Xuefeng Mao Department of Mathematics, Shanghai University, - - PowerPoint PPT Presentation

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Cone length for DG modules and global dimension of DG algebras

Xuefeng Mao

Department of Mathematics, Shanghai University, xuefengmao@shu.edu.cn

Fudan University, September 16, 2011

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

1

Motivation and inspiration

2

Invariants on DG modules

3

Global dimension of DG algebras

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

1

Motivation and inspiration

2

Invariants on DG modules

3

Global dimension of DG algebras

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Motivations How to define finite global dimension of DG algebras? — P . Jørgensen, (2006) Amplitude inequalities for DG modules, Forum Math. (2010) A list of homological invariants on DG modules invariants definiens years rhdAM, fhdAM

• D. Apassov

1999 k.pdAM, k.idAM, P . Jørgensen, A. Frankild 2003 proj.dimAM, flat.dimAM

• A. Yekutieli,
• J. J. Zhang

2006

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Motivations How to define finite global dimension of DG algebras? — P . Jørgensen, (2006) Amplitude inequalities for DG modules, Forum Math. (2010) A list of homological invariants on DG modules invariants definiens years rhdAM, fhdAM

• D. Apassov

1999 k.pdAM, k.idAM, P . Jørgensen, A. Frankild 2003 proj.dimAM, flat.dimAM

• A. Yekutieli,
• J. J. Zhang

2006

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

‘free class’ for solvable free differential graded modules over a graded polynomial ring — G. Carlesson, (1983) On the homology of finite free (Z/2)k-complexes, Invent. Math. free class, projective class and flat class for differential modules

• ver a commutative ring

— Avramov, Buchweitz and Iyengar Class and rank of differential modules,

• Invent. Math.

(2007) ‘class’

• shortest length of a filtration with sub-quotients of certain type

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

‘free class’ for solvable free differential graded modules over a graded polynomial ring — G. Carlesson, (1983) On the homology of finite free (Z/2)k-complexes, Invent. Math. free class, projective class and flat class for differential modules

• ver a commutative ring

— Avramov, Buchweitz and Iyengar Class and rank of differential modules,

• Invent. Math.

(2007) ‘class’

• shortest length of a filtration with sub-quotients of certain type

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

‘free class’ for solvable free differential graded modules over a graded polynomial ring — G. Carlesson, (1983) On the homology of finite free (Z/2)k-complexes, Invent. Math. free class, projective class and flat class for differential modules

• ver a commutative ring

— Avramov, Buchweitz and Iyengar Class and rank of differential modules,

• Invent. Math.

(2007) ‘class’

• shortest length of a filtration with sub-quotients of certain type

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

‘free class’ for solvable free differential graded modules over a graded polynomial ring — G. Carlesson, (1983) On the homology of finite free (Z/2)k-complexes, Invent. Math. free class, projective class and flat class for differential modules

• ver a commutative ring

— Avramov, Buchweitz and Iyengar Class and rank of differential modules,

• Invent. Math.

(2007) ‘class’

• shortest length of a filtration with sub-quotients of certain type

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Fundamental definitions in DG homological algebra Let A be a DG algebra. A DG module over A is called DG free, if it is isomorphic to a direct sum of suspensions of A. A DG module F over A is called semi-free if there is a sequence of DG submodules 0 = F(−1) ⊆ F(0) ⊆ · · · ⊆ F(n) ⊂ · · · such that F = ∪n≥0 F(n) and that each F(n)/F(n − 1) is DG free on a cocycle basis.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Fundamental definitions in DG homological algebra Let A be a DG algebra. A DG module over A is called DG free, if it is isomorphic to a direct sum of suspensions of A. A DG module F over A is called semi-free if there is a sequence of DG submodules 0 = F(−1) ⊆ F(0) ⊆ · · · ⊆ F(n) ⊂ · · · such that F = ∪n≥0 F(n) and that each F(n)/F(n − 1) is DG free on a cocycle basis.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Fundamental definitions in DG homological algebra Let A be a DG algebra. A DG module over A is called DG free, if it is isomorphic to a direct sum of suspensions of A. A DG module F over A is called semi-free if there is a sequence of DG submodules 0 = F(−1) ⊆ F(0) ⊆ · · · ⊆ F(n) ⊂ · · · such that F = ∪n≥0 F(n) and that each F(n)/F(n − 1) is DG free on a cocycle basis.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

1

Motivation and inspiration

2

Invariants on DG modules

3

Global dimension of DG algebras

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Definition of DG free class for semi-free DG modules Let 0 = F(−1) ⊆ F(0) ⊆ · · · ⊆ F(n) ⊆ · · · be a semi-free filtration of a left semi-free DG module F over A.

1

It is called strictly increasing, if F(i − 1) = F(i) when F(i − 1) = F, i ≥ 0.

2

If ∃n ∈ N such that F(n) = F and F(n − 1) = F, then we say that this strictly increasing semi-free filtration has length n. If ∃ such n, then we say the length is +∞.

3

The DG free class of F is defined to be the minimal length

• f all its strictly increasing semi-free filtrations. We denote

it as DGfree classAF.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Definition of DG free class for semi-free DG modules Let 0 = F(−1) ⊆ F(0) ⊆ · · · ⊆ F(n) ⊆ · · · be a semi-free filtration of a left semi-free DG module F over A.

1

It is called strictly increasing, if F(i − 1) = F(i) when F(i − 1) = F, i ≥ 0.

2

If ∃n ∈ N such that F(n) = F and F(n − 1) = F, then we say that this strictly increasing semi-free filtration has length n. If ∃ such n, then we say the length is +∞.

3

The DG free class of F is defined to be the minimal length

• f all its strictly increasing semi-free filtrations. We denote

it as DGfree classAF.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Definition of DG free class for semi-free DG modules Let 0 = F(−1) ⊆ F(0) ⊆ · · · ⊆ F(n) ⊆ · · · be a semi-free filtration of a left semi-free DG module F over A.

1

It is called strictly increasing, if F(i − 1) = F(i) when F(i − 1) = F, i ≥ 0.

2

If ∃n ∈ N such that F(n) = F and F(n − 1) = F, then we say that this strictly increasing semi-free filtration has length n. If ∃ such n, then we say the length is +∞.

