(slides)

Ideals of cubic algebras and an

invariant ring of the Weyl

algebra

talk at the Antwerp Mini Workshop on Noncommutative Geometry Koen De Naeghel∗ University of Hasselt January 18, 2006 ∗_{website: http://alpha.luc.ac.be/} elucp1324

Based on joined work with N. Marconnet math.AG/0601096

Contents:

1. Introduction and main results 2. Noncommutative quadrics

3. From reflexive ideals to line bundles

4. From line bundles to quiver representations 5. Generic type A

6. The enveloping algebra 7. Hilbert series of ideals

1. Introduction and main results

• k = C _{field of complex numbers} • First Weyl algebra

A₁ = khx, yi/(xy − yx − 1) Question: describe right ideals of A₁

• Cannings and Holland (1994), Wilson (1998) R(A₁) = { right A₁-ideals}/₌∼ _←→ a

n

C_n where

C_n={(X_,Y_{)∈ Mn(k)}2_{| rk(}YX₋XY₋I_{)≤ 1}/Gln(k)} is the n-th Calogero-Moser space

Le Bruyn (1995) proposed an alternative method • Beilinson (1979) Db(coh(P2₎₎ RHom_P2(E,–) > < –⊗L_∆E Db(mod(∆))

where E = O(2) ⊕ O(1) ⊕ O and ∆ is the quiver −2 X₋₂ > Y₋₂ > Z₋₂ > −1 X₋₁ > Y₋₁ > Z₋₁ > 0

with relations reflecting the relations in k[x, y, z]

     Y₋₁X₋₂ = X₋₁Y₋₂ Z₋₁Y₋₂ = Y₋₁Z₋₂ X₋₁Z₋₂ = Z₋₁X₋₂

• Hulek, Barth (1977-1980): (stable) vector bundles on P2 _{are determined by certain} (stable) representations of the quiver ∆

• The picture survives when we replace

– k[x, y, z] by homogenized Weyl algebra

H = khx, y, zi .(zx − xz, zy − yz, yx − xy − z2) a noncommutative analogue of k[x, y, z]

– P2 _{by Proj H (sense of Artin and Zhang)} – relations of ∆ by the ones induced by H

• Equality A₁ = H[z−1]₀ induces bijection between sets

R(A₁) = { right A₁-ideals}/₌∼

and

R(H) = { reflexive graded right H-ideals }/_=,sh∼

They correspond to “line bundles” on P2

q

By derived equivalence line bundles are de-termined by certain stable representations of the quiver ∆

• Berest and Wilson (2002) used these ideas to reprove

Theorem A.

Aut A₁ has a natural action on R(A₁) – orbits indexed by N_,

– the n-th orbit is in bijection with n-th Calogero-Moser space

C_n={(X_,Y_{)∈ Mn(k)}2_{| rk(}YX₋XY₋I_{)≤ 1}/Gln(k)} smooth connected affine variety of

There are many more k-algebras inducing a P2_q. Interesting class: by Artin and Schelter (1986)

• Artin-Schelter algebra of dimension 3 is (i) graded k-algebra A = k ⊕ A₁ ⊕ A₂ ⊕ . . .

global dimension 3

(ii) A has polynomial growth

(iii) A is Gorenstein, i.e. for some l ∈ Z Exti_A(k_A, A) ∼=

(

Ak(l) if i = 3,

0 otherwise.

• Classified by Artin, Tate and Van den bergh (1990) and Stephenson (1994).

• They are all noetherian domains GK-dim 3 have all expected nice homological proper-ties

• Assume A generated in degree one. Two possibilities:

– A is quadratic

0 → A(−3) → A(−2)3 → A(−1)3 → A → kA → 0

h_A(t) = X

n

dim_k Antn = 1

(1 − t)3 – A is cubic

0 → A(−4) → A(−3)2 → A(−1)2 → A → k_A → 0

h_A(t) = X

n

dim_k Antn = 1

(1 − t)2(1 − t2)

• Generic class: called type A algebras quadratic:      ayz + bzy + cx2 = 0 azx + bxz + cy2 = 0 axy + byx + cz2 = 0 cubic: ( ay2x + byxy + axy2 + cx3 = 0 ax2y + bxyx + ayx2 + cy3 = 0 where a, b, c ∈ k generic.

• A is determined by a triple (E, σ, j) where either – A is linear: j : E ∼= P2 _{if A is quadratic} j : E ∼= P1 _× P1 _{if A is cubic} – A is elliptic: j : E ,→ P2_{, E divisor degree 3} if A is quadratic j : E ,→ P1_×P1_{, E divisor bidegree (2, 2)} if A is cubic

Generic case: A is type A and E is smooth elliptic curve (called generic type A)

• Define the set

• Van den Bergh and De Naeghel (2002) Theorem B.

