Tilting theory and Cohen-Macaulay representations Osamu Iyama - - PowerPoint PPT Presentation

Tilting theory and Cohen-Macaulay representations

Osamu Iyama

Nagoya University

Osamu Iyama (Nagoya) Tilting theory and CM representations 1 / 27

Introduction Results in dimension d ≤ 2 Examples from d-representation-infinite algebras Geigle-Lenzing complete intersections

Osamu Iyama (Nagoya) Tilting theory and CM representations 2 / 27

T : triangulated category

Definition

U ∈ T : tilting object ⇐ ⇒

• ∀i = 0 HomT (U, U[i]) = 0
• T = thick U (:=the smallest triangulated subcategory
• f T closed under direct summands and containing U)

Example: Λ ∈ Kb(proj Λ) is a tilting object

Theorem (Rickard, Keller)

U ∈ T : tilting object If T satisfies mild conditions (algebraic and idempotent complete), then T ≃ Kb(proj EndT (U))

Osamu Iyama (Nagoya) Tilting theory and CM representations 3 / 27

Definition

Λ : Iwanaga-Gorenstein ring ⇐ ⇒ Noetherian ring with inj.dim ΛΛ = inj.dim ΛΛ < ∞ Examples :

• Finite dimensional selfinjective algebra over a field k
• Commutative Gorenstein local ring
• Gorenstein order

Definition

Λ : Iwanaga-Gorenstein ring

• X ∈ mod Λ : (maximal) Cohen-Macaulay ⇐

⇒ ∀i > 0 Exti

Λ(X, Λ) = 0

• CM Λ : category of Cohen-Macaulay Λ-modules
• CM Λ : stable category

Osamu Iyama (Nagoya) Tilting theory and CM representations 4 / 27

Properties (Happel, Buchweitz, Orlov)

• CM Λ is a Frobenius category
• CM Λ is an algebraic triangulated category
• CM Λ ≃ Db(mod Λ)/Kb(proj Λ)

Question

When does CM Λ has a tilting object?

Observation

Λ : semiperfect CM Λ has a tilting object ⇐ ⇒ CM Λ = 0

Osamu Iyama (Nagoya) Tilting theory and CM representations 5 / 27

G : abelian group Assume Λ is G-graded, i.e. Λ =

g∈G Λg and

ΛgΛh ⊂ Λg+h

• modG Λ : category of G-graded Λ-modules
• CMG Λ := {X ∈ modG Λ | ∀i > 0 Exti

Λ(X, Λ) = 0}

Properties

• CMG Λ is a Frobenius category
• CMG Λ is an algebraic triangulated category
• CMG Λ ≃ Db(modG Λ)/Kb(projG Λ)

Question

When does CMG Λ has a tilting object?

Osamu Iyama (Nagoya) Tilting theory and CM representations 6 / 27

First examples

k : field k[x]/(x2) : Z-graded by deg x = 1

Observation

• modZ (k[x]/(x2)) ≃ Cb(mod k)
• modZ (k[x]/(x2)) ≃ Db(mod k)

Osamu Iyama (Nagoya) Tilting theory and CM representations 7 / 27

• Λ : finite dimensional k-algebra
• T(Λ) = Λ ⊕ DΛ : trivial extension algebra
• T(Λ) is a Z-graded symmetric k-algebra

with T(Λ)0 = Λ and T(Λ)1 = DΛ

Theorem (Happel)

Assume gl.dim Λ is finite

• Λ ∈ modZ T(Λ) is a tilting object
• modZ T(Λ) ≃ Db(mod Λ)

Application (Tachikawa)

Q : Dynkin quiver = ⇒ T(kQ) is representation-finite

Osamu Iyama (Nagoya) Tilting theory and CM representations 8 / 27

Dimension zero

• Λ =

i≥0 Λi : Z-graded selfinjective k-algebra

• For X ∈ modZ Λ, let X≥0 :=

i≥0 Xi

Theorem (Yamaura)

• gl.dim Λ < ∞ ⇐

⇒ modZ Λ has a tilting object

• In this case

i≥0 Λ(i)≥0 gives a tilting object

Π : preprojective algebra of an n-representation-finite algebra Λ (e.g. kQ for a Dynkin quiver Q)

Corollary (I-Oppermann, Yamaura)

modZ Π ≃ Db(mod EndΛ(Π))

Osamu Iyama (Nagoya) Tilting theory and CM representations 9 / 27

Dimension one

• R =

i≥0 Ri : commutative Gorenstein ring in

dimension one

• Assume R0 = k and has a Gorenstein parameter a ≤ 0

i.e. Ext1

R(k, R) ≃ k(a) in modZ R

Theorem (Buchweitz-I-Yamaura)