3

The DG free class of F is defined to be the minimal length

• f all its strictly increasing semi-free filtrations. We denote

it as DGfree classAF.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Definition of DG free class for semi-free DG modules Let 0 = F(−1) ⊆ F(0) ⊆ · · · ⊆ F(n) ⊆ · · · be a semi-free filtration of a left semi-free DG module F over A.

1

It is called strictly increasing, if F(i − 1) = F(i) when F(i − 1) = F, i ≥ 0.

2

If ∃n ∈ N such that F(n) = F and F(n − 1) = F, then we say that this strictly increasing semi-free filtration has length n. If ∃ such n, then we say the length is +∞.

3

The DG free class of F is defined to be the minimal length

• f all its strictly increasing semi-free filtrations. We denote

it as DGfree classAF.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

A well-known result in DG homological algebra For any DG module M over a DG algebra A, there is a semi-free DG module P and a quasi-isomorphism θ : P → M. Here, θ or P is called a semi-free resolution (or semi-free model) of M. Definition of cone length for a DG module Let M be a left DG module over a DG algebra A. The cone length of M is defined to be the number clAM = inf{ DGfree classAF | F is a semi-free resolution of M}.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

A well-known result in DG homological algebra For any DG module M over a DG algebra A, there is a semi-free DG module P and a quasi-isomorphism θ : P → M. Here, θ or P is called a semi-free resolution (or semi-free model) of M. Definition of cone length for a DG module Let M be a left DG module over a DG algebra A. The cone length of M is defined to be the number clAM = inf{ DGfree classAF | F is a semi-free resolution of M}.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

A well-known result in DG homological algebra For any DG module M over a DG algebra A, there is a semi-free DG module P and a quasi-isomorphism θ : P → M. Here, θ or P is called a semi-free resolution (or semi-free model) of M. Definition of cone length for a DG module Let M be a left DG module over a DG algebra A. The cone length of M is defined to be the number clAM = inf{ DGfree classAF | F is a semi-free resolution of M}.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

A well-known result in DG homological algebra For any DG module M over a DG algebra A, there is a semi-free DG module P and a quasi-isomorphism θ : P → M. Here, θ or P is called a semi-free resolution (or semi-free model) of M. Definition of cone length for a DG module Let M be a left DG module over a DG algebra A. The cone length of M is defined to be the number clAM = inf{ DGfree classAF | F is a semi-free resolution of M}.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras 1

Let C be a subcategory or a set of some objects of D(A).

2

smd(C) : the minimal strictly full subcategory of D(A) which contains C and is closed under taking direct summands.

3

add(C): the intersection of all strict and full subcategories of D(A) that contain C and are closed under finite direct sums and all suspensions. add(C): if finite arbitrary.

4

Let A and B be two strict and full subcategories of D(A). A ⋆ B: M ∈ A ⋆ B if and only if ∃L ∈ A, N ∈ B and an exact triangle L → M → N → Σ−1L. A ⋄ B := smd(A ⋆ B).

5

A⋆n =          n = 0; A n = 1;

n copies

• A ⋆ · · · ⋆ A

n ≥ 1. ,          A1 = smd(add(A)) An = An−1 ⋄ A1, n ≥ 2; A1 = smd(add(A)) An = An−1 ⋄ A1, n ≥ 2.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras 1

Let C be a subcategory or a set of some objects of D(A).

2

smd(C) : the minimal strictly full subcategory of D(A) which contains C and is closed under taking direct summands.

3

add(C): the intersection of all strict and full subcategories of D(A) that contain C and are closed under finite direct sums and all suspensions. add(C): if finite arbitrary.

4

Let A and B be two strict and full subcategories of D(A). A ⋆ B: M ∈ A ⋆ B if and only if ∃L ∈ A, N ∈ B and an exact triangle L → M → N → Σ−1L. A ⋄ B := smd(A ⋆ B).

5

A⋆n =          n = 0; A n = 1;

n copies

• A ⋆ · · · ⋆ A

n ≥ 1. ,          A1 = smd(add(A)) An = An−1 ⋄ A1, n ≥ 2; A1 = smd(add(A)) An = An−1 ⋄ A1, n ≥ 2.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras 1

Let C be a subcategory or a set of some objects of D(A).

2

smd(C) : the minimal strictly full subcategory of D(A) which contains C and is closed under taking direct summands.

3

add(C): the intersection of all strict and full subcategories of D(A) that contain C and are closed under finite direct sums and all suspensions. add(C): if finite arbitrary.

4

Let A and B be two strict and full subcategories of D(A). A ⋆ B: M ∈ A ⋆ B if and only if ∃L ∈ A, N ∈ B and an exact triangle L → M → N → Σ−1L. A ⋄ B := smd(A ⋆ B).

5

A⋆n =          n = 0; A n = 1;

n copies

• A ⋆ · · · ⋆ A

n ≥ 1. ,          A1 = smd(add(A)) An = An−1 ⋄ A1, n ≥ 2; A1 = smd(add(A)) An = An−1 ⋄ A1, n ≥ 2.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras 1

Let C be a subcategory or a set of some objects of D(A).

2

smd(C) : the minimal strictly full subcategory of D(A) which contains C and is closed under taking direct summands.

3

add(C): the intersection of all strict and full subcategories of D(A) that contain C and are closed under finite direct sums and all suspensions. add(C): if finite arbitrary.

4

Let A and B be two strict and full subcategories of D(A). A ⋆ B: M ∈ A ⋆ B if and only if ∃L ∈ A, N ∈ B and an exact triangle L → M → N → Σ−1L. A ⋄ B := smd(A ⋆ B).

5

A⋆n =          n = 0; A n = 1;

n copies

• A ⋆ · · · ⋆ A

n ≥ 1. ,          A1 = smd(add(A)) An = An−1 ⋄ A1, n ≥ 2; A1 = smd(add(A)) An = An−1 ⋄ A1, n ≥ 2.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras 1

Let C be a subcategory or a set of some objects of D(A).

2

smd(C) : the minimal strictly full subcategory of D(A) which contains C and is closed under taking direct summands.

3

add(C): the intersection of all strict and full subcategories of D(A) that contain C and are closed under finite direct sums and all suspensions. add(C): if finite arbitrary.

4

Let A and B be two strict and full subcategories of D(A). A ⋆ B: M ∈ A ⋆ B if and only if ∃L ∈ A, N ∈ B and an exact triangle L → M → N → Σ−1L. A ⋄ B := smd(A ⋆ B).