Let _{A be elliptic quadratic and o(σ) = ∞.}

Then the set _{R(A) is in bijection with} `_n∈NDn where D_n is smooth locally closed variety dimension 2n.

If _{A is of generic type A then Dn} is affine. • Similar result by Nevins and Stafford (2002). • Aim of the talk:

analogue of Theorem B for cubic A.

N := {(n_e, n_o) ∈ N2 _{| ne − (ne − no)}2 _{≥ 0}} Theorem 1.

Let A be elliptic cubic and o(σ) = ∞. Then the set _{R(A) is in bijection with}

`

(n_e,no)∈N D(ne,no) where D(ne,no) is smooth

locally closed variety of dimension 2(n_e − (n_e − n_o)2).

If A is of generic type A then D_(ne,no) is affine.

• Application: enveloping algebra

Hc = khx, y, zi/(yz − zy, xz − zx, xy − yx − z)

= khx, yi/([y, [y, x]], [x, [x, y]])

Let ϕ ∈ Aut(A₁), ϕ(x) = −x, ϕ(y) = −y. Equality Ahϕi₁ = H_c[z−1]₀ induces bijection between sets

R(Ahϕi₁ ) = { right Ahϕi₁ -ideals}/₌∼

and

R(Hc) = { reflexive graded right Hc-ideals }/_=,sh∼

By Theorem 1 and further investigation Theorem 2. The set R(Ahϕi₁ ) is in bijection with `_(n e,no)∈N D(ne,no) where D_(ne,no) = {(X_,Y_,X0_,Y0_{) ∈ Mn} e×no(k)2×Mno×ne(k)2 | Y0X ₋ X0Y ₌ I _and rank(YX0₋XY0₋I_{) ≤ 1} /Gln} e(k) × Glno(k)

are smooth affine varieties of dimension 2(n_e − (n_e − n_o)2).

2. Noncommutative quadrics • A cubic Artin-Schelter algebra

graded k algebra A = k ⊕ A₁ ⊕ A₂ ⊕ . . . Hilbert series 1 + 2t + 4t2 + 6t3 + 9t4 + . . . • graded right A-module M = ⊕_i∈ZM_i then

degree 0 ↓

M = · · · ⊕ M₋₁⊕M₀ ⊕ M₁ ⊕ . . . M (1) := · · · ⊕ M₀ ⊕M₁ ⊕ M₂ ⊕ . . . • - GrMod(A) graded right A-modules

- Tors(A) direct limits of fdim modules

- GrMod(A) −→ GrMod(A)/ Tors(A) = Tails(A)π M 7→ M

A 7→ O

- M 7→ M (1) induces sh : M 7→ M(1)

- grmod(A), tors(A), tails(A) noeth. obj. - X := Proj(A) := (tails(A), O, sh)

A is determined by triple (E, σ, j)

• P = ⊕_nPn ∈ grmod(A) is a point module if

A _ P and h_P(t) = X n dim_k Pntn = 1 1 − t Choosing a k-basis e₀, e₁, . . . in P₀, P₁, . . . ( e₀ · x = α₀e₁ e₀ · y = β₀e₁ , ( e₁ · x = α₁e₂ e₁ · y = β₁e₂ , ( e₂ · x = α₂e₃ e₂ · y = β₂e₃ , . . . for some α_i, β_i ∈ k. If r is relation in A then e₀ · r = 0. Leads to equation in (α₀, β₀), (α₁, β₁)

E is the zero locus in P1 _×P1 _{of equation.} • Example: if A is type A then E is

(c2−b2)α₀β₀α₁β₁+aα2₀(cα2₁−bβ₁2)+aβ₀2(cβ₁2−bα2₁) = 0 • Example: if A = H_c enveloping algebra

then E is

(α₀β₁ − β₀α₁)2 = 0

the double diagonal 2D on P1 _× P1

• Point module P 7→ p closed point in E P (1)_≥0 7→ σ(p)

Artin, Tate and Van den Bergh:

A −→ (E, σ, j) −→ B = B(E, σ, j) • A linear: B ∼= A

A elliptic: B ∼= A/gA with g ∈ A₄ normal • Tails B and Qcoh E are equivalent

Tails A −⊗_AB −→ ←− (−)_A Tails B ˜ (−) −→ ←− Γ_∗ Qcoh E > i∗ < i∗

i∗ is right exact (restriction functor) i∗ is exact

3. From reflexive ideals to line bundles

• A cubic Artin-Schelter algebra, X = Proj A • For I ∈ grmod(A) graded right ideal

m 7→ dim_k Am − dim_k Im

is linear in m.

