Generalized Koszul duality applied to complete intersection rings - PowerPoint PPT Presentation

Examples 1) A arbitrary A ∞ -algebra, can form curved dg-coalgebra Bar A Bar A → A is an acyclic twisting cochain

Examples 1) A arbitrary A ∞ -algebra, can form curved dg-coalgebra Bar A Bar A → A is an acyclic twisting cochain 2) ( Q , n , k ) local ring, R = Q / I quotient A ≃ − → R minimal Q -free resolution

Examples 1) A arbitrary A ∞ -algebra, can form curved dg-coalgebra Bar A Bar A → A is an acyclic twisting cochain 2) ( Q , n , k ) local ring, R = Q / I quotient A ≃ − → R minimal Q -free resolution A has an A ∞ -algebra structure (-) (but not necessarily dg-algebra (Avramov))

Examples 1) A arbitrary A ∞ -algebra, can form curved dg-coalgebra Bar A Bar A → A is an acyclic twisting cochain 2) ( Q , n , k ) local ring, R = Q / I quotient A ≃ − → R minimal Q -free resolution A has an A ∞ -algebra structure (-) (but not necessarily dg-algebra (Avramov)) if Q → R Golod, Bar A is minimal, hence only acyclic twisting cochain; proves several new results about Golod maps and consolidates existing theory

Examples 1) A arbitrary A ∞ -algebra, can form curved dg-coalgebra Bar A Bar A → A is an acyclic twisting cochain 2) ( Q , n , k ) local ring, R = Q / I quotient A ≃ − → R minimal Q -free resolution A has an A ∞ -algebra structure (-) (but not necessarily dg-algebra (Avramov)) if Q → R Golod, Bar A is minimal, hence only acyclic twisting cochain; proves several new results about Golod maps and consolidates existing theory For non-Golod rings, deform Koszul dual of algebra underlying minimal model?

3) g restricted Lie algebra with k -basis ( x 1 , . . . , x n ); set y i = x [ p ] − x p i ∈ U ( g ) i O ( g ) := Sym k ( g (1) ) ∼ = k [ y 1 , . . . , y n ] ⊆ U ( g ) U is a finitely generated free O module.

3) g restricted Lie algebra with k -basis ( x 1 , . . . , x n ); set y i = x [ p ] − x p i ∈ U ( g ) i O ( g ) := Sym k ( g (1) ) ∼ = k [ y 1 , . . . , y n ] ⊆ U ( g ) U is a finitely generated free O module. O ( g ) u ( g ) = U ( g ) ⊗ O ( y 1 , . . . , y n ) restricted enveloping algebra; set A = Kos( y 1 , . . . , y n ) so U ⊗ O A ≃ − → u ( g ) is quasi-isomorphism.

We know A has Koszul dual C W ; if U has Koszul dual D (over O !), then C W ⊗ D is Koszul dual of u ( g ) ≃ U ⊗ A .

We know A has Koszul dual C W ; if U has Koszul dual D (over O !), then C W ⊗ D is Koszul dual of u ( g ) ≃ U ⊗ A . For example, g = g a , so U = k [ x ] and O = k [ x p ]. What is Koszul dual of k [ x ] over k [ x p ]?

We know A has Koszul dual C W ; if U has Koszul dual D (over O !), then C W ⊗ D is Koszul dual of u ( g ) ≃ U ⊗ A . For example, g = g a , so U = k [ x ] and O = k [ x p ]. What is Koszul dual of k [ x ] over k [ x p ]? More generally, are trying to study the family of algebras O ( g ) A χ = U ⊗ O ( y 1 − χ ( y 1 ) , . . . , y n − χ ( y n )) for character χ : g (1) → k .

(Back to) complete intersection rings f = f 1 , . . . , f c ⊆ Q A = Kos( f 1 , . . . , f c )

(Back to) complete intersection rings f = f 1 , . . . , f c ⊆ Q A = Kos( f 1 , . . . , f c ) Assume H i ( A ) = 0 for i > 0, so A ≃ − → Q / ( f ) =: R is a quasi-isomorphism.

