Ideals of three dimensional Artin-Schelter regular algebras

Ideals of three dimensional

Artin-Schelter regular algebras

Koen De Naeghel

Thesis Supervisor: Michel Van den Bergh February 17, 2006

Polynomial ring

Put k = C. Commutative polynomial ring S = k[x, y, z] = khx, y, zi/(f₁, f₂, f₃)      f₁ : xy − yx = 0 f₂ : yz − zy = 0 f₃ : zx − xz = 0

Polynomial ring

Put k = C. Commutative polynomial ring S = k[x, y, z] = khx, y, zi/(f₁, f₂, f₃)      f₁ : xy − yx = 0 f₂ : yz − zy = 0 f₃ : zx − xz = 0

Noncommutative polynomial rings How to define them?

Polynomial ring

Put k = C. Commutative polynomial ring S = k[x, y, z] = khx, y, zi/(f₁, f₂, f₃)      f₁ : xy − yx = 0 f₂ : yz − zy = 0 f₃ : zx − xz = 0

Noncommutative polynomial rings

Artin-Schelter (1986) defined class of algebras. • A is quadratic: A = khx, y, zi/(g₁, g₂, g₃) Generic:      g₁ : ayz + bzy + cx2 = 0 g₂ : azx + bxz + cy2 = 0 g₃ : axy + byx + cz2 = 0

Polynomial ring

Put k = C. Commutative polynomial ring S = k[x, y, z] = khx, y, zi/(f₁, f₂, f₃)      f₁ : xy − yx = 0 f₂ : yz − zy = 0 f₃ : zx − xz = 0

Noncommutative polynomial rings

Artin-Schelter (1986) defined class of algebras. • A is quadratic: A = khx, y, zi/(g₁, g₂, g₃) Generic:      g₁ : ayz + bzy + cx2 = 0 g₂ : azx + bxz + cy2 = 0 g₃ : axy + byx + cz2 = 0 • A is cubic: A = khx, yi/(g₁, g₂) Generic: ( g₁ : ay2x + byxy + axy2 + cx3 = 0 g₂ : ax2y + bxyx + ayx2 + cy3 = 0

Projective plane

Consider the projective plane P2.

Projective plane

Consider the projective plane P2.

Homogeneous coordinate ring S = k[x, y, z]

What has S = k[x, y, z] to do with P2?

For any homogeneous polynomial f ∈ k[x, y, z] {p ∈ P2 | f (p) = 0}

is a curve on P2.

Example:

Projective plane

Consider the projective plane P2.

Homogeneous coordinate ring S = k[x, y, z] S_d = {homogeneous polynomials degree d}

Projective plane

Consider the projective plane P2.

Homogeneous coordinate ring S = k[x, y, z] S_d = {homogeneous polynomials degree d}

S = k ⊕ S₁ ⊕ S₂ ⊕ . . . graded k-algebra P2 _{is completely determined by S.}

Theorem of Serre (1955)

Projective plane

Consider the projective plane P2.

Homogeneous coordinate ring S = k[x, y, z] S_d = {homogeneous polynomials degree d}

S = k ⊕ S₁ ⊕ S₂ ⊕ . . . graded k-algebra P2 is completely determined by S.

Theorem of Serre (1955)

Qcoh P2 _{' GrMod S/ Tors S}

What is GrMod S?

An object of GrMod S is

Projective plane

Consider the projective plane P2.

Homogeneous coordinate ring S = k[x, y, z] S_d = {homogeneous polynomials degree d}

S = k ⊕ S₁ ⊕ S₂ ⊕ . . . graded k-algebra P2 is completely determined by S.

Theorem of Serre (1955)

Qcoh P2 _{' GrMod S/ Tors S}

What is Tors S?

Generated by modules M ∈ GrMod S for which ∀m ∈ M : mS_≥d = 0 for some d

Typical:

Projective plane

Consider the projective plane P2.

Homogeneous coordinate ring S = k[x, y, z] S_d = {homogeneous polynomials degree d}

S = k ⊕ S₁ ⊕ S₂ ⊕ . . . graded k-algebra P2 is completely determined by S.