• For J ∈ grmod(A) of rank one, ∃! d ∈ Z _s.t. dim_k A_m − dim_k J(d)_m =

(

n_e for m  0 even n_o for m  0 odd for some integers n_e, n_o. We say

- J(d) is normalized

- (n_e, no) are the invariants of J

• For n_e, no ∈ Z define full subcategory of

grmod(A)

R_(ne,no)(A) = {J ∈ grmod(A) | J rank one, reflexive, normalized with invariants (n_e, n_o)}

• Apply exact quotient functor

π : grmod(A) → coh(X) I 7→ I

Let R_(ne,no)(X) be image of R_(ne,no)(A). Objects in R_(ne,no)(X) are called

normalized line bundles.

• R_(ne,no)(A) and R_(ne,no)(X) are groupoids • Natural bijection between set

R(A) = { reflexive graded right A-ideals }/_=,sh∼

and isoclasses of a (ne,no)∈Z2 R_(ne,no)(A) and isoclasses of a (ne,no)∈Z2 R_(ne,no)(X)

• For I ∈ R_(ne,no)(X) its cohomology groups Hi(X, I) := Exti_X(O, I)

are partially computed (for I 6∼= O)

l . . . −5 −4 −3 −2 −1 0 1 . .

dim_k H0(X, I(l)) . . . 0 0 0 0 0 0 ∗ . .

dim_k H1(X, I(l)) . . . ∗ ne − 1 no ne no ne − 1 ∗ . .

dim_k H2(X, I(l)) . . . ∗ 0 0 0 0 0 0 . .

Thus n_e > 0, no ≥ 0 and easy to prove

R_(0,0) = {O}

• By standard computation on the Euler form dim_k Ext1_X(I, I) = 2(n_e − (n_e − n_o)2)

hence for any integers ne, no

R_(ne,no)(X) 6= ∅ ⇒ (n_e, n_o) ∈ N Converse also true! See below.

4. From line bundles to quiver representa-tions

• A cubic Artin-Schelter algebra, X = Proj A • Due to a Theorem of Bondal (1990)

Db(coh X) RHom_X(E,–) > < –⊗L_ΓE Db(mod Γ)

where E = O(3) ⊕ O(2) ⊕ O(1) ⊕ O and Γ is the quiver

−3 X₋₃ > Y₋₃ >−2 X₋₂ > Y₋₂ >−1 X₋₁ > Y₋₁ >0

with relations reflecting the relations in A.

Assume (n_e, n_o) 6= (0, 0). Fix I ∈ R_(ne,no)(X).

• Consider I as complex of degree zero

• By previous its image is in degree one RHom_X(E, I) = M [−1]

where M = Ext1_X(E, I)

• M is given by a representation of Γ H1(X, I(−3)) X > Y >H 1_{(X, I(−2))} X0 > Y0 >H 1_{(X, I(−1))} X00 > Y00 >H 1_{(X, I)}

dimM = (n_o, ne, no, ne − 1) and relations.

For example, if A is type A then



X00 Y 00  · aY 0Y + cX0X bX0Y + aY 0X bY 0X + aX0Y aX0X + cY 0Y

!

How is “I is line bundle” translated? • Let P be point module, P = πP .

Cohomology groups given by

l . . . −5 −4 −3 −2 −1 0 1 . . .

dim_k H0(X, P(l)) . . . 1 1 1 1 1 1 1 . . . dim_k H1(X, P(l)) . . . 0 0 0 0 0 0 0 . . . dim_k H2(X, P(l)) . . . 0 0 0 0 0 0 0 . . .

P determines the representation of Γ k α₋₃ > β₋₃ >k α₋₂ > β₋₂ >k α₋₁ > β₋₁ >k where p = ((α₀, β₀), (α₁, β₁)) ∈ E σip = ((α_i, β_i), (α_i+1, β_i+1))

• I is line bundle means Ext1_X(P, I) = 0 0 = Ext1_X(P, I) = H0(RHom_X(P, I[1]))

∼

= H0(RHom_Γ(p, M )) = Hom_Γ(p, M )

• Trivially,

Hom_∆(M, p) = H0(RHom_Γ(M, p))

= H0(RHom_X(I[1], P)) = 0

• These properties characterize M . Theorem.