5

A⋆n =          n = 0; A n = 1;

n copies

• A ⋆ · · · ⋆ A

n ≥ 1. ,          A1 = smd(add(A)) An = An−1 ⋄ A1, n ≥ 2; A1 = smd(add(A)) An = An−1 ⋄ A1, n ≥ 2.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras 1

Let C be a subcategory or a set of some objects of D(A).

2

smd(C) : the minimal strictly full subcategory of D(A) which contains C and is closed under taking direct summands.

3

add(C): the intersection of all strict and full subcategories of D(A) that contain C and are closed under finite direct sums and all suspensions. add(C): if finite arbitrary.

4

Let A and B be two strict and full subcategories of D(A). A ⋆ B: M ∈ A ⋆ B if and only if ∃L ∈ A, N ∈ B and an exact triangle L → M → N → Σ−1L. A ⋄ B := smd(A ⋆ B).

5

A⋆n =          n = 0; A n = 1;

n copies

• A ⋆ · · · ⋆ A

n ≥ 1. ,          A1 = smd(add(A)) An = An−1 ⋄ A1, n ≥ 2; A1 = smd(add(A)) An = An−1 ⋄ A1, n ≥ 2.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras 1

Let C be a subcategory or a set of some objects of D(A).

2

smd(C) : the minimal strictly full subcategory of D(A) which contains C and is closed under taking direct summands.

3

add(C): the intersection of all strict and full subcategories of D(A) that contain C and are closed under finite direct sums and all suspensions. add(C): if finite arbitrary.

4

Let A and B be two strict and full subcategories of D(A). A ⋆ B: M ∈ A ⋆ B if and only if ∃L ∈ A, N ∈ B and an exact triangle L → M → N → Σ−1L. A ⋄ B := smd(A ⋆ B).

5

A⋆n =          n = 0; A n = 1;

n copies

• A ⋆ · · · ⋆ A

n ≥ 1. ,          A1 = smd(add(A)) An = An−1 ⋄ A1, n ≥ 2; A1 = smd(add(A)) An = An−1 ⋄ A1, n ≥ 2.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras 1

Let C be a subcategory or a set of some objects of D(A).

2

smd(C) : the minimal strictly full subcategory of D(A) which contains C and is closed under taking direct summands.

3

add(C): the intersection of all strict and full subcategories of D(A) that contain C and are closed under finite direct sums and all suspensions. add(C): if finite arbitrary.

4

Let A and B be two strict and full subcategories of D(A). A ⋆ B: M ∈ A ⋆ B if and only if ∃L ∈ A, N ∈ B and an exact triangle L → M → N → Σ−1L. A ⋄ B := smd(A ⋆ B).

5

A⋆n =          n = 0; A n = 1;

n copies

• A ⋆ · · · ⋆ A

n ≥ 1. ,          A1 = smd(add(A)) An = An−1 ⋄ A1, n ≥ 2; A1 = smd(add(A)) An = An−1 ⋄ A1, n ≥ 2.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras 1

Let C be a subcategory or a set of some objects of D(A).

2

smd(C) : the minimal strictly full subcategory of D(A) which contains C and is closed under taking direct summands.

3

add(C): the intersection of all strict and full subcategories of D(A) that contain C and are closed under finite direct sums and all suspensions. add(C): if finite arbitrary.

4

Let A and B be two strict and full subcategories of D(A). A ⋆ B: M ∈ A ⋆ B if and only if ∃L ∈ A, N ∈ B and an exact triangle L → M → N → Σ−1L. A ⋄ B := smd(A ⋆ B).

5

A⋆n =          n = 0; A n = 1;

n copies

• A ⋆ · · · ⋆ A

n ≥ 1. ,          A1 = smd(add(A)) An = An−1 ⋄ A1, n ≥ 2; A1 = smd(add(A)) An = An−1 ⋄ A1, n ≥ 2.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

A characterization of clAM For any left DG A-module M, clAM = n if and only if M is an

• bject in add(A) ⋆n+1 but not in add(A) ⋆n.

Definition of level of DG modules The A-level of a left DG A-module M is defined to be levelA

D(A)(M) = inf{n ∈ N ∪ {0}|M ∈ An}.

— Avramov, Buchweitz, Iyengar and Miller Homology of perfect complexes, Adv. Math. (2010)

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

A characterization of clAM For any left DG A-module M, clAM = n if and only if M is an

• bject in add(A) ⋆n+1 but not in add(A) ⋆n.

Definition of level of DG modules The A-level of a left DG A-module M is defined to be levelA

D(A)(M) = inf{n ∈ N ∪ {0}|M ∈ An}.

— Avramov, Buchweitz, Iyengar and Miller Homology of perfect complexes, Adv. Math. (2010)

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

A characterization of clAM For any left DG A-module M, clAM = n if and only if M is an

• bject in add(A) ⋆n+1 but not in add(A) ⋆n.

Definition of level of DG modules The A-level of a left DG A-module M is defined to be levelA

D(A)(M) = inf{n ∈ N ∪ {0}|M ∈ An}.

— Avramov, Buchweitz, Iyengar and Miller Homology of perfect complexes, Adv. Math. (2010)

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Ghost length of a DG module (K. Kuribayashi)

1

A morphism f : M → N in D(A) is called a ghost morphism if H(f) = 0.

2

A left DG module M over A is said to have ghost length n, written gh.len.M = n, if every composite M

f1

→ I1

f2

→ · · ·

fn+1

→ In+1

• f n + 1 ghosts is 0 in D(A), and there is a composite of n

ghosts from M that is not 0 in D(A).

3

Set gh.len.M = −1, if M is a zero object in D(A).

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Ghost length of a DG module (K. Kuribayashi)

1

A morphism f : M → N in D(A) is called a ghost morphism if H(f) = 0.

2

A left DG module M over A is said to have ghost length n, written gh.len.M = n, if every composite M

f1

→ I1

f2

→ · · ·

fn+1

→ In+1

• f n + 1 ghosts is 0 in D(A), and there is a composite of n

ghosts from M that is not 0 in D(A).

3

Set gh.len.M = −1, if M is a zero object in D(A).

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Ghost length of a DG module (K. Kuribayashi)

1

A morphism f : M → N in D(A) is called a ghost morphism if H(f) = 0.

