Generic modules and rational invariants for gentle algebras Andrew - PowerPoint PPT Presentation

Generic modules and rational invariants for gentle algebras Andrew T. Carroll University of Missouri, Columbia, MO Number Theory and Representation Theory September 27, 2012

Plan of the talk: Gentle algebras 1 Generic Decomposition 2 Semi-invariants 3

Definition Given a character χ : GL( d ) → k ∗ , SI( A, C ) χ = { f ∈ k [ C ] | g · f = χ ( g ) f } is called the space of semi-invariants of weight χ . The scheme � M ( A, C, χ ) := Proj SI( A, C ) n · χ n ≥ 0 is a GIT quotient of the subset of χ -semi-stable points in C .

Conjecture 1. Weyman A is tame if and only if for each irreducible component and each weight M ( A, C, χ ) is simply a product of projective spaces. (Note: in wild type, can get any conceivable projective variety) 2. Chindris A is tame if and only if for each irreducible component C in which the generic module is indecomposable, k ( C ) GL( d ) = k ( t ) . 1,2 have been shown in case A = kQ is a path algebra [Chindris]; 1,2 ( ⇒ ) have been shown in this case and when A is a so-called quasi-tilted algebra; (C.-Chindris) forward implications when kQ/I is a gentle algebra.

Gentle algebras

Colorings A coloring c : Q 1 → { 1 , . . . , m } is a surjection with c − 1 ( i ) is a path for each i . I c := � a 2 a 1 | a 1 → a 2 − − → , c ( a 1 ) = c ( a 2 ) � Example 1 2 3 4 5 6 7 8

Definition/Proposition If kQ/I is an acyclic gentle algebra, then there is a coloring c of Q with I = I c . Definition Fix A = kQ/I c , and d ∈ N Q 0 rank function is a map r : Q 1 → N s.t. x ⇒ r ( a 1 ) + r ( a 2 ) ≤ d ( x ) mod( A, d, r ) ⊂ mod( A, d ) the algebraic set { V ∈ mod d ( A ) | rank V a ≤ r ( a ) ∀ a ∈ Q 1 }

Proposition (Corollary to DeConcini-Strickland) mod A ( d, r ) is an irreducible component of mod A ( d ) whenever r is maximal; mod A ( d, r ) is normal. Goal Given an irreducible component mod( A, d, r ) , determine the structure of the generic module; Determine k (mod( A, d, r )) GL( d ) , and a transcendence basis; Show that M ( A, mod A ( d, r ) , χ ) is a product of projective spaces.

Generic Decomposition

Goal: In each irreducible component C ⊂ mod( A, d ) determine a dense subset U ⊂ C a decomposition d = d 1 + . . . + d m such that for all M ∈ U , M ∼ = M 1 ⊕ . . . ⊕ M m where M i is indecomposable of dimension d i . [Kac] Such a decomposition of d exists; [Gabriel] If Ext 1 A ( M, M ) = 0 , then U = GL( d ) · M ;

Krull-Schmidt-type property If C 1 , . . . , C m are irreducible components, C i ⊂ mod A ( d i ) , consider � C 1 ⊕ . . . ⊕ C m := GL( d ) · ( M 1 ⊕ . . . ⊕ M m ) M i ∈ C i Theorem (Crawley-Boevey Schr¨ oer) If C is an irreducible component, then C = C 1 ⊕ . . . ⊕ C m where C i are indecomposable irreducible components and min { dim Ext 1 A ( M i , M j ) | M l ∈ C l } = 0 for all i � = j . Moreover, this decomposition is unique

Fix A = kQ/I c a gentle algebra X = { ( x, s ) ∈ Q 0 × S | ∃ a ∈ Q 1 with c ( a ) = s } sign function ǫ : X → {± 1 } with ǫ ( x, s 1 ) = − ǫ ( x, s 2 ) when s 1 � = s 2 − + − + + − − + − + − +

For fixed d and r , construct a digraph Γ = Γ Q,c ( d, r, ǫ ) : Γ 0 = { v x j | x ∈ Q 0 , j = 1 , . . . , d ( x ) } j | a ∈ Q 1 , j = 1 , . . . , r ( a ) } : suppose x a Γ 1 = { f a − → y ∈ Q 1