Let _{A be elliptic cubic and o(σ) = ∞.} Then equivalence R_(ne,no)(X) Ext1_X(E,–) > < TorΓ₁(–,E) C_(ne,no)(Γ) where C_(ne,no)(Γ) = {M ∈ mod(Γ) | dimM = (n_o, ne, no, ne−1) Hom_Γ(M, p) = Hom_Γ(p, M ) = 0 ∀p ∈ E}

Pick up another idea of Le Bruyn. I ∈ R_(ne,no)(X) is determined by both H1(X, I(−3)) X > Y >H 1_{(X, I(−2))} X0 > Y0 >H 1_{(X, I(−1))} X00 > Y00 >H 1_{(X, I)} H1(X, I(−4)) 0_X > 0_Y >H 1_{(X, I(−3))} X > Y >H 1_{(X, I(−2))} X0 > Y0 >H 1_{(X, I(−1))}

So I is actually determined by repr. M0

H1(X, I(−3)) X > Y >H 1_{(X, I(−2))} X0 > Y0 >H 1_{(X, I(−1))}

of the full subquiver Γ0 of Γ −3 X₋₃ > Y₋₃ >−2 X₋₂ > Y₋₂ >−1

Characterizing properties of M0?

• As M0 is the restriction of M :

certain rank condition involving X, Y, X0, Y 0 For example, if A is type A then

 X00 Y 00  · aY 0Y + cX0X bX0Y + aY 0X bY 0X + aX0Y aX0X + cY 0Y ! = 0 yields rank aY 0Y + cX0X bX0Y + aY 0X bY 0X + aX0Y aX0X + cY 0Y ! ≤ dim ker  X00 Y 00  = 2n_o − (n_e − 1) • M0 is θ-stable for θ = (−1, 0, 1). Indeed:

∀I ∈ R_(ne,no)(X) : ∃ v ∈ A₂ : Hom_X(I, π(A/vA)) = 0 ⇒ Hom_Γ0(M0, Q0) = 0

⇒M0 ⊥ Q0

• These properties characterize M0 for (n_e, no) 6= (1, 1)

Theorem.

Let _{A be elliptic cubic and o(σ) = ∞.} Then equivalence C_(ne,no)(Γ) Res > < Ind D_(ne,no)(Γ0) where D_(ne,no)(Γ0) = {F ∈ mod(Γ0) | dimF = (n_o, ne, no),

F is θ-stable, dim_k(Ind F )₀ ≥ n_e − 1}

• Put α = (n_o, n_e, n_o) and

D_(ne,no) := {F ∈ Rep_α(Γ0) | F ∈ D_(ne,no)(Γ0)}// Gl_α(k) Locally closed follows.

Smooth of dimension 2(n_e − (n_e − n_o)2) is proved.

5. Generic type A

Assume A is generic type A.

`

(ne,no) R(ne,no)(X) equivalent to

{M ∈ coh(X) | i∗M is line bundle on E deg. zero} By picking a suitable line bundle V on E

∀I ∈ R_(ne,no)(X) : RHom_E(i∗I, V) = 0 ⇒ RHom_X(I, i∗V) = 0

⇒ RHom_Γ0(M0, V 0) = 0

⇒M0 ⊥ V 0

for some representation V 0 ∈ mod(Γ0). Leads to

D_(ne,no)(Γ0) = {F ∈ mod(Γ0) | dimF = (n_o, n_e, n_o),

F ⊥ V 0, dim_k(Ind F )₀ ≥ n_e − 1} As {F ∈ Rep_α(Γ0) | F ⊥ V 0} is affine

6. The enveloping algebra

Assume A = H_c is the enveloping algebra

H_c = khx, y, zi/(yz − zy, xz − zx, xy − yx − z) = khx, yi/([y, [y, x]], [x, [x, y]])

Work with E_red = D, the diagonal on P1 _× P1_.

`

(ne,no) R(ne,no)(X) equivalent to

{M ∈ coh(X) | i∗M line bundle on D deg. zero} ={M ∈ coh(X) | i∗M ∼= O_D}

Translates into

∀I ∈ R_(ne,no)(X) : Hom_X(I, π(A/zA)) = 0 ⇒ Hom_Γ0(M0, Q0) = 0 ⇒M0 ⊥ Q0 Leads to D_(ne,no)(Γ0) = {F ∈ mod(Γ0) | dimF = (n_o, n_e, n_o), F ⊥ Q0, dim_k(Ind F )₀ ≥ n_e − 1} Thus D_(ne,no) is affine.