2

A left DG module M over A is said to have ghost length n, written gh.len.M = n, if every composite M

f1

→ I1

f2

→ · · ·

fn+1

→ In+1

• f n + 1 ghosts is 0 in D(A), and there is a composite of n

ghosts from M that is not 0 in D(A).

3

Set gh.len.M = −1, if M is a zero object in D(A).

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Ghost length of a DG module (K. Kuribayashi)

1

A morphism f : M → N in D(A) is called a ghost morphism if H(f) = 0.

2

A left DG module M over A is said to have ghost length n, written gh.len.M = n, if every composite M

f1

→ I1

f2

→ · · ·

fn+1

→ In+1

• f n + 1 ghosts is 0 in D(A), and there is a composite of n

ghosts from M that is not 0 in D(A).

3

Set gh.len.M = −1, if M is a zero object in D(A).

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Ghost length of a DG module (K. Kuribayashi)

1

A morphism f : M → N in D(A) is called a ghost morphism if H(f) = 0.

2

A left DG module M over A is said to have ghost length n, written gh.len.M = n, if every composite M

f1

→ I1

f2

→ · · ·

fn+1

→ In+1

• f n + 1 ghosts is 0 in D(A), and there is a composite of n

ghosts from M that is not 0 in D(A).

3

Set gh.len.M = −1, if M is a zero object in D(A).

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Relations between ghost length, cone length and level

1

• K. Kuribayashi (2010)

gh.len.M + 1 ≤ levelA

D(A)(M)

2

Define the complete A-level of a left DG module M as LevelA

D(A)(M) = inf{n ∈ N ∪ {0}|M ∈ An}.

3

Mao (2011) gh.len.M + 1 = LevelA

D(A)(M),

LevelA

D(A)(M) = n if and only if M is a retract of some DG

A-module of cone length n − 1.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Relations between ghost length, cone length and level

1

• K. Kuribayashi (2010)

gh.len.M + 1 ≤ levelA

D(A)(M)

2

Define the complete A-level of a left DG module M as LevelA

D(A)(M) = inf{n ∈ N ∪ {0}|M ∈ An}.

3

Mao (2011) gh.len.M + 1 = LevelA

D(A)(M),

LevelA

D(A)(M) = n if and only if M is a retract of some DG

A-module of cone length n − 1.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Relations between ghost length, cone length and level

1

• K. Kuribayashi (2010)

gh.len.M + 1 ≤ levelA

D(A)(M)

2

Define the complete A-level of a left DG module M as LevelA

D(A)(M) = inf{n ∈ N ∪ {0}|M ∈ An}.

3

Mao (2011) gh.len.M + 1 = LevelA

D(A)(M),

LevelA

D(A)(M) = n if and only if M is a retract of some DG

A-module of cone length n − 1.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Relations between ghost length, cone length and level

1

• K. Kuribayashi (2010)

gh.len.M + 1 ≤ levelA

D(A)(M)

2

Define the complete A-level of a left DG module M as LevelA

D(A)(M) = inf{n ∈ N ∪ {0}|M ∈ An}.

3

Mao (2011) gh.len.M + 1 = LevelA

D(A)(M),

LevelA

D(A)(M) = n if and only if M is a retract of some DG

A-module of cone length n − 1.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Relations between ghost length, cone length and level

1

• K. Kuribayashi (2010)

gh.len.M + 1 ≤ levelA

D(A)(M)

2

Define the complete A-level of a left DG module M as LevelA

D(A)(M) = inf{n ∈ N ∪ {0}|M ∈ An}.

3

Mao (2011) gh.len.M + 1 = LevelA

D(A)(M),

LevelA

D(A)(M) = n if and only if M is a retract of some DG

A-module of cone length n − 1.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Relations between ghost length, cone length and level

1

• K. Kuribayashi (2010)

gh.len.M + 1 ≤ levelA

D(A)(M)

2

Define the complete A-level of a left DG module M as LevelA

D(A)(M) = inf{n ∈ N ∪ {0}|M ∈ An}.

3

Mao (2011) gh.len.M + 1 = LevelA

D(A)(M),

LevelA

D(A)(M) = n if and only if M is a retract of some DG

A-module of cone length n − 1.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Notation

1

A is a connected cochain DG algebra over a field k;

2

A

• p:

the opposite DG algebra of A;

3

left DG module over A ↔ DG A-module;

4

right DG module over A ↔ DG A

• p-module;

5

i-th suspension Σi: (ΣiM)j = Mi+j;

6

M#: the underlying graded module of a DG A-module M

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Notation

1

A is a connected cochain DG algebra over a field k;

2

A

• p:

the opposite DG algebra of A;

3

left DG module over A ↔ DG A-module;

4

right DG module over A ↔ DG A

• p-module;

5

i-th suspension Σi: (ΣiM)j = Mi+j;

6

M#: the underlying graded module of a DG A-module M

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Notation

1

A is a connected cochain DG algebra over a field k;

2

A

• p:

the opposite DG algebra of A;

3

left DG module over A ↔ DG A-module;

4

right DG module over A ↔ DG A

• p-module;

5

i-th suspension Σi: (ΣiM)j = Mi+j;

6

M#: the underlying graded module of a DG A-module M

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Notation

1

A is a connected cochain DG algebra over a field k;

2

A

• p:

the opposite DG algebra of A;

3

left DG module over A ↔ DG A-module;

4

right DG module over A ↔ DG A

• p-module;

5

i-th suspension Σi: (ΣiM)j = Mi+j;

6

M#: the underlying graded module of a DG A-module M

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Notation

1

A is a connected cochain DG algebra over a field k;

2

A

• p:

the opposite DG algebra of A;

3

left DG module over A ↔ DG A-module;

4

right DG module over A ↔ DG A

• p-module;

5

i-th suspension Σi: (ΣiM)j = Mi+j;

6

M#: the underlying graded module of a DG A-module M

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Notation

1

A is a connected cochain DG algebra over a field k;

2

A

• p:

the opposite DG algebra of A;

3

left DG module over A ↔ DG A-module;

4

right DG module over A ↔ DG A

• p-module;

5

i-th suspension Σi: (ΣiM)j = Mi+j;

6

M#: the underlying graded module of a DG A-module M

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Notation

1

A is a connected cochain DG algebra over a field k;

2

A

• p:

the opposite DG algebra of A;

3

left DG module over A ↔ DG A-module;