Theorem of Serre (1955)

Qcoh P2 _{' GrMod S/ Tors S}

Noncommutative projective plane

Model of noncommutative projective plane P2_q

Projective plane

Consider the projective plane P2.

Homogeneous coordinate ring S = k[x, y, z] S_d = {homogeneous polynomials degree d}

S = k ⊕ S₁ ⊕ S₂ ⊕ . . . graded k-algebra P2 is completely determined by S.

Theorem of Serre (1955)

Qcoh P2 _{' GrMod S/ Tors S}

Noncommutative projective plane

Model of noncommutative projective plane P2_q

Artin-Zhang (1994)

• Replace S by noncommutative k-algebra A • Define Qcoh P2_q := GrMod A/ Tors A

Artin-The points on P Point p ∈ P2

p

The points on P

Point p ∈ P2 _{↔ two linear forms l1}, l₂ ∈ S₁

p

x

l₁

The points on P

Point p ∈ P2 ↔ two linear forms l₁, l₂ ∈ S₁

p x l₁ l₂ represented by 0 → S(−2) −l₂ l₁ ! · −−−−−→ S(−1)2  l₁ l₂· −−−−−−→ S → P → 0 where • P = P₀ ⊕ P₁ ⊕ P₂ ⊕ . . . ∈ GrMod S

The points on P

Correspondence is reversible

point p on P2 _{↔ S-module P = P0}S, h_P(t) = 1 1 − t

The points on P

Correspondence is reversible

point p on P2 _{↔ S-module P = P0S, hP}(t) = 1 1 − t

The points on P2_q: by definition

“point” p on P2_q := right A-module P = P₀A, h_P(t) = 1 1 − t

The points on P

Correspondence is reversible

point p on P2 _{↔ S-module P = P0S, hP(t) =} 1 1 − t

The points on P2_q: by definition

“point” p on P2_q := right A-module P = P₀A, h_P(t) = 1 1 − t

Artin, Tate and Van den Bergh (1990):

• There is divisor E ⊂ P2 of deg 3 such that (closed) point p on E ↔ “point” on P2_q

• A, P2_q determined by the “points” on P2_q

Generic: E is smooth elliptic curve

The points on P versus graded _S-ideals Point p ∈ P2 _{↔ two linear forms l1, l2 ∈ S1}

p x l1 l2 0 → S(−2) −l₂ l₁ ! · −−−−−→ S(−1)2  l₁ l₂· −−−−−−→ S → P → 0

The points on P versus graded _S-ideals Point p ∈ P2 _{↔ two linear forms l1, l2 ∈ S1}

p x l1 l2 0 → S(−2) −l₂ l₁ ! · −−−−−→ S(−1)2  l₁ l₂· −−−−−−→ S → P → 0 & % I

The points on P versus graded _S-ideals Point p ∈ P2 _{↔ two linear forms l1, l2 ∈ S1}

p x l1 l2 0 → S(−2) −l₂ l₁ ! · −−−−−→ S(−1)2  l₁ l₂· −−−−−−→ S → P → 0 & % I

I = l₁S + l₂S ideal polynomials vanishing at p. In general: any graded ideal I, I  _{S(d) is (up} to Tors S) the ideal of polynomials vanishing at some points.

The points on P versus graded _S-ideals Point p ∈ P2 _{↔ two linear forms l1, l2 ∈ S1}

p x l1 l2 0 → S(−2) −l₂ l₁ ! · −−−−−→ S(−1)2  l₁ l₂· −−−−−−→ S → P → 0 & % I

I = l₁S + l₂S ideal polynomials vanishing at p. In general: any graded ideal I, I  _{S(d) is (up} to Tors S) the ideal of polynomials vanishing at some points.

If I ⊂ S graded S-ideal: - put J = ωπI

- Either J ∼= S(d) or pd J = 1.