(Back to) complete intersection rings f = f 1 , . . . , f c ⊆ Q A = Kos( f 1 , . . . , f c ) Assume H i ( A ) = 0 for i > 0, so A ≃ − → Q / ( f ) =: R is a quasi-isomorphism. E.g. Q = k [ z 1 , . . . , z c ] and f i = z p i .

� � � � (Back to) complete intersection rings f = f 1 , . . . , f c ⊆ Q A = Kos( f 1 , . . . , f c ) Assume H i ( A ) = 0 for i > 0, so A ≃ − → Q / ( f ) =: R is a quasi-isomorphism. E.g. Q = k [ z 1 , . . . , z c ] and f i = z p i . R D f D f cdg ( S W ) dg ( A ) ∼ = L ∼ ∼ = = D f ( R ) If M is an R -module, what are representatives in these categories?

Fix Q and R free resolutions: ≃ ≃ G − → M R ⊗ Q F − → M with G ♯ , F ♯ graded free Q -modules.

Fix Q and R free resolutions: ≃ ≃ G − → M R ⊗ Q F − → M with G ♯ , F ♯ graded free Q -modules. Proposition (Eisenbud, 1980) There exists a system of higher homotopies { σ a | a ∈ N c } on G, with σ a : G → G a degree 2 | a | − 1 endomorphism. These determine a differential d on S ⊗ G such that ( S ⊗ G , d ) ∈ D f cdg ( S W ) .

Example of higher homotopies f = ( x 3 , y 3 , z 3 ) Q = Z / 3 Z [ x , y , z ] W = x 3 T 1 + y 3 T 2 + z 3 T 3 . S = Q [ T 1 , T 1 , T 3 ]

Example of higher homotopies f = ( x 3 , y 3 , z 3 ) Q = Z / 3 Z [ x , y , z ] W = x 3 T 1 + y 3 T 2 + z 3 T 3 . S = Q [ T 1 , T 1 , T 3 ] M = Q / ( xz + yz , y 2 + z 2 , x 2 , y 3 , z 3 ) G = 0 → Q 3 → Q 6 → Q 4 → Q 1 → 0 S (1) 4 ⊕ S (3) 3 � P = S ⊗ G ∼ � S ⊕ S (2) 6 � � ⊕ =

d 0 � S (2) 4 ⊕ S (4) 3 S ⊕ S (2) 6 d 1 S (1) 4 ⊕ S (3) 3 � S (1) ⊕ S (3) 6 − y 2 − z 2    z 2 xz − z 2  0 − T 1 x − T 3 z 0 0 yz x 2 − xz + z 2 − z 2 − 2 − T 2 y 0 0 0     y 2 + z 2  − 2   T 3 x + T 3 z − yz − xz − z 2 − x 2 − xy − yz  0     d 0 =  − 2   T 2 0 0 x − z y 0 − z         T 3 x 2 + T 2 yz − T 3 z 2 − T 1 x 2 + T 1 xz − T 3 xz + T 3 z 2  − 2 0 0 − T 1 xz − T 2 yz − T 3 z 2 − T 2 xy + T 2 yz − T 2 y 2     − T 1 x 2 − T 1 xz + T 3 xz + T 3 z 2    − T 2 3 x − T 2 T 2 xy − T 3 xz + T 2 yz − T 3 z 2 − T 3 y 2 + T 1 xz − T 2 yz T 2 y 2 + T 3 yz T 3 x 2 + T 2 yz  − 2 3 z T 3 xy + T 3 yz     T 2 y 2 − T 2 z 2 − T 1 x 2 − T 1 xz − 2 − T 3 xy − T 2 xz − T 3 yz − T 2 z 2 T 1 xy + T 3 yz + T 2 z 2 − T 1 xz − T 2 yz − T 3 z 2 0 0    x 2 y 2 + z 2 xz − z 2 yz 2  0 0 0 0 T 2 y − T 1 x − T 3 z z 0 0 0 0 0        − T 3 x − T 3 z − T 2 y − T 3 z x + z − z  0 0 0 0        − T 1 x 2 − T 1 xz  d 1 = 0 − T 2 x − T 2 z 0 − T 3 y − T 2 z 0 − y z        − T 2 y 2  − 2 0 − T 2 y T 3 x − T 3 z z x − z 0         − 2 T 3 x + T 3 z 0 T 1 x 0 0 − z − y     − 2 T 2 z T 2 z T 1 xz + T 2 yz + T 3 z 2 y x − z 0 0

matrix factorization = G + (higher) homotopies

matrix factorization = G + (higher) homotopies constant terms = differential of G ; e.g.

