � s ∈ S X s , ≤ + ) • • • • • • s ∈ S X s , ≤ − ) � � � ������� � � ������� � � � � � � � � � � � • • • • • • • � � � � � � � � � � � � � � ������� � � ������� � � � � � ������� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � • • • • • • • � � � � ������� � � ������� � � � � � � � � � � • • • • • • 13
Posets, sheaves, and their derived equivalences Mates of triangular matrix algebras Let k be a field, R and S k -algebras and R M S bimodule. Consider the triangular matrix algebras � � � � R M S DM � Λ = and Λ = 0 0 S R where DM = Hom k ( M, k ). Theorem. D b (mod Λ) ≃ D b (mod � Λ), under the assumptions: • dim k R < ∞ , dim k S < ∞ , dim k M < ∞ • gl . dim R < ∞ , gl . dim S < ∞ 14
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