4

right DG module over A ↔ DG A

• p-module;

5

i-th suspension Σi: (ΣiM)j = Mi+j;

6

M#: the underlying graded module of a DG A-module M

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Definition of minimal semi-free resolution A semi-free resolution F of M is called minimal if dF(F) ⊆ mF, where m: · · · → 0 → A1 d1

A

→ A2 d2

A

→ · · · is the maximal DG ideal of A. Existence of minimal semi-free resolution Let M be a DG A-module such that H(M) is bounded below. Then M admits a minimal semi-free resolution. — Mao and Wu (2008)

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Definition of minimal semi-free resolution A semi-free resolution F of M is called minimal if dF(F) ⊆ mF, where m: · · · → 0 → A1 d1

A

→ A2 d2

A

→ · · · is the maximal DG ideal of A. Existence of minimal semi-free resolution Let M be a DG A-module such that H(M) is bounded below. Then M admits a minimal semi-free resolution. — Mao and Wu (2008)

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Compact DG modules and homologically smooth DGAs A DG A-module M is called compact, if it is in the smallest triangulated subcategory of D(A) containing AA — Dc(A). A DG A-module M is compact if and only if it admits a minimal semi-free resolution with a finite semi-basis. A is called homologically smooth, if A is compact as a DG module over the enveloping DG algebra Ae = A ⊗ Aop.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Compact DG modules and homologically smooth DGAs A DG A-module M is called compact, if it is in the smallest triangulated subcategory of D(A) containing AA — Dc(A). A DG A-module M is compact if and only if it admits a minimal semi-free resolution with a finite semi-basis. A is called homologically smooth, if A is compact as a DG module over the enveloping DG algebra Ae = A ⊗ Aop.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Compact DG modules and homologically smooth DGAs A DG A-module M is called compact, if it is in the smallest triangulated subcategory of D(A) containing AA — Dc(A). A DG A-module M is compact if and only if it admits a minimal semi-free resolution with a finite semi-basis. A is called homologically smooth, if A is compact as a DG module over the enveloping DG algebra Ae = A ⊗ Aop.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Compact DG modules and homologically smooth DGAs A DG A-module M is called compact, if it is in the smallest triangulated subcategory of D(A) containing AA — Dc(A). A DG A-module M is compact if and only if it admits a minimal semi-free resolution with a finite semi-basis. A is called homologically smooth, if A is compact as a DG module over the enveloping DG algebra Ae = A ⊗ Aop.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Compact DG modules and homologically smooth DGAs A DG A-module M is called compact, if it is in the smallest triangulated subcategory of D(A) containing AA — Dc(A). A DG A-module M is compact if and only if it admits a minimal semi-free resolution with a finite semi-basis. A is called homologically smooth, if A is compact as a DG module over the enveloping DG algebra Ae = A ⊗ Aop.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Let θ : P → M be a semi-free resolution of a DG module

• M. Let 0 = P(−1) ⊂ P(0) ⊆ · · · ⊆ P(i) ⊆ · · · be a

semi-free filtration of P such that P(i)/P(i − 1) = A ⊗ Vi. Set F −iP = P(i), then P admits a filtration: 0 = F 1P ⊆ F 0P ⊆ F −1P ⊆ · · · ⊆ F −iP ⊆ · · · . This filtration induces a spectral sequence (Er(P), dr)r≥0. Let F 1M = 0 and F iM = M, i ≤ 0. We get a filtration of M: 0 = F 1M ⊂ F 0M = F −1M = · · · . This filtration induces a spectral sequence (Er(M), dr)r≥0. θ : P → M preserves the filtration. By E1(θ), we have a complex of graded H(A)-modules, · · · → H(A) ⊗ Σ−iV−i → · · · → H(A) ⊗ V0 → H(M) → 0. If this complex is exact, then it is a free resolution of H(M). And θ : P ≃ → M is an Eilenberg-Moore resolution of M.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Let θ : P → M be a semi-free resolution of a DG module

• M. Let 0 = P(−1) ⊂ P(0) ⊆ · · · ⊆ P(i) ⊆ · · · be a

semi-free filtration of P such that P(i)/P(i − 1) = A ⊗ Vi. Set F −iP = P(i), then P admits a filtration: 0 = F 1P ⊆ F 0P ⊆ F −1P ⊆ · · · ⊆ F −iP ⊆ · · · . This filtration induces a spectral sequence (Er(P), dr)r≥0. Let F 1M = 0 and F iM = M, i ≤ 0. We get a filtration of M: 0 = F 1M ⊂ F 0M = F −1M = · · · . This filtration induces a spectral sequence (Er(M), dr)r≥0. θ : P → M preserves the filtration. By E1(θ), we have a complex of graded H(A)-modules, · · · → H(A) ⊗ Σ−iV−i → · · · → H(A) ⊗ V0 → H(M) → 0. If this complex is exact, then it is a free resolution of H(M). And θ : P ≃ → M is an Eilenberg-Moore resolution of M.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Let θ : P → M be a semi-free resolution of a DG module

• M. Let 0 = P(−1) ⊂ P(0) ⊆ · · · ⊆ P(i) ⊆ · · · be a

semi-free filtration of P such that P(i)/P(i − 1) = A ⊗ Vi. Set F −iP = P(i), then P admits a filtration: 0 = F 1P ⊆ F 0P ⊆ F −1P ⊆ · · · ⊆ F −iP ⊆ · · · . This filtration induces a spectral sequence (Er(P), dr)r≥0. Let F 1M = 0 and F iM = M, i ≤ 0. We get a filtration of M: 0 = F 1M ⊂ F 0M = F −1M = · · · . This filtration induces a spectral sequence (Er(M), dr)r≥0. θ : P → M preserves the filtration. By E1(θ), we have a complex of graded H(A)-modules, · · · → H(A) ⊗ Σ−iV−i → · · · → H(A) ⊗ V0 → H(M) → 0. If this complex is exact, then it is a free resolution of H(M). And θ : P ≃ → M is an Eilenberg-Moore resolution of M.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Let θ : P → M be a semi-free resolution of a DG module