∃ p > 0 s.t. p

i=1 R(i)≥0 is a tilting object in CMZ R

Osamu Iyama (Nagoya) Tilting theory and CM representations 10 / 27

Dimension two

• Q : extended Dynkin quiver
• Π = kQ/aa∗ − a∗a | a ∈ Q1 : preprojective algebra

Π is Z-graded by deg (a) = 0, deg (a∗) = 1

• e ∈ Q0 : extended vertex

Theorem (Auslander-Reiten, Geigle-Lenzing, Kajiura-Saito-Takahashi)

• R := eΠe is a simple singularity in dimension two
• R is CM-finite with CM R = add eΠ and

has an Auslander algebra EndR(eΠ) = Π

• CMZ R ≃ Db(kQ/(e))

Osamu Iyama (Nagoya) Tilting theory and CM representations 11 / 27

d-representation-infinite algebras

(Joint work with Amiot and Reiten)

• k : algebraically closed field
• Λ : finite dimensional k-algebra of global dimension d
• νd := DΛ[−d]

L

⊗Λ − : Db(mod Λ) → Db(mod Λ) : derived d-AR translation

Definition (Herschend-I-Oppermann)

Λ d-representation-infinite ⇐ ⇒ ∀i ≥ 0, ν−i

d (Λ) ∈ mod Λ

Osamu Iyama (Nagoya) Tilting theory and CM representations 12 / 27

• Γ =

i≥0 Γi : positively graded k-algebra s.t. ∀i ≥ 0

dimk Γi < ∞

• Γe := Γ ⊗k Γop

Definition (cf. Ginzburg)

Γ : n-Calabi-Yau algebra of Gorenstein parameter a ⇐ ⇒ • Γ ∈ Kb(projZ Γe)

• RHomΓe(Γ, Γe) ≃ Γ[−n](a) in D(ModZ Γe)

Theorem (Minamoto-Mori, Keller)

∃ bijection given by Λ → Π {d-reresentation-infinite algebras} ≃ {(d + 1)-Calabi-Yau algebras of Gorenstein parameter 1}

Osamu Iyama (Nagoya) Tilting theory and CM representations 13 / 27

Setting

• Λ : d-representation-infinite algebra
• Assume Π := Π(Λ) is noetherian
• e ∈ Λ : idempotent s.t. dimk(Π/(e)) < ∞ and

eΛ(1 − e) = 0

Theorem (Amiot-I-Reiten)

Let R := eΠe and Λ := Λ/(e)

• R is Iwanaga-Gorenstein
• gl.dim Λ ≤ d
• CMZ R has a tilting object eΠ
• CMZ R ≃ Db(mod Λ)

Osamu Iyama (Nagoya) Tilting theory and CM representations 14 / 27

Definition

• M ∈ CM R : d-cluster tilting ⇐

⇒ add M = {X ∈ CM R | ∀i ∈ [1, d − 1] Exti

R(M, X) = 0}

= {X ∈ CM R | ∀i ∈ [1, d − 1] Exti

R(X, M) = 0}

• R : d-CM-finite ⇐

⇒ ∃ M ∈ CM R : d-cluster tilting Cd(Λ) : d-cluster category (Amiot-Guo-Keller)

Theorem (Amiot-I-Reiten)

• R is d-CM-finite with a d-cluster tilting module eΠ
• R has a d-Auslander algebra EndR(eΠ) = Π
• CM R ≃ Cd(Λ)

Osamu Iyama (Nagoya) Tilting theory and CM representations 15 / 27

• 2 ≤ d ≤ n
• (a1, . . . , ad) s.t. 0 < ai < n, (n, ai) = 1, d

i=1 ai = n

• G := diag(ζa1, . . . , ζad) ⊂ SLd(k)
• Π := k[x1, . . . , xd] ∗ G : skew group algebra

vertices : {1, 2, . . . , n} ≃ Z/nZ arrows : Q1 := {xj : i → i + aj | i ∈ Q0, 1 ≤ j ≤ d} relations : xjxj′ = xj′xj grading : deg (xj : i → i + aj) =

• 0 (i ≤ i + aj)

1 (i > i + aj)

d = n = 4 a1 = a2 = a3 = a4 = 1 Π 1 2 3 4

x1

• x2
• x3
• x4
• x1
• x2
• x3
• x4
• x1
• x2
• x3
• x4
• x1
• x2
• x3
• x4
• Osamu Iyama (Nagoya)