S-ideals of projective dimension one Let I ⊂ S graded ideal, pd I = 1

0 → ⊕_iS(−i)bi _{−−→ ⊕}M·

S-ideals of projective dimension one Let I ⊂ S graded ideal, pd I = 1

0 → ⊕_iS(−i)bi _{−−→ ⊕}M·

iS(−i)ai → I → 0

Hilbert-Burch (1890)

I is generated by the maximal minors of M (whose zero’s determine configuration of points)

S-ideals of projective dimension one Let I ⊂ S graded ideal, pd I = 1

0 → ⊕_iS(−i)bi _{−−→ ⊕}M·

iS(−i)ai → I → 0

Hilbert-Burch (1890)

I is generated by the maximal minors of M (whose zero’s determine configuration of points) Known:

Given a_i, b_i there is such an ideal I (up to shift) if and only if deg⊕S(−i)bi _{→ ⊕} iS(−i)ai  =          × × . . . × × × × × × × × × . . . × × × ×          ×0 s > 0

S-ideals of projective dimension one Let I ⊂ S graded ideal, pd I = 1

0 → ⊕_iS(−i)bi _{−−→ ⊕}M·

iS(−i)ai → I → 0

Hilbert-Burch (1890)

I is generated by the maximal minors of M (whose zero’s determine configuration of points) Known:

Given a_i, b_i there is such an ideal I (up to shift) if and only if deg⊕S(−i)bi _{→ ⊕} iS(−i)ai  =           × × . . . × × × × × × × × × . . . × × × × ×           ×0 s > 0 if and only if h_I(t) = 1 (1 − t)3 − s(t) 1 − t

A Castelnuovo polynomial is of the form

s(t) = 1 + 2t + 3t2 +· · · + utu−1+ sutu +· · · + svtv

u ≥ s_u ≥ . . . ≥ s_v ≥ 0 for some integers u, v ≥ 0.

A Castelnuovo polynomial is of the form

s(t) = 1 + 2t + 3t2 +· · · + utu−1+ sutu +· · · + svtv

u ≥ s_u ≥ . . . ≥ s_v ≥ 0 for some integers u, v ≥ 0.

Visualized in form of a stair

Example:

The “points” on P_q versus right _A-ideals “point” on P2_q

p

E

The “points” on P_q versus right _A-ideals “point” on P2_{q ↔ two linear forms l1, l2 ∈ A1}

intersecting at E p l1 x l2 E

The “points” on P_q versus right _A-ideals “point” on P2_{q ↔ two linear forms l1, l2 ∈ A1}

intersecting at E p l1 x l2 E 0 → A(−2) w₁ w₂ ! · −−−−−→ A(−1)2  l₁ l₂· −−−−−−→ A → P → 0 & % I

The “points” on P_q versus right _A-ideals “point” on P2_{q ↔ two linear forms l1, l2 ∈ A1}

intersecting at E p l1 x l2 E v1 x v2 0 → A(−2) w₁ w₂ ! · −−−−−→ A(−1)2  l₁ l₂· −−−−−−→ A → P → 0 & % I If v₁, v₂ ∈ A₁ not intersecting at E 0 → A(−2) v₁ v₂ ! · −−−−→ A(−1)2 → I0 → 0

Right _{A-ideals of projective dimension one}

Let I ⊂ A graded right ideal, pd I = 1 0 → ⊕_iA(−i)bi _{−−→ ⊕}M·

Right _{A-ideals of projective dimension one}

Let I ⊂ A graded right ideal, pd I = 1 0 → ⊕_iA(−i)bi _{−−→ ⊕}M·

iA(−i)ai → I → 0

Given a_i, b_i there is such I (up to shift) if and only if (Theorem 6)

deg⊕S(−i)bi _{→ ⊕} iS(−i)ai  =           × × . . . × × × × × × × × × . . . × × × × ×           ×0 s > 0

Right _{A-ideals of projective dimension one}

Let I ⊂ A graded right ideal, pd I = 1 0 → ⊕_iA(−i)bi _{−−→ ⊕}M·

iA(−i)ai → I → 0

Given a_i, b_i there is such I (up to shift) if and only if (Theorem 6)

deg⊕S(−i)bi _{→ ⊕} iS(−i)ai  =           × × . . . × × × × × × × × × . . . × × × × ×           ×0 s > 0

if and only if (Theorem 4)