• M. Let 0 = P(−1) ⊂ P(0) ⊆ · · · ⊆ P(i) ⊆ · · · be a

semi-free filtration of P such that P(i)/P(i − 1) = A ⊗ Vi. Set F −iP = P(i), then P admits a filtration: 0 = F 1P ⊆ F 0P ⊆ F −1P ⊆ · · · ⊆ F −iP ⊆ · · · . This filtration induces a spectral sequence (Er(P), dr)r≥0. Let F 1M = 0 and F iM = M, i ≤ 0. We get a filtration of M: 0 = F 1M ⊂ F 0M = F −1M = · · · . This filtration induces a spectral sequence (Er(M), dr)r≥0. θ : P → M preserves the filtration. By E1(θ), we have a complex of graded H(A)-modules, · · · → H(A) ⊗ Σ−iV−i → · · · → H(A) ⊗ V0 → H(M) → 0. If this complex is exact, then it is a free resolution of H(M). And θ : P ≃ → M is an Eilenberg-Moore resolution of M.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Let θ : P → M be a semi-free resolution of a DG module

• M. Let 0 = P(−1) ⊂ P(0) ⊆ · · · ⊆ P(i) ⊆ · · · be a

semi-free filtration of P such that P(i)/P(i − 1) = A ⊗ Vi. Set F −iP = P(i), then P admits a filtration: 0 = F 1P ⊆ F 0P ⊆ F −1P ⊆ · · · ⊆ F −iP ⊆ · · · . This filtration induces a spectral sequence (Er(P), dr)r≥0. Let F 1M = 0 and F iM = M, i ≤ 0. We get a filtration of M: 0 = F 1M ⊂ F 0M = F −1M = · · · . This filtration induces a spectral sequence (Er(M), dr)r≥0. θ : P → M preserves the filtration. By E1(θ), we have a complex of graded H(A)-modules, · · · → H(A) ⊗ Σ−iV−i → · · · → H(A) ⊗ V0 → H(M) → 0. If this complex is exact, then it is a free resolution of H(M). And θ : P ≃ → M is an Eilenberg-Moore resolution of M.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Let θ : P → M be a semi-free resolution of a DG module

• M. Let 0 = P(−1) ⊂ P(0) ⊆ · · · ⊆ P(i) ⊆ · · · be a

semi-free filtration of P such that P(i)/P(i − 1) = A ⊗ Vi. Set F −iP = P(i), then P admits a filtration: 0 = F 1P ⊆ F 0P ⊆ F −1P ⊆ · · · ⊆ F −iP ⊆ · · · . This filtration induces a spectral sequence (Er(P), dr)r≥0. Let F 1M = 0 and F iM = M, i ≤ 0. We get a filtration of M: 0 = F 1M ⊂ F 0M = F −1M = · · · . This filtration induces a spectral sequence (Er(M), dr)r≥0. θ : P → M preserves the filtration. By E1(θ), we have a complex of graded H(A)-modules, · · · → H(A) ⊗ Σ−iV−i → · · · → H(A) ⊗ V0 → H(M) → 0. If this complex is exact, then it is a free resolution of H(M). And θ : P ≃ → M is an Eilenberg-Moore resolution of M.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Let θ : P → M be a semi-free resolution of a DG module

• M. Let 0 = P(−1) ⊂ P(0) ⊆ · · · ⊆ P(i) ⊆ · · · be a

semi-free filtration of P such that P(i)/P(i − 1) = A ⊗ Vi. Set F −iP = P(i), then P admits a filtration: 0 = F 1P ⊆ F 0P ⊆ F −1P ⊆ · · · ⊆ F −iP ⊆ · · · . This filtration induces a spectral sequence (Er(P), dr)r≥0. Let F 1M = 0 and F iM = M, i ≤ 0. We get a filtration of M: 0 = F 1M ⊂ F 0M = F −1M = · · · . This filtration induces a spectral sequence (Er(M), dr)r≥0. θ : P → M preserves the filtration. By E1(θ), we have a complex of graded H(A)-modules, · · · → H(A) ⊗ Σ−iV−i → · · · → H(A) ⊗ V0 → H(M) → 0. If this complex is exact, then it is a free resolution of H(M). And θ : P ≃ → M is an Eilenberg-Moore resolution of M.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Existence of Eilenberg-Moore resolution Every DG A-module M admits an Eilenberg-Moore resolution. More precisely, every free resolution of H(M) can be realized as an E1-term of some Eilenberg-Moore resolution of M. Relations between clAM and pdH(A)H(M)

1

In general, clAM ≤ pdH(A)H(M). If M admits a minimal Eilenberg-Moore resolution, then clAM = pdH(A)H(M).

2

If clAM is finite, then we have gradeH(A)H(M) ≤ clAM. Furthermore, if gradeH(A)H(M) = clAM, then pdH(A)H(M) = gradeH(A)H(M) = clAM.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Existence of Eilenberg-Moore resolution Every DG A-module M admits an Eilenberg-Moore resolution. More precisely, every free resolution of H(M) can be realized as an E1-term of some Eilenberg-Moore resolution of M. Relations between clAM and pdH(A)H(M)

1

In general, clAM ≤ pdH(A)H(M). If M admits a minimal Eilenberg-Moore resolution, then clAM = pdH(A)H(M).

2

If clAM is finite, then we have gradeH(A)H(M) ≤ clAM. Furthermore, if gradeH(A)H(M) = clAM, then pdH(A)H(M) = gradeH(A)H(M) = clAM.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Existence of Eilenberg-Moore resolution Every DG A-module M admits an Eilenberg-Moore resolution. More precisely, every free resolution of H(M) can be realized as an E1-term of some Eilenberg-Moore resolution of M. Relations between clAM and pdH(A)H(M)

1

In general, clAM ≤ pdH(A)H(M). If M admits a minimal Eilenberg-Moore resolution, then clAM = pdH(A)H(M).

2

If clAM is finite, then we have gradeH(A)H(M) ≤ clAM. Furthermore, if gradeH(A)H(M) = clAM, then pdH(A)H(M) = gradeH(A)H(M) = clAM.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Existence of Eilenberg-Moore resolution Every DG A-module M admits an Eilenberg-Moore resolution. More precisely, every free resolution of H(M) can be realized as an E1-term of some Eilenberg-Moore resolution of M. Relations between clAM and pdH(A)H(M)

1

In general, clAM ≤ pdH(A)H(M). If M admits a minimal Eilenberg-Moore resolution, then clAM = pdH(A)H(M).