matrix factorization = G + (higher) homotopies constant terms = differential of G ; e.g. − y 2 − z 2  − T 1 x − T 3 z z 2 xz − z 2 yz  0 0 − T 2 y x 2 − xz + z 2 0 − z 2 0 0    y 2 + z 2  T 3 x + T 3 z 0 − yz − xz − z 2 − x 2 − xy − yz     d 0 = T 2 0 0 x − z y 0 − z   T 3 x 2 + T 2 yz − T 3 z 2 − T 1 x 2 + T 1 xz − T 3 xz + T 3 z 2  − T 1 xz − T 2 yz − T 3 z 2 − T 2 y 2  0 0 − T 2 xy + T 2 yz   − T 3 y 2 + T 1 xz − T 2 yz T 2 y 2 + T 3 yz − T 1 x 2 − T 1 xz + T 3 xz + T 3 z 2 T 3 x 2 + T 2 yz  − T 2 3 x − T 2 3 z T 2 xy − T 3 xz + T 2 yz − T 3 z 2 T 3 xy + T 3 yz    T 2 y 2 − T 2 z 2 − T 1 x 2 − T 1 xz 0 − T 3 xy − T 2 xz − T 3 yz − T 2 z 2 T 1 xy + T 3 yz + T 2 z 2 − T 1 xz − T 2 yz − T 3 z 2 0

matrix factorization = G + (higher) homotopies constant terms = differential of G ; e.g. − y 2 − z 2  − T 1 x − T 3 z z 2 xz − z 2 yz  0 0 − T 2 y x 2 − xz + z 2 0 − z 2 0 0    y 2 + z 2  T 3 x + T 3 z 0 − yz − xz − z 2 − x 2 − xy − yz     d 0 = T 2 0 0 x − z y 0 − z   T 3 x 2 + T 2 yz − T 3 z 2 − T 1 x 2 + T 1 xz − T 3 xz + T 3 z 2  − T 1 xz − T 2 yz − T 3 z 2 − T 2 y 2  0 0 − T 2 xy + T 2 yz   − T 3 y 2 + T 1 xz − T 2 yz T 2 y 2 + T 3 yz − T 1 x 2 − T 1 xz + T 3 xz + T 3 z 2 T 3 x 2 + T 2 yz  − T 2 3 x − T 2 3 z T 2 xy − T 3 xz + T 2 yz − T 3 z 2 T 3 xy + T 3 yz    T 2 y 2 − T 2 z 2 − T 1 x 2 − T 1 xz 0 − T 3 xy − T 2 xz − T 3 yz − T 2 z 2 T 1 xy + T 3 yz + T 2 z 2 − T 1 xz − T 2 yz − T 3 z 2 0

matrix factorization = G + (higher) homotopies constant terms = differential of G ; e.g. − y 2 − z 2  − T 1 x − T 3 z z 2 xz − z 2 yz  0 0 − T 2 y x 2 − xz + z 2 0 − z 2 0 0    y 2 + z 2 − xz − z 2 − x 2  T 3 x + T 3 z 0 − yz − xy − yz     d 0 = T 2 0 0 x − z y 0 − z   T 3 x 2 + T 2 yz − T 3 z 2 − T 1 x 2 + T 1 xz − T 3 xz + T 3 z 2  − T 1 xz − T 2 yz − T 3 z 2 − T 2 xy + T 2 yz − T 2 y 2  0 0   − T 3 y 2 + T 1 xz − T 2 yz T 2 y 2 + T 3 yz − T 1 x 2 − T 1 xz + T 3 xz + T 3 z 2 T 3 x 2 + T 2 yz  − T 2 3 x − T 2 3 z T 2 xy − T 3 xz + T 2 yz − T 3 z 2 T 3 xy + T 3 yz    T 2 y 2 − T 2 z 2 − T 1 x 2 − T 1 xz 0 − T 3 xy − T 2 xz − T 3 yz − T 2 z 2 T 1 xy + T 3 yz + T 2 z 2 − T 1 xz − T 2 yz − T 3 z 2 0 1 ⊗ d G S (2) 6 ∼ → S (2) ⊗ Q G 1 ∼ = S (2) 4 2 = S (2) ⊗ Q G 2 − − −