Right _{A-ideals of projective dimension one}

Let I ⊂ A graded right ideal, pd I = 1 0 → ⊕_iA(−i)bi _{−−→ ⊕}M·

iA(−i)ai → I → 0

Given a_i, b_i there is such I (up to shift) if and only if (Theorem 6)

deg⊕S(−i)bi _{→ ⊕} iS(−i)ai  =           × × . . . × × × × × × × × × . . . × × × × ×           ×0 s > 0

if and only if (Theorem 4)

h_I(t) = 1

(1 − t)3 −

s(t) 1 − t

for some Castelnuovo polynomial s(t).

Hilbert scheme of points on P

Classify all possible configurations of n points on P2. Can be done by parameterspace.

Formally: parameter space for subschemes of P2 of dimension zero and degree n.

Hilbert scheme of points on P

Classify all possible configurations of n points on P2. Can be done by parameterspace.

Formally: parameter space for subschemes of P2 of dimension zero and degree n.

moduli problem

Hilb_n(P2_{) : Noeth /k → Sets}

R 7→ Hilb_n(P2_)(R) Hilb_n(P2_{)(R) ={N ⊂ P}2

R | N is R-flat and ∀x ∈ Spec R

Hilbert scheme of points on P

Classify all possible configurations of n points on P2. Can be done by parameterspace.

Formally: parameter space for subschemes of P2 of dimension zero and degree n.

moduli problem

Hilb_n(P2_{) : Noeth /k → Sets}

R 7→ Hilb_n(P2_)(R) Hilb_n(P2_{)(R) ={N ⊂ P}2

R | N is R-flat and ∀x ∈ Spec R

Hilbert scheme of points on P_q

Initial problem: P2_q has few zero-dimensional noncommutative subschemes

Hilbert scheme of points on P_q

Initial problem: P2_q has few zero-dimensional noncommutative subschemes

Hilbert scheme of points on P_q

Initial problem: P2_q has few zero-dimensional noncommutative subschemes

Solution: consider ideal sheaves instead moduli problem

Hilb_n(P2_{q) : Noeth /k → Sets}

R 7→ Hilb_n(P2_q)(R)

Hilb_n(P2_{q)(R) ={I ∈ cohP}2

q,R | I is R-flat and ∀x ∈ Spec R

Ix ∈ cohP2_q,k(x) torsion free, pd 1, normalized, rk 1}/ Pic R

Hilbert scheme of points on P_q

Initial problem: P2_q has few zero-dimensional noncommutative subschemes

Solution: consider ideal sheaves instead moduli problem

Hilb_n(P2_{q) : Noeth /k → Sets}

R 7→ Hilb_n(P2_q)(R)

Hilb_n(P2_{q)(R) ={I ∈ cohP}2

q,R | I is R-flat and ∀x ∈ Spec R

Ix ∈ cohP2_q,k(x) torsion free, pd 1, normalized, rk 1}/ Pic R

In case A = S: agrees with Hilb_n(P2_).

- A graded (right) ideal I is reflexive if Ext1(P, I) = 0 for all point modules P - If I is reflexive then pd I ≤ 1.

- A graded (right) ideal I is reflexive if Ext1(P, I) = 0 for all point modules P - If I is reflexive then pd I ≤ 1.

- A graded (right) ideal I is reflexive if Ext1(P, I) = 0 for all point modules P - If I is reflexive then pd I ≤ 1.

reflexive graded S-ideals: S(d)

reflexive graded right A-ideals In case A is generic: Theorems 1,2

R(A) = {reflexive graded right A-ideals}/iso,shift l

a n

Dn where Dn is smooth affine variety of dim 2n

points D_n are given by stable representations

kn X > Y > Z k n _{with rank}    cX aZ bY bZ cY aX aY bX cZ    ≤ 2n+1

Picture in case A is generic:

Hilb_n(P2_q)

D_n

• Hilb_n(P2_q) parameterizes

{graded right A-ideals I, pd I = 1 h_I(t) is 1

(1 − t)3 −

s(t)

1 − t up to shift}/iso, shift • Dn parameterizes

For any graded right ideal J with pd J = 1 0 → J →J₁ → P₁(d₁) → 0

0 → J₁ →J₂ → P₂(d₂) → 0 ...