− + − + + − − + − + − +

v (1) v (2) v (3) 1 1 1 v (1) v (2) 2 2 v (1) v (2) 3 3 v (2) 4 v (4) v (5) v (6) 1 1 1 v (4) v (5) v (6) 2 2 2 v (5) 3

v (1) v (3) v (2) 1 1 1 v (1) v (2) 2 2 v (2) v (1) 3 3 v (2) 4 v (4) v (5) v (6) 1 1 1 λ v (4) v (5) v (6) 2 2 2 v (5) 3

After decorating Γ( d, r, ǫ ) with scalars λ ∈ ( k ∗ ) B , we get a representation M ( d, r, ǫ ) λ Theorem (C.) The generic module in mod A ( d, r ) is isomorphic to M ( d, r, ǫ ) λ . I.e., � λ ∈ ( k ∗ ) B GL( d ) M ( d, r, ǫ ) λ is dense in mod A ( d, r ) . Sketch: ∂ 0 (1) Find an explicit projective resolution . . . → P 1 − → P 0 → M λ ; (2) Ext 1 ( M λ , M λ ′ ) = 0 whenever λ, λ ′ share no common entries; (3) Ext 1 ( M λ , M λ ) = 1 when Γ consists of a single cycle (so M λ is indecomposable); (4) From Kraft, there is an injective map: → Ext 1 ( X, X ) T X ( C ) /T X (GL( d ) · X ) ֒ and C = mod( A, d, r ) is smooth at M λ .

Semi-invariants

∂ 0 ( X ) P 0 ( X ) Suppose P 1 ( X ) X is a minimal projective presentation of X in mod A . Hom A ( ∂ 0 ( X ) ,M ) Consider Hom A ( P 0 ( X ) , M ) − − − − − − − − − − → Hom A ( P 1 ( X ) , M ) obtained by applying Hom A ( − , M ) to the presentation. If Hom A ( ∂ 0 ( X ) , M ) is a square matrix, define c X ( M ) := det Hom( ∂ 0 ( X ) , M ) c X : mod A (dim M ) → k is a semi-invariant function (Schofield). Definition An irreducible component is called regular if the generic module is the sum of band modules ( Γ ’s connected components are cycles).

Proposition (C.-Chindris) If mod A ( d, r ) is regular then the following hold: mod A ( d, r ′ ) regular implies r = r ′ ; If mod A ( d, r ) = mod A ( d 1 , r 1 ) m 1 ⊕ . . . ⊕ mod A ( d n , r n ) m n then c M ( d i ,r i ) λ : mod A ( d, r ) → k is a well-defined (non-trivial) semi-invariant of weight θ d i . Thus c M ( d i ,r i ) λ c M ( d i ,r i ) µ ∈ k (mod A ( d, r )) GL( d ) .

Theorem (C.-Chindris) Let { λ ( i, j ) | i = 1 , . . . , n, j = 0 , . . . , m i } be distinct fixed elements of k ∗ . Then � � � c M ( d i ,r i ) λ ( i,j ) � f i,j = � i = 1 , . . . , n, j = 0 , . . . , m i − 1 � c M ( d i ,r i ) λ ( i,j +1) � is a transcendental basis for k (mod( d, r )) GL( d ) .

Sketch: Calin next week: If C is Schur ( min M ∈ C dim k End( M ) = 1 ), and A is tame then k ( C ) GL( d ) is purely transcendental of transcendence degree 1; if mod A ( d, r ) is indecomposable, then it is Schur (combinatorics of a certain bilinear form); Since d i � = d j for any summands of mod A ( d, r ) , k (mod A ( d, r )) GL( d ) is purely transcendental of transcendence degree equal to the number of direct summands N ; i,j D ( c M ( d i ,r i ) λ ( i,j ) ) The f i,j separate orbits in the open set � Kraft: k (mod( d, r )) GL( d ) = k ( { f i,j } ) # { f i,j | i = 1 , . . . , n, j = 0 , . . . , m i − 1 } = N .

Thank You!