We now simplify this expression. Let F ∈ Rep_α(Γ0) i.e.

kno X > Y >k n_e X0 > Y 0 >k n_o • F ⊥ Q0 means Y 0X −X0Y is an isomorphism Defining ( X _{= X} Y _{= Y} and ( X0 _{= (Y} 0_{X − X}0_{Y )}−1_X0 Y0 _{= (Y} 0_{X − X}0_{Y )}−1_Y 0 this means YX0 ₋ X0Y0 ₌ I

• dim_k(Ind F )₀ ≥ n_e − 1 means rank Y 0Y X0Y − 2Y 0X

Y 0X − 2X0Y X0X

!

≤ 2n_o−(n_e−1)

7. Hilbert series of ideals

A is three-dimensional Artin-Schelter algebra. Question.

If _{I = ⊕nIn} ∈ grmod(A) is a graded right ideal, how does _hI(t) = P_n dim_k Intn look like?

• Macaulay (1927): for A = k[x, y, z] Restrict to pd I ≤ 1.

m 7→ dim_k A_m − dim_k I_m is linear in m, i.e. ∃! d ∈ Z _s.t.

dim_k A_m − dim_k I(d)_m = n for m  0 for some integer n.

In terms of formal power series h_I(d)(t) = h_k[x,y,z](t) − s(t) 1 − t = 1 (1 − t)3 − s(t) 1 − t

Turns out s(t) is a Castelnuovo polynomial s(t) = 1 + 2t + 3t2 +· · · + utu−1+ s_utu +· · · + s_vtv

u ≥ su ≥ . . . ≥ sv ≥ 0

for some integers u, v ≥ 0. s(1) = n ≥ 0.

Visualized in form of a stair Example:

1 2 3 4 5 5 3 2 1 1 1 1

Answer (commutative case). h(t) ∈ Z_{((t)) is of the form h}

I(d)(t) for some

graded ideal _{I with pd I ≤ 1 if and only if}

h(t) = 1

(1 − t)3 −

s(t) 1 − t

• Van den Bergh and De Naeghel (2004): Answer (quadratic case).

Let _{A be quadratic Artin-Schelter algebra.} h(t) ∈ Z_{((t)) is of the form h}

I(d)(t) for

some graded right ideal _{I with pd I ≤ 1} iff

h(t) = 1

(1 − t)3 −

s(t) 1 − t

for some Castelnuovo polynomial _s(t).

If in addition _{A is elliptic and o(σ) = ∞} then _{I may be chosen reflexive.}

• What is the answer for cubic A? As indicated above: ∃! d ∈ Z _s.t. dim_k A_m − dim_k I(d)_m =

(

ne for m  0 even

n_o for m  0 odd for some integers n_e, no ≥ 0.

In terms of formal power series

h_I(t) = 1

(1 − t)2(1 − t2) −

s(t) 1 − t2 for some s(t) = P_i s_iti ∈ Z_{[t, t}−1_]

with P_i s_2i = ne and P_i s_2i+1 = no.

Answer (cubic case).

Let _{A be cubic Artin-Schelter algebra.} h(t) ∈ Z_{((t)) is of the form h}

I(d)(t) for

some graded right ideal _{I with pd I ≤ 1} iff

h(t) = 1

(1 − t)2(1 − t2) −

s(t) 1 − t2 for some Castelnuovo polynomial _s(t).

If in addition _{A is elliptic and o(σ) = ∞} then _{I may be chosen reflexive.}

Example:

1 2 3 4 5 5 3 2 1 1 1 1

P

i s2i = 14 even weight

P

i s2i+1 = 15 odd weight.

As a consequence, for all integers n_e, n_o

R_(ne,no)(X) 6= ∅ ⇔ ∃ Castelnuovo polynomial s(t) with even weight n_e and odd weight n_o But recall

R_(ne,no)(X) 6= ∅ ⇒ (n_e, no) ∈ N

where

N = {(ne, no) ∈ N2 | ne − (ne − no)2 ≥ 0}

For (n_e, n_o) ∈ N easy to construct s(t), hence R_(ne,no)(X) 6= ∅ ⇔ (n_e, n_o) ∈ N

We end with combinatorical by-product. Shift rows of s(t) to left

For any partition λ, put draughts colouring. Number of _ is called even weight of λ

Number of _ is called odd weight of λ Bijection between

{Castelnuovo polynomials

with even weight n_e and odd weight n_o} and

{partitions in distinct parts

Theorem. Let n_e, no ∈ N. There is a partition

λ in distinct parts with even weight ne and odd

weight n_o iff

ne − (ne − no)2 ≥ 0

Special case of Chung’s Conjecture (1951) Proved by G. de B. Robinson (1951)

In particular, the number of such partitions λ is the number of partitions of n_e − (n_e − n_o)2. We may extend Theorem to all partitions