Tilting theory and CM representations 16 / 27

Corollary

Let R := k[x1, . . . , xd]G and Λ := Π0/(ed)

• CMZ R ≃ Db(mod Λ)
• CM R ≃ Cd(Λ)

Π 1 2 3 4

x1

• x2
• x3
• x4
• x1
• x2
• x3
• x4
• x1
• x2
• x3
• x4
• x1
• x2
• x3
• x4
• Π0

1 2 3 4

x1

• x2
• x3
• x4
• x1
• x2
• x3
• x4
• x1
• x2
• x3
• x4
• Λ

1 2 3

x1

• x2
• x3
• x4
• x1
• x2
• x3
• x4
• Osamu Iyama (Nagoya)

Tilting theory and CM representations 17 / 27

Corollary

Let R := k[x1, . . . , xd]G and Λ := Π0/(ed)

• CMZ R ≃ Db(mod Λ)
• CM R ≃ Cd(Λ)

Π 1 2 3 4

x1

• x2
• x3
• x4
• x1
• x2
• x3
• x4
• x1
• x2
• x3
• x4
• x1
• x2
• x3
• x4
• Π0

1 2 3 4

x1

• x2
• x3
• x4
• x1
• x2
• x3
• x4
• x1
• x2
• x3
• x4
• Λ

1 2 3

x1

• x2
• x3
• x4
• x1
• x2
• x3
• x4
• Osamu Iyama (Nagoya)

Tilting theory and CM representations 17 / 27

Corollary

Let R := k[x1, . . . , xd]G and Λ := Π0/(ed)

• CMZ R ≃ Db(mod Λ)
• CM R ≃ Cd(Λ)

Π 1 2 3 4

x1

• x2
• x3
• x4
• x1
• x2
• x3
• x4
• x1
• x2
• x3
• x4
• x1
• x2
• x3
• x4
• Π0

1 2 3 4

x1

• x2
• x3
• x4
• x1
• x2
• x3
• x4
• x1
• x2
• x3
• x4
• Λ

1 2 3

x1

• x2
• x3
• x4
• x1
• x2
• x3
• x4
• Osamu Iyama (Nagoya)

Tilting theory and CM representations 17 / 27

Corollary

Let R := k[x1, . . . , xd]G and Λ := Π0/(ed)

• CMZ R ≃ Db(mod Λ)
• CM R ≃ Cd(Λ)

Π 1 2 3 4

x1

• x2
• x3
• x4
• x1
• x2
• x3
• x4
• x1
• x2
• x3
• x4
• x1
• x2
• x3
• x4
• Π0

1 2 3 4

x1

• x2
• x3
• x4
• x1
• x2
• x3
• x4
• x1
• x2
• x3
• x4
• Λ

1 2 3

x1

• x2
• x3
• x4
• x1
• x2
• x3
• x4
• Osamu Iyama (Nagoya)

Tilting theory and CM representations 17 / 27

Geigle-Lenzing complete intersection

(Joint work with Herschend, Minamoto, Oppermann)

• k : field
• d ≥ 0
• C := k[T0, . . . , Td] : polynomial algebra
• n ≥ 0
• ℓ1, . . . , ℓn ∈ C : linear forms
• p1, . . . , pn ≥ 2 : integers (weights)

Definition

• R := C[X1, . . . , Xn]/(Xpi

i − ℓi | 1 ≤ i ≤ n)

• L :=

c, x1, . . . , xn/pi xi − c | 1 ≤ i ≤ n

Osamu Iyama (Nagoya) Tilting theory and CM representations 18 / 27

• L is an abelian group of rank one
• R is an L-graded k-algebra : deg Tj :=

c, deg Xi := xi

• R is a complete intersection ring in dimension d + 1
• R has a Gorenstein parameter −

ω for

• ω := (n − d − 1)

c − n

i=1

xi i.e. Extd+1

R (k, R) ≃ k(−

ω) in modZ R

Definition

(R, L) : Geigle-Lenzing (GL) complete intersection ⇐ ⇒ ∀ at most d + 1 elements from ℓ1, . . . , ℓn are linearly independent

Osamu Iyama (Nagoya) Tilting theory and CM representations 19 / 27

• n ≤ d + 1 ⇐

⇒ R is regular

• n = d + 2 ⇐

⇒ R is a hypersurface In this case R ≃ k[X1, . . . , Xd+2]/(n

i=1 αiXpi i )

Example

• When d = 1, (R, L) is the weighted projective line of

Geigle-Lenzing

• When d = 0 and n = 2, R ≃ k[X1, X2]/(Xp1

1 − Xp2 2 )

is studied by Jensen-King-Su and Baur-King-Marsh to cluster algebra structure of the coordinate rings of Grassmannians

