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 Motivation and inspiration 1 Invariants on DG modules 2 Global dimension of DG algebras 3 logo
Motivation and inspiration Invariants on DG modules Global dimension of DG algebras Motivation and inspiration 1 Invariants on DG modules 2 Global dimension of DG algebras 3 logo
Motivation and inspiration Invariants on DG modules Global dimension of DG algebras Motivations How to define finite global dimension of DG algebras? — P . J ø rgensen, (2006) Amplitude inequalities for DG modules, Forum Math. (2010) A list of homological invariants on DG modules invariants definiens years r hd A M , f hd A M D. Apassov 1999 k . pd A M , k . id A M , P . J ø rgensen, A. Frankild 2003 proj . dim A M , flat . dim A M 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 . J ø rgensen, (2006) Amplitude inequalities for DG modules, Forum Math. (2010) A list of homological invariants on DG modules invariants definiens years r hd A M , f hd A M D. Apassov 1999 k . pd A M , k . id A M , P . J ø rgensen, A. Frankild 2003 proj . dim A M , flat . dim A M 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 over 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 over 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 over 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 over 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 Motivation and inspiration 1 Invariants on DG modules 2 Global dimension of DG algebras 3 logo
Motivation and inspiration Invariants on DG modules Global dimension of DG algebras Definition of DG free class for semi-free DG modules 0 = F ( − 1 ) ⊆ F ( 0 ) ⊆ · · · ⊆ F ( n ) ⊆ · · · Let be a semi-free filtration of a left semi-free DG module F over A . It is called strictly increasing, if F ( i − 1 ) � = F ( i ) when 1 F ( i − 1 ) � = F , i ≥ 0. If ∃ n ∈ N such that F ( n ) = F and F ( n − 1 ) � = F , then we 2 say that this strictly increasing semi-free filtration has length n . If � ∃ such n , then we say the length is + ∞ . The DG free class of F is defined to be the minimal length 3 of all its strictly increasing semi-free filtrations. We denote it as DGfree class A F . logo
Motivation and inspiration Invariants on DG modules Global dimension of DG algebras Definition of DG free class for semi-free DG modules 0 = F ( − 1 ) ⊆ F ( 0 ) ⊆ · · · ⊆ F ( n ) ⊆ · · · Let be a semi-free filtration of a left semi-free DG module F over A . It is called strictly increasing, if F ( i − 1 ) � = F ( i ) when 1 F ( i − 1 ) � = F , i ≥ 0. If ∃ n ∈ N such that F ( n ) = F and F ( n − 1 ) � = F , then we 2 say that this strictly increasing semi-free filtration has length n . If � ∃ such n , then we say the length is + ∞ . The DG free class of F is defined to be the minimal length 3 of all its strictly increasing semi-free filtrations. We denote it as DGfree class A F . logo
Motivation and inspiration Invariants on DG modules Global dimension of DG algebras Definition of DG free class for semi-free DG modules 0 = F ( − 1 ) ⊆ F ( 0 ) ⊆ · · · ⊆ F ( n ) ⊆ · · · Let be a semi-free filtration of a left semi-free DG module F over A . It is called strictly increasing, if F ( i − 1 ) � = F ( i ) when 1 F ( i − 1 ) � = F , i ≥ 0. If ∃ n ∈ N such that F ( n ) = F and F ( n − 1 ) � = F , then we 2 say that this strictly increasing semi-free filtration has length n . If � ∃ such n , then we say the length is + ∞ . The DG free class of F is defined to be the minimal length 3 of all its strictly increasing semi-free filtrations. We denote it as DGfree class A F . logo
Motivation and inspiration Invariants on DG modules Global dimension of DG algebras Definition of DG free class for semi-free DG modules 0 = F ( − 1 ) ⊆ F ( 0 ) ⊆ · · · ⊆ F ( n ) ⊆ · · · Let be a semi-free filtration of a left semi-free DG module F over A . It is called strictly increasing, if F ( i − 1 ) � = F ( i ) when 1 F ( i − 1 ) � = F , i ≥ 0. If ∃ n ∈ N such that F ( n ) = F and F ( n − 1 ) � = F , then we 2 say that this strictly increasing semi-free filtration has length n . If � ∃ such n , then we say the length is + ∞ . The DG free class of F is defined to be the minimal length 3 of all its strictly increasing semi-free filtrations. We denote it as DGfree class A F . 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 cl A M = inf { DGfree class A F | 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 cl A M = inf { DGfree class A F | F is a semi-free resolution of M } . logo
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