0 → J_r−1 →Jr → Pr(dr) → 0

where J_r is reflexive. Note 0 ≤ r ≤ n.

· J · · J₁ . . . J_r−1 ` n Hilbn(P2q) · Jr ` n Dn

For any graded right ideal J with pd J = 1 0 → J →J₁ → P₁(d₁) → 0

0 → J₁ →J₂ → P₂(d₂) → 0 ...

0 → J_r−1 →Jr → Pr(dr) → 0

where J_r is reflexive. Note 0 ≤ r ≤ n.

· J · · J₁ . . . J_r−1 ` n Hilbn(P2q) · Jr ` n Dn

Let Hilb≥d_n (P2_q) be the J ∈ Hilbn(P2_q) with r ≥ d.

For any graded right ideal J with pd J = 1 0 → J →J₁ → P₁(d₁) → 0

0 → J₁ →J₂ → P₂(d₂) → 0 ...

0 → J_r−1 →Jr → Pr(dr) → 0

where J_r is reflexive. Note 0 ≤ r ≤ n.

· J · · J₁ . . . J_r−1 ` n Hilbn(P2q) · Jr ` n Dn

Let Hilb≥d_n (P2_q) be the J ∈ Hilbn(P2q) with r ≥ d.

Theorem 7

- Hilb≥d_n (P2_{q) projective variety of dimension 2n−d}

Stratification of Hilb_n(P_q)

Consider all points of Hilb_n(P2) parameterizing ideals of A with same Hilbert series.

Hilbn(P2_q)

Stratification of Hilb_n(P_q)

Consider all points of Hilb_n(P2) parameterizing ideals of A with same Hilbert series.

Hilbn(P2_q)

. . . Hilb_h(P2_q)

For an appearing Hilbert series h(t) = 1 (1 − t)3 − s(t) 1 − t s(t) is Castelnuovo polynomial, s(1) = n put Hilb_h(P2_q) = {I ∈ Hilbn(P2_q) | h_I(t) = h(t)}

Hilbn(Pq)

. . . Hilb_h(P2_q)

Chapter 3

Hilb_h(P2_q) ⊂ Hilb_n(P2_q) is locally closed subvariety - smooth

- connected

Hilbn(Pq)

. . . Hilb_h(P2_q)

Chapter 3

Hilb_h(P2_q) ⊂ Hilb_n(P2_q) is locally closed subvariety - smooth

- connected

In case A = S: Proved by Gotzmann (1988)

Formula for dim Hilb_h(P2_q):

constant term of (t−1− t−2)s(t−1)s(t) + n + 1 ⇒ There is an unique stratum with maximal dimension 2n

Hilbn(Pq)

. . . Hilb_h(P2_q)

Chapter 3

Hilb_h(P2_q) ⊂ Hilb_n(P2_q) is locally closed subvariety - smooth

- connected

In case A = S: Proved by Gotzmann (1988)

Formula for dim Hilb_h(P2_q):

Hilb_n(P ) and Hilb_n(P2_q) analogous strata Hilb_n(P2) Hilb_n(P2_q) H_ϕ H_ψ H_ϕ H_ψ • Incidence problem:

Hilb_n(P ) and Hilb_n(P2_q) analogous strata Hilb_n(P2) Hilb_n(P2_q) H_ϕ H_ψ H_ϕ H_ψ • Incidence problem:

for which ϕ, ψ do we have H_ϕ ⊂ H_ψ ?

Hilb_n(P ) and Hilb_n(P2_q) analogous strata Hilb_n(P2) Hilb_n(P2_q) H_ϕ H_ψ H_ϕ H_ψ • Incidence problem:

for which ϕ, ψ do we have H_ϕ ⊂ H_ψ ?