Osamu Iyama (Nagoya) Tilting theory and CM representations 20 / 27

• L ⊃ L+ :=

c, x1, . . . , xnmonoid

• L is a poset:

x ≥ y ⇐ ⇒ x − y ∈ L+

Definition

δ := d c + 2 ω ∈ L

• ACM := (R

x− y)0≤ x, y≤ δ : CM-canonical algebra

k-algebra by product in R and matrix multiplication

• n = d + 2 =

⇒ δ = n

i=1(pi − 2)

xi and ACM = n

i=1 kApi−1

• d = 1, weights (2, 2, 2, 3)

ACM

• x4
• x1
• x2
• x3

2 x4

• x1 +

x4

• x2 +

x4

• x3 +

x4

• c
• δ =

x4 + c

• Osamu Iyama (Nagoya)

Tilting theory and CM representations 21 / 27

Theorem (HIMO)

CML R ≃ Db(mod ACM )

• Hypsersurface case was shown by

Kussin-Meltzer-Lenzing (d = 1) and Futaki-Ueda

• Idea of proof : Realize both sides as subcategories of

Db(modL R)

Corollary (Kn¨

• rrer periodicity)

(R, L), (R′, L′) : GL hypersurfaces with weights (p1, . . . , pd+2) and (2, p1, . . . , pd+2) resp. = ⇒ CML R ≃ CML′ R′

Osamu Iyama (Nagoya) Tilting theory and CM representations 22 / 27

Definition

(R, L) : CM-finite ⇐ ⇒ there are only finitely many indecomposable objects in CML R up to degree shift

• Define a group homomorphism d : L → Z by

d( c) = 1, d( xi) = 1

pi

• d(

ω) = n − d − 1 − n

i=1 1 pi

d( ω) < 0 = 0 > 0 d = 1 domestic tubular wild (R, L) Fano Calabi-Yau anti-Fano

• domestic ⇔ (p, q), (2, 2, p), (2, 3, 3), (2, 3, 4), (2, 3, 5)
• tubular ⇐

⇒ (2, 3, 6), (2, 4, 4), (3, 3, 3), (2, 2, 2, 2)

Osamu Iyama (Nagoya) Tilting theory and CM representations 23 / 27

Theorem (Geigle-Lenzing)

Assume d = 1

• (R, L) : CM-finite ⇐

⇒ domestic

• In this case ∃ Dynkin quiver Q s.t.

CML R ≃ Db(mod kQ)

Theorem (HIMO)

(R, L) : GL complete intersection is CM-finite ⇐ ⇒ • n ≤ d + 1 or

• n = d + 2 and weights are (2, . . . , 2, 2, p),

(2, . . . , 2, 3, 3), (2, . . . , 2, 3, 4), (2, . . . , 2, 3, 5)

Osamu Iyama (Nagoya) Tilting theory and CM representations 24 / 27

Definition

C ⊂ CML R : n-cluster tilting subcategory ⇐ ⇒ C = {X ∈ CML R | ∀i ∈ [1, n − 1] Exti

modL R(C, X) = 0}

= {X ∈ CML R | ∀i ∈ [1, n − 1] Exti

modL R(X, C) = 0}

and C is functorially finite

• C = C(

ω)

• higher Auslander-Reiten theory works in C

Definition

(R, L) : d-CM-finite ⇐ ⇒ ∃ C ⊂ CML R : d-cluster tilting subcategory s.t. C contains only finitely many isoclasses of indecomposable objects up to shift by Z ω

• d = 1 =

⇒ CM-finite=1-CM-finite

Osamu Iyama (Nagoya) Tilting theory and CM representations 25 / 27

Theorem (HIMO)

(R, L) : GL complete intersection Then (b)⇒(a)(c) holds (a) (R, L) is d-CM-finite (b) ∃ finite dimensional k-algebra A s.t. gl.dim A ≤ d and CML R ≃ Db(mod A) (c) (R, L) is Fano Proof of (b)⇒(a) : Use the category U(A) ⊂ Db(mod A)

Conjecture

All conditions are equivalent

Osamu Iyama (Nagoya) Tilting theory and CM representations 26 / 27

Example (HIMO)

The condition (b) holds if

• n ≤ d + 1 or
• n = d + 2 and weights are (2, 2, p3, p4, . . . , pn),

(2, 3, 3, p4, . . . , pn), (2, 3, 4, p4, . . . , pn), (2, 3, 5, p4, . . . , pn)

Osamu Iyama (Nagoya) Tilting theory and CM representations 27 / 27