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:

1

Gentle algebras

2

Generic Decomposition

3

Semi-invariants

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

• n≥0

SI(A, C)n·χ 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 : Q1 → {1, . . . , m} is a surjection with c−1(i) is a path for each i. Ic := a2a1 | a1 − → a2 − →, c(a1) = c(a2)

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 = Ic.

Definition

Fix A = kQ/Ic, and d ∈ NQ0 rank function is a map r : Q1 → N s.t. x ⇒ r(a1) + r(a2) ≤ d(x) mod(A, d, r) ⊂ mod(A, d) the algebraic set {V ∈ modd(A) | rank Va ≤ r(a) ∀a ∈ Q1}

Proposition (Corollary to DeConcini-Strickland)

modA(d, r) is an irreducible component of modA(d) whenever r is maximal; modA(d, r) is normal.

Goal

Given an irreducible component mod(A, d, r), determine the structure

• f the generic module;

Determine k(mod(A, d, r))GL(d), and a transcendence basis; Show that M(A, modA(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 = d1 + . . . + dm such that for all M ∈ U, M ∼ = M1 ⊕ . . . ⊕ Mm where Mi is indecomposable of dimension di. [Kac] Such a decomposition of d exists; [Gabriel] If Ext1

A(M, M) = 0, then U = GL(d) · M;

Krull-Schmidt-type property

If C1, . . . , Cm are irreducible components, Ci ⊂ modA(di), consider C1 ⊕ . . . ⊕ Cm :=

• Mi∈Ci

GL(d) · (M1 ⊕ . . . ⊕ Mm)

Theorem (Crawley-Boevey Schr¨

• er)

If C is an irreducible component, then C = C1 ⊕ . . . ⊕ Cm where Ci are indecomposable irreducible components and min{dim Ext1

A(Mi, Mj) | Ml ∈ Cl} = 0 for all i = j.

Moreover, this decomposition is unique

Fix A = kQ/Ic a gentle algebra X = {(x, s) ∈ Q0 × S | ∃a ∈ Q1 with c(a) = s} sign function ǫ : X → {±1} with ǫ(x, s1) = −ǫ(x, s2) when s1 = s2 −+ −+ +− −+ −+ −+

For fixed d and r, construct a digraph Γ = ΓQ,c(d, r, ǫ): Γ0 = {vx

j | x ∈ Q0, j = 1, . . . , d(x)}

Γ1 = {f a

j | a ∈ Q1, j = 1, . . . , r(a)}: suppose x a

− → y ∈ Q1

−+ −+ +− −+ −+ −+

v(1)

1

v(2)

1

v(3)

1

v(1)

2

v(2)

2

v(1)

3

v(2)

3

v(2)

4

v(4)

1

v(5)

1

v(6)

1

v(4)

2

v(5)

2

v(6)

2

v(5)

3

v(1)

1

v(2)

1

v(3)

1

v(1)

2

v(2)

2

v(1)

3

v(2)

3

v(2)

4

v(4)

1

v(5)

1 λ

v(6)

1

v(4)

2

v(5)

2

v(6)

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 modA(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 modA(d, r).

Sketch: (1) Find an explicit projective resolution . . . → P1

∂0

− → P0 → Mλ; (2) Ext1(Mλ, Mλ′) = 0 whenever λ, λ′ share no common entries; (3) Ext1(Mλ, Mλ) = 1 when Γ consists of a single cycle (so Mλ is indecomposable); (4) From Kraft, there is an injective map: TX(C)/TX(GL(d) · X) ֒ → Ext1(X, X) and C = mod(A, d, r) is smooth at Mλ.

Semi-invariants

Suppose P1(X)

∂0(X) P0(X)

X is a minimal projective presentation of X in modA. Consider HomA(P0(X), M)

HomA(∂0(X),M)

− − − − − − − − − − → HomA(P1(X), M)

• btained by applying HomA(−, M) to the presentation.

If HomA(∂0(X), M) is a square matrix, define cX(M) := det Hom(∂0(X), M) cX : modA(dim M) → k is a semi-invariant function (Schofield).

Definition

An irreducible component is called regular if the generic module is the sum

• f band modules (Γ’s connected components are cycles).

Proposition (C.-Chindris)

If modA(d, r) is regular then the following hold: modA(d, r′) regular implies r = r′; If modA(d, r) = modA(d1, r1)m1 ⊕ . . . ⊕ modA(dn, rn)mn then cM(di,ri)λ : modA(d, r) → k is a well-defined (non-trivial) semi-invariant of weight θdi. Thus cM(di,ri)λ cM(di,ri)µ ∈ k(modA(d, r))GL(d).

Theorem (C.-Chindris)

Let {λ(i, j) | i = 1, . . . , n, j = 0, . . . , mi} be distinct fixed elements of k∗. Then

• fi,j =

cM(di,ri)λ(i,j) cM(di,ri)λ(i,j+1)

• i = 1, . . . , n, j = 0, . . . , mi − 1
• is a transcendental basis for k(mod(d, r))GL(d).

Sketch:

Calin next week: If C is Schur (minM∈C dimk End(M) = 1), and A is tame then k(C)GL(d) is purely transcendental of transcendence degree 1; if modA(d, r) is indecomposable, then it is Schur (combinatorics of a certain bilinear form); Since di = dj for any summands of modA(d, r), k(modA(d, r))GL(d) is purely transcendental of transcendence degree equal to the number

• f direct summands N;

The fi,j separate orbits in the open set

i,j D(cM(di,ri)λ(i,j))

Kraft: k(mod(d, r))GL(d) = k({fi,j}) #{fi,j | i = 1, . . . , n, j = 0, . . . , mi − 1} = N.

Thank You!