General incidence problem for Hilb_n(P2): unknown - Guerimand (2002)

solved case ϕ and ψ are “as close as possible” under a technical condition

“as close as possible” means: writing ϕ(t) = 1 (1 − t)3 − s_ϕ 1 − t, ψ(t) = 1 (1 − t)3 − s_ψ 1 − t s_ψ is obtained from s_ϕ by minimal movement of one square to the left.

“as close as possible” means: writing ϕ(t) = 1 (1 − t)3 − s_ϕ 1 − t, ψ(t) = 1 (1 − t)3 − s_ψ 1 − t s_ψ is obtained from s_ϕ by minimal movement of one square to the left.

Examples:

n = 17: ϕ and ψ as close as possible

“as close as possible” means: writing ϕ(t) = 1 (1 − t)3 − s_ϕ 1 − t, ψ(t) = 1 (1 − t)3 − s_ψ 1 − t s_ψ is obtained from s_ϕ by minimal movement of one square to the left.

Examples:

n = 17: ϕ and ψ as close as possible

sϕ sψ

Hilb_n(P ) and Hilb_n(P2_q) analogous strata Hilb_n(P2) Hilb_n(P2_q) H_ϕ H_ψ H_ϕ H_ψ • Incidence problem:

for which ϕ, ψ do we have H_ϕ ⊂ H_ψ ?

General incidence problem for Hilb_n(P2): unknown - Guerimand (2002)

solved case ϕ and ψ are “as close as possible” under a technical condition

- Theorem 9

If ϕ, ψ Hilbert series “as close as possible”: We have Hϕ ⊂ H_ψ if and only if

6 ? - ≥ 2 ≥ 0 ≥ 1 - 3 6 ? 6 ?≥ 0 - 3 6 ? - ≥ 1 ≥ 2 6_{≥ 0} 6 ≥ 1 6 ?≥ 0 6 ? 6 ? - C ≥ 1 D ≥ 0 ≥ 1 - 2 6 ?≥ 0 6 ? 6 ? C ≥ 1 D ≥ 0 6 ? 6 ? A ≥ 0 B ≥ 1 C > D C > D A < B I II III IV

• Incidence problem:

for which ϕ, ψ do we have H_ϕ ⊂ H_ψ ?

• Incidence problem:

for which ϕ, ψ do we have H_ϕ ⊂ H_ψ ?

• Same solution for Hilb_n(P2_q) as for Hilb_n(P2)? No for generic _A

s_ϕ s_ψ

Generic I ∈ Hϕ

0 → A(−4)⊕A(−7) → A(−2)⊕A(−3)⊕A(−6) → I → 0 V = {f : I _ F | h_F = ϕ − ψ}

• Incidence problem:

for which ϕ, ψ do we have H_ϕ ⊂ H_ψ ?

• Same solution for Hilb_n(P2_q) as for Hilb_n(P2)? No for generic _A s_ϕ s_ψ Generic I ∈ Hϕ 0 → A(−4)⊕A(−7) → A(−2)⊕A(−3)⊕A(−6) → I → 0 V = {f : I _ F | h_F = ϕ − ψ} - commutative case: V = P2

V → N : f 7→ dim_k Ext1_A(ker f, ker f ) non-constant hence H_ϕ ⊂ H_ψ

- smooth elliptic case: V = three points on E V → N _{: f 7→ dimk} Ext1_A(ker f, ker f ) constant

If ϕ, ψ Hilbert series “as close as possible”: We have H_ϕ ⊂ H_ψ if and only if

6 ? - ≥ 2 ≥ 0 ≥ 1 - 3 6 ? 6 ?≥ 0 - 3 6 ? - ≥ 1 ≥ 2 6 ?≥ 0 ≥ 1 6 ≥ 1 6 6 ? ≥ 1 6 ?≥ 0 6 ? 6 ? - C ≥ 1 D ≥ 0 ≥ 1 - 2 6 ?≥ 0 6 ? 6 ? C ≥ 1 D ≥ 0 6 ? 6 ? A ≥ 0 B ≥ 1 C > D C > D A < B I II III IV