matrix factorization = G + (higher) homotopies linear terms = homotopies for multiplication by f i ; e.g.

matrix factorization = G + (higher) homotopies linear terms = homotopies for multiplication by f i ; e.g. − y 2 − z 2  − T 1 x − T 3 z z 2 xz − z 2 yz  0 0 − T 2 y x 2 − xz + z 2 0 − z 2 0 0    y 2 + z 2  T 3 x + T 3 z 0 − yz − xz − z 2 − x 2 − xy − yz     d 0 = T 2 0 0 x − z y 0 − z   T 3 x 2 + T 2 yz − T 3 z 2 − T 1 x 2 + T 1 xz − T 3 xz + T 3 z 2  − T 1 xz − T 2 yz − T 3 z 2 − T 2 y 2  0 0 − T 2 xy + T 2 yz   − T 3 y 2 + T 1 xz − T 2 yz T 2 y 2 + T 3 yz − T 1 x 2 − T 1 xz + T 3 xz + T 3 z 2 T 3 x 2 + T 2 yz  − T 2 3 x − T 2 3 z T 2 xy − T 3 xz + T 2 yz − T 3 z 2 T 3 xy + T 3 yz    T 2 y 2 − T 2 z 2 − T 1 x 2 − T 1 xz 0 − T 3 xy − T 2 xz − T 3 yz − T 2 z 2 T 1 xy + T 3 yz + T 2 z 2 − T 1 xz − T 2 yz − T 3 z 2 0

matrix factorization = G + (higher) homotopies linear terms = homotopies for multiplication by f i ; e.g. − y 2 − z 2  − T 1 x − T 3 z z 2 xz − z 2 yz  0 0 − T 2 y x 2 − xz + z 2 0 − z 2 0 0    y 2 + z 2  T 3 x + T 3 z 0 − yz − xz − z 2 − x 2 − xy − yz     d 0 = T 2 0 0 x − z y 0 − z   T 3 x 2 + T 2 yz − T 3 z 2 − T 1 x 2 + T 1 xz − T 3 xz + T 3 z 2  − T 1 xz − T 2 yz − T 3 z 2 − T 2 y 2  0 0 − T 2 xy + T 2 yz   − T 3 y 2 + T 1 xz − T 2 yz T 2 y 2 + T 3 yz − T 1 x 2 − T 1 xz + T 3 xz + T 3 z 2 T 3 x 2 + T 2 yz  − T 2 3 x − T 2 3 z T 2 xy − T 3 xz + T 2 yz − T 3 z 2 T 3 xy + T 3 yz    T 2 y 2 − T 2 z 2 − T 1 x 2 − T 1 xz 0 − T 3 xy − T 2 xz − T 3 yz − T 2 z 2 T 1 xy + T 3 yz + T 2 z 2 − T 1 xz − T 2 yz − T 3 z 2 0