2

If clAM is finite, then we have gradeH(A)H(M) ≤ clAM. Furthermore, if gradeH(A)H(M) = clAM, then pdH(A)H(M) = gradeH(A)H(M) = clAM.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Existence of Eilenberg-Moore resolution Every DG A-module M admits an Eilenberg-Moore resolution. More precisely, every free resolution of H(M) can be realized as an E1-term of some Eilenberg-Moore resolution of M. Relations between clAM and pdH(A)H(M)

1

In general, clAM ≤ pdH(A)H(M). If M admits a minimal Eilenberg-Moore resolution, then clAM = pdH(A)H(M).

2

If clAM is finite, then we have gradeH(A)H(M) ≤ clAM. Furthermore, if gradeH(A)H(M) = clAM, then pdH(A)H(M) = gradeH(A)H(M) = clAM.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Existence of Eilenberg-Moore resolution Every DG A-module M admits an Eilenberg-Moore resolution. More precisely, every free resolution of H(M) can be realized as an E1-term of some Eilenberg-Moore resolution of M. Relations between clAM and pdH(A)H(M)

1

In general, clAM ≤ pdH(A)H(M). If M admits a minimal Eilenberg-Moore resolution, then clAM = pdH(A)H(M).

2

If clAM is finite, then we have gradeH(A)H(M) ≤ clAM. Furthermore, if gradeH(A)H(M) = clAM, then pdH(A)H(M) = gradeH(A)H(M) = clAM.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Existence of Eilenberg-Moore resolution Every DG A-module M admits an Eilenberg-Moore resolution. More precisely, every free resolution of H(M) can be realized as an E1-term of some Eilenberg-Moore resolution of M. Relations between clAM and pdH(A)H(M)

1

In general, clAM ≤ pdH(A)H(M). If M admits a minimal Eilenberg-Moore resolution, then clAM = pdH(A)H(M).

2

If clAM is finite, then we have gradeH(A)H(M) ≤ clAM. Furthermore, if gradeH(A)H(M) = clAM, then pdH(A)H(M) = gradeH(A)H(M) = clAM.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Existence of minimal Eilenberg-Moore resolution A DG A-module M admits a minimal Eilenberg-Moore resolution if any one of the following conditions holds. dA = 0 pdH(A)H(M) ≤ 1 M = Ak and gl.dimH(A) ≤ 2 H(M) admits a free resolution · · ·

∂i+1

→ H(A) ⊗ Vi

∂i

→ · · ·

∂1

→ H(A) ⊗ V0

ε

→ H(M) → 0 with inf{|v| | v ∈ Vi} > sup{|v| | v ∈ Vi−1}, i ≥ 1

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Existence of minimal Eilenberg-Moore resolution A DG A-module M admits a minimal Eilenberg-Moore resolution if any one of the following conditions holds. dA = 0 pdH(A)H(M) ≤ 1 M = Ak and gl.dimH(A) ≤ 2 H(M) admits a free resolution · · ·

∂i+1

→ H(A) ⊗ Vi

∂i

→ · · ·

∂1

→ H(A) ⊗ V0

ε

→ H(M) → 0 with inf{|v| | v ∈ Vi} > sup{|v| | v ∈ Vi−1}, i ≥ 1

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Existence of minimal Eilenberg-Moore resolution A DG A-module M admits a minimal Eilenberg-Moore resolution if any one of the following conditions holds. dA = 0 pdH(A)H(M) ≤ 1 M = Ak and gl.dimH(A) ≤ 2 H(M) admits a free resolution · · ·

∂i+1

→ H(A) ⊗ Vi

∂i

→ · · ·

∂1

→ H(A) ⊗ V0

ε

→ H(M) → 0 with inf{|v| | v ∈ Vi} > sup{|v| | v ∈ Vi−1}, i ≥ 1

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Existence of minimal Eilenberg-Moore resolution A DG A-module M admits a minimal Eilenberg-Moore resolution if any one of the following conditions holds. dA = 0 pdH(A)H(M) ≤ 1 M = Ak and gl.dimH(A) ≤ 2 H(M) admits a free resolution · · ·

∂i+1

→ H(A) ⊗ Vi

∂i

→ · · ·

∂1

→ H(A) ⊗ V0

ε

→ H(M) → 0 with inf{|v| | v ∈ Vi} > sup{|v| | v ∈ Vi−1}, i ≥ 1

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Existence of minimal Eilenberg-Moore resolution A DG A-module M admits a minimal Eilenberg-Moore resolution if any one of the following conditions holds. dA = 0 pdH(A)H(M) ≤ 1 M = Ak and gl.dimH(A) ≤ 2 H(M) admits a free resolution · · ·

∂i+1

→ H(A) ⊗ Vi

∂i

→ · · ·

∂1

→ H(A) ⊗ V0

ε

→ H(M) → 0 with inf{|v| | v ∈ Vi} > sup{|v| | v ∈ Vi−1}, i ≥ 1

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Counterexample Let (A, dA) be a connected DG algebra with A# = k < x, y > /(xy + yx), |x| = |y| = 1 and dA(x) = y2, dA(y) = 0.

Ak admit a minimal semi-free resolution θ : F → Ak, with

F # = A# ⊕ A#Σey ⊕ A#Σez ⊕ A#Σet, where z = x + yΣey, t = xΣey + yΣez, and dF is defined by dF(Σey) = y, dF(Σez) = z, dF(Σet) = t. θ is defined by θ(a + ayΣey + azΣez + atΣet) = ε(a), where ε : A → k is the augmented map. clAk ≤ 3 but H(A) = k[x2, y]/(y2). Hence Ak has no minimal Eilenberg-Moore resolution.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Counterexample Let (A, dA) be a connected DG algebra with A# = k < x, y > /(xy + yx), |x| = |y| = 1 and dA(x) = y2, dA(y) = 0.

Ak admit a minimal semi-free resolution θ : F → Ak, with

F # = A# ⊕ A#Σey ⊕ A#Σez ⊕ A#Σet, where z = x + yΣey, t = xΣey + yΣez, and dF is defined by dF(Σey) = y, dF(Σez) = z, dF(Σet) = t. θ is defined by θ(a + ayΣey + azΣez + atΣet) = ε(a), where ε : A → k is the augmented map. clAk ≤ 3 but H(A) = k[x2, y]/(y2). Hence Ak has no minimal Eilenberg-Moore resolution.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Counterexample Let (A, dA) be a connected DG algebra with A# = k < x, y > /(xy + yx), |x| = |y| = 1 and dA(x) = y2, dA(y) = 0.

Ak admit a minimal semi-free resolution θ : F → Ak, with

F # = A# ⊕ A#Σey ⊕ A#Σez ⊕ A#Σet, where z = x + yΣey, t = xΣey + yΣez, and dF is defined by dF(Σey) = y, dF(Σez) = z, dF(Σet) = t. θ is defined by θ(a + ayΣey + azΣez + atΣet) = ε(a), where ε : A → k is the augmented map. clAk ≤ 3 but H(A) = k[x2, y]/(y2). Hence Ak has no minimal Eilenberg-Moore resolution.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Counterexample Let (A, dA) be a connected DG algebra with A# = k < x, y > /(xy + yx), |x| = |y| = 1 and dA(x) = y2, dA(y) = 0.

Ak admit a minimal semi-free resolution θ : F → Ak, with

F # = A# ⊕ A#Σey ⊕ A#Σez ⊕ A#Σet, where z = x + yΣey, t = xΣey + yΣez, and dF is defined by dF(Σey) = y, dF(Σez) = z, dF(Σet) = t. θ is defined by θ(a + ayΣey + azΣez + atΣet) = ε(a), where ε : A → k is the augmented map. clAk ≤ 3 but H(A) = k[x2, y]/(y2). Hence Ak has no minimal Eilenberg-Moore resolution.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

1

Motivation and inspiration

2

Invariants on DG modules

3

Global dimension of DG algebras

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Definition of global dimension of DG algebra The left global dimension of a DG algebra A is defined by l.Gl.dimA = sup{clAM|M ∈ D(A)}. Similarly, the right global dimension of A is defined by r.Gl.dimA = sup{clA

• pM|M ∈ D(A
• p)}.

Simple results on l.Gl.dimA Let A be a connected DG algebra with zero differential. Then l.Gl.dimA = gl.dimA# = r.Gl.dimA. l.Gl.dimA = 0 if and only if A ≃ → k. l.Gl.dimA = 1 if and

• nly if gl.dimH(A) = 1.

l.Gl.dimA ≤ gl.dimH(A). If gl.dimH(A) = n ≤ 2, then l.Gl.dimA = n.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Definition of global dimension of DG algebra The left global dimension of a DG algebra A is defined by l.Gl.dimA = sup{clAM|M ∈ D(A)}. Similarly, the right global dimension of A is defined by r.Gl.dimA = sup{clA

• pM|M ∈ D(A
• p)}.

Simple results on l.Gl.dimA Let A be a connected DG algebra with zero differential. Then l.Gl.dimA = gl.dimA# = r.Gl.dimA. l.Gl.dimA = 0 if and only if A ≃ → k. l.Gl.dimA = 1 if and

• nly if gl.dimH(A) = 1.

l.Gl.dimA ≤ gl.dimH(A). If gl.dimH(A) = n ≤ 2, then l.Gl.dimA = n.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Definition of global dimension of DG algebra The left global dimension of a DG algebra A is defined by l.Gl.dimA = sup{clAM|M ∈ D(A)}. Similarly, the right global dimension of A is defined by r.Gl.dimA = sup{clA

• pM|M ∈ D(A
• p)}.

Simple results on l.Gl.dimA Let A be a connected DG algebra with zero differential. Then l.Gl.dimA = gl.dimA# = r.Gl.dimA. l.Gl.dimA = 0 if and only if A ≃ → k. l.Gl.dimA = 1 if and

• nly if gl.dimH(A) = 1.

l.Gl.dimA ≤ gl.dimH(A). If gl.dimH(A) = n ≤ 2, then l.Gl.dimA = n.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Definition of global dimension of DG algebra The left global dimension of a DG algebra A is defined by l.Gl.dimA = sup{clAM|M ∈ D(A)}. Similarly, the right global dimension of A is defined by r.Gl.dimA = sup{clA

• pM|M ∈ D(A
• p)}.

Simple results on l.Gl.dimA Let A be a connected DG algebra with zero differential. Then l.Gl.dimA = gl.dimA# = r.Gl.dimA. l.Gl.dimA = 0 if and only if A ≃ → k. l.Gl.dimA = 1 if and

• nly if gl.dimH(A) = 1.

l.Gl.dimA ≤ gl.dimH(A). If gl.dimH(A) = n ≤ 2, then l.Gl.dimA = n.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Definition of global dimension of DG algebra The left global dimension of a DG algebra A is defined by l.Gl.dimA = sup{clAM|M ∈ D(A)}. Similarly, the right global dimension of A is defined by r.Gl.dimA = sup{clA

• pM|M ∈ D(A
• p)}.

Simple results on l.Gl.dimA Let A be a connected DG algebra with zero differential. Then l.Gl.dimA = gl.dimA# = r.Gl.dimA. l.Gl.dimA = 0 if and only if A ≃ → k. l.Gl.dimA = 1 if and

• nly if gl.dimH(A) = 1.

l.Gl.dimA ≤ gl.dimH(A). If gl.dimH(A) = n ≤ 2, then l.Gl.dimA = n.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Theorem Mao and Wu (2011) Suppose that depthH(A)H(A) = gl.dimH(A) < +∞. Then l.Gl.dimA = gl.dimH(A) = clAk. Theorem Mao and Wu (2011) Assume that A is homologically smooth and X is the minimal semi-free resolution of the DG Ae-module A. Then dim Dc(A) ≤ l.Gl.dimA ≤ DGfree.classAeX < ∞, where dimDc(A) is defined by R. Rouquier as the invariant min{n ∈ N ∪ {+∞}|Xn+1 = Dc(A), for some X ∈ Dc(A)}.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

Theorem Mao and Wu (2011) Suppose that depthH(A)H(A) = gl.dimH(A) < +∞. Then l.Gl.dimA = gl.dimH(A) = clAk. Theorem Mao and Wu (2011) Assume that A is homologically smooth and X is the minimal semi-free resolution of the DG Ae-module A. Then dim Dc(A) ≤ l.Gl.dimA ≤ DGfree.classAeX < ∞, where dimDc(A) is defined by R. Rouquier as the invariant min{n ∈ N ∪ {+∞}|Xn+1 = Dc(A), for some X ∈ Dc(A)}.

logo Motivation and inspiration Invariants on DG modules Global dimension of DG algebras

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