matrix factorization = G + (higher) homotopies linear terms = homotopies for multiplication by f i ; e.g. − y 2 − z 2  z 2 xz − z 2  − T 1 x − T 3 z 0 0 yz x 2 − xz + z 2 − z 2 − T 2 y 0 0 0   y 2 + z 2  T 3 x + T 3 z − yz − xz − z 2 − x 2 − xy − yz  0   d 0 =  T 2 0 0 x − z y 0 − z     T 3 x 2 + T 2 yz − T 3 z 2 − T 1 x 2 + T 1 xz − T 3 xz + T 3 z 2  0 0 − T 1 xz − T 2 yz − T 3 z 2 − T 2 xy + T 2 yz − T 2 y 2   − T 1 x 2 − T 1 xz + T 3 xz + T 3 z 2  − T 2 3 x − T 2 T 2 xy − T 3 xz + T 2 yz − T 3 z 2 − T 3 y 2 + T 1 xz − T 2 yz T 2 y 2 + T 3 yz T 3 x 2 + T 2 yz  3 z T 3 xy + T 3 yz   T 2 y 2 − T 2 z 2 − T 1 x 2 − T 1 xz − T 3 xy − T 2 xz − T 3 yz − T 2 z 2 T 1 xy + T 3 yz + T 2 z 2 − T 1 xz − T 2 yz − T 3 z 2 0 0       − x 0 − z 0 − y 0       T 1  + T 2  + T 3       0 1 x + z     0 0 0 � = T i ⊗ σ i : S ⊗ G 0 → S (1) ⊗ G 1 σ i : G 0 → G 1

matrix factorization = G + (higher) homotopies quadratic term = higher homotopy

matrix factorization = G + (higher) homotopies quadratic term = higher homotopy − y 2 − z 2  − T 1 x − T 3 z z 2 xz − z 2 yz  0 0 − T 2 y x 2 − xz + z 2 − z 2 0 0 0    y 2 + z 2  T 3 x + T 3 z 0 − yz − xz − z 2 − x 2 − xy − yz     d 0 = T 2 0 0 x − z y 0 − z   T 3 x 2 + T 2 yz − T 3 z 2 − T 1 x 2 + T 1 xz − T 3 xz + T 3 z 2  − T 1 xz − T 2 yz − T 3 z 2 − T 2 y 2  0 0 − T 2 xy + T 2 yz   − T 3 y 2 + T 1 xz − T 2 yz T 2 y 2 + T 3 yz − T 1 x 2 − T 1 xz + T 3 xz + T 3 z 2 T 3 x 2 + T 2 yz  − T 2 3 x − T 2 3 z T 2 xy − T 3 xz + T 2 yz − T 3 z 2 T 3 xy + T 3 yz    T 2 y 2 − T 2 z 2 − T 1 x 2 − T 1 xz 0 − T 3 xy − T 2 xz − T 3 yz − T 2 z 2 T 1 xy + T 3 yz + T 2 z 2 − T 1 xz − T 2 yz − T 3 z 2 0

matrix factorization = G + (higher) homotopies quadratic term = higher homotopy − y 2 − z 2  − T 1 x − T 3 z z 2 xz − z 2 yz  0 0 − T 2 y x 2 − xz + z 2 − z 2 0 0 0    y 2 + z 2  T 3 x + T 3 z 0 − yz − xz − z 2 − x 2 − xy − yz     d 0 = T 2 0 0 x − z y 0 − z   T 3 x 2 + T 2 yz − T 3 z 2 − T 1 x 2 + T 1 xz − T 3 xz + T 3 z 2  − T 1 xz − T 2 yz − T 3 z 2 − T 2 y 2  0 0 − T 2 xy + T 2 yz   − T 3 y 2 + T 1 xz − T 2 yz T 2 y 2 + T 3 yz − T 1 x 2 − T 1 xz + T 3 xz + T 3 z 2 T 3 x 2 + T 2 yz  − T 2 3 x − T 2 3 z T 2 xy − T 3 xz + T 2 yz − T 3 z 2 T 3 xy + T 3 yz    T 2 y 2 − T 2 z 2 − T 1 x 2 − T 1 xz 0 − T 3 xy − T 2 xz − T 3 yz − T 2 z 2 T 1 xy + T 3 yz + T 2 z 2 − T 1 xz − T 2 yz − T 3 z 2 0

matrix factorization = G + (higher) homotopies quadratic term = higher homotopy − y 2 − z 2  z 2 xz − z 2  − T 1 x − T 3 z 0 0 yz x 2 − xz + z 2 − z 2 − T 2 y 0 0 0   y 2 + z 2  T 3 x + T 3 z − yz − xz − z 2 − x 2 − xy − yz  0   d 0 =  T 2 0 0 x − z y 0 − z     T 3 x 2 + T 2 yz − T 3 z 2 − T 1 x 2 + T 1 xz − T 3 xz + T 3 z 2  0 0 − T 1 xz − T 2 yz − T 3 z 2 − T 2 xy + T 2 yz − T 2 y 2   − T 1 x 2 − T 1 xz + T 3 xz + T 3 z 2  − T 2 3 x − T 2 T 2 xy − T 3 xz + T 2 yz − T 3 z 2 − T 3 y 2 + T 1 xz − T 2 yz T 2 y 2 + T 3 yz T 3 x 2 + T 2 yz  3 z T 3 xy + T 3 yz   T 2 y 2 − T 2 z 2 − T 1 x 2 − T 1 xz − T 3 xy − T 2 xz − T 3 yz − T 2 z 2 T 1 xy + T 3 yz + T 2 z 2 − T 1 xz − T 2 yz − T 3 z 2 0 0   0 T 2  = T 2 − x − z 3 ⊗ σ (0 , 0 , 2) : S ⊗ G 0 → S (4) ⊗ G 3 . 3  0

matrix factorization = G + (higher) homotopies quadratic term = higher homotopy − y 2 − z 2  z 2 xz − z 2  − T 1 x − T 3 z 0 0 yz − T 2 y x 2 − xz + z 2 − z 2 0 0 0   y 2 + z 2  T 3 x + T 3 z − yz − xz − z 2 − x 2 − xy − yz  0     d 0 = T 2 0 0 x − z y 0 − z    T 3 x 2 + T 2 yz − T 3 z 2 − T 1 x 2 + T 1 xz − T 3 xz + T 3 z 2  0 0 − T 1 xz − T 2 yz − T 3 z 2 − T 2 xy + T 2 yz − T 2 y 2   − T 3 y 2 + T 1 xz − T 2 yz T 2 y 2 + T 3 yz − T 1 x 2 − T 1 xz + T 3 xz + T 3 z 2 T 3 x 2 + T 2 yz  − T 2 3 x − T 2 T 2 xy − T 3 xz + T 2 yz − T 3 z 2  3 z T 3 xy + T 3 yz   T 2 y 2 − T 2 z 2 − T 1 x 2 − T 1 xz − T 3 xy − T 2 xz − T 3 yz − T 2 z 2 T 1 xy + T 3 yz + T 2 z 2 − T 1 xz − T 2 yz − T 3 z 2 0 0   0 T 2  = T 2 − x − z 3 ⊗ σ (0 , 0 , 2) : S ⊗ G 0 → S (4) ⊗ G 3 . 3  0 σ (0 , 0 , 2) only nonzero σ J with | J | ≥ 2

R -free resolution from higher homotopies S ∗ ⊗ G ⊗ R ≃ − → R M an R -free resolution; differentials given by higher homotopies.

R -free resolution from higher homotopies S ∗ ⊗ G ⊗ R ≃ − → R M an R -free resolution; differentials given by higher homotopies. 0 ← G 0 ← G 1 ← ( S 2 ) ∗ ⊗ G 0 ← ( S 2 ) ∗ ⊗ G 1 ← ( S 4 ) ∗ ⊗ G 0 ( S 2 ) ∗ ⊗ G 2 ← . . . G 2 G 3 with − ⊗ R applied to above.

Explanation for higher homotopies: we can transfer the R -module ≃ structure on M to an A ∞ A -module structure on G − → M .

Explanation for higher homotopies: we can transfer the R -module ≃ structure on M to an A ∞ A -module structure on G − → M . This is encoded in an extended Bar A -comodule structure on Bar A ⊗ G . But by Koszul duality, Bar A ≃ C W is a homotopy equivalence, and so Bar A ⊗ G ≃ C W ⊗ G . Now dualize C to S .

Proposition (-, Eisenbud, Schreyer) There exists a system of higher operators { t i 1 ,..., i j | 1 ≤ i 1 < . . . < i j ≤ c } , with t i 1 ,..., i j : F → F a degree j endomorphism. These determine a derivation d on A ⊗ F such that ( A ⊗ F , d ) is a dg A-module quasi-isomorphic to M.

Proposition (-, Eisenbud, Schreyer) There exists a system of higher operators { t i 1 ,..., i j | 1 ≤ i 1 < . . . < i j ≤ c } , with t i 1 ,..., i j : F → F a degree j endomorphism. These determine a derivation d on A ⊗ F such that ( A ⊗ F , d ) is a dg A-module quasi-isomorphic to M. These are dual to the higher homotopies, via the generalized BGG correspondence.

Representatives of M

� � � � Representatives of M R S ⊗ G ∈ D f D f cdg ( S W ) dg ( A ) ∋ A ⊗ F ∼ = L ∼ ∼ = = M ∈ D f ( R )

� � � � Representatives of M R S ⊗ G ∈ D f D f cdg ( S W ) dg ( A ) ∋ A ⊗ F ∼ = L ∼ ∼ = = M ∈ D f ( R ) Want to use this BGG to study numerical invariants of M .

Assume ( Q , n , k ) is local and the resolutions G , R ⊗ F are minimal .

Assume ( Q , n , k ) is local and the resolutions G , R ⊗ F are minimal . Guiding questions: what are the shapes and sizes of G and F ? How are they related?

Assume ( Q , n , k ) is local and the resolutions G , R ⊗ F are minimal . Guiding questions: what are the shapes and sizes of G and F ? How are they related? Set β Q M ( i ) = dim k G i ⊗ k = dim k Tor Q i ( M , k ) β R M ( i ) = dim k F i ⊗ k = dim k Ext n R ( M , k ) � P Q β Q M ( n ) t n M ( t ) := n ≥ 0 � P R β R M ( n ) t n M ( t ) := n ≥ 0

� Apply − ⊗ Q k to BGG diagram: R � D f S ⊗ ¯ ¯ dg ( ¯ dg (¯ Λ) ∋ ¯ Λ ⊗ ¯ G ∈ D f S ) F ∼ = L

� Apply − ⊗ Q k to BGG diagram: R � D f S ⊗ ¯ ¯ dg ( ¯ dg (¯ Λ) ∋ ¯ Λ ⊗ ¯ G ∈ D f S ) F ∼ = L We have R Hom R ( M , k ) ∼ = ¯ S ⊗ ¯ G Q k ∼ = ¯ Λ ⊗ ¯ M ⊗ L F

� Apply − ⊗ Q k to BGG diagram: R � D f S ⊗ ¯ ¯ dg ( ¯ dg (¯ Λ) ∋ ¯ Λ ⊗ ¯ G ∈ D f S ) F ∼ = L We have since S ∗ ⊗ G ⊗ R R Hom R ( M , k ) ∼ = ¯ S ⊗ ¯ ≃ G − → R M Q k ∼ = ¯ Λ ⊗ ¯ ≃ M ⊗ L F since A ⊗ F − → Q M

� Apply − ⊗ Q k to BGG diagram: R � D f S ⊗ ¯ ¯ dg ( ¯ dg (¯ Λ) ∋ ¯ Λ ⊗ ¯ G ∈ D f S ) F ∼ = L We have R Hom R ( M , k ) ∼ = ¯ S ⊗ ¯ G ∼ Λ ( k , M ⊗ L = R Hom ¯ Q k ) (by BGG) Q k ∼ = ¯ Λ ⊗ ¯ ≃ M ⊗ L F since A ⊗ F − → Q M

� Apply − ⊗ Q k to BGG diagram: R � D f S ⊗ ¯ ¯ dg ( ¯ dg (¯ Λ) ∋ ¯ Λ ⊗ ¯ G ∈ D f S ) ∼ F = L We have R Hom R ( M , k ) ∼ = ¯ S ⊗ ¯ G ∼ Λ ( k , M ⊗ L = R Hom ¯ Q k ) (by BGG) Q k ∼ = ¯ Λ ⊗ ¯ F ∼ M ⊗ L = k ⊗ L S R Hom R ( M , k ) (by BGG) ¯

Eilenberg-Moore spectral sequence

Eilenberg-Moore spectral sequence For dg-modules M , N over dg-algebra B , have Eilenberg-Moore spectral sequence: E 2 = Ext ∗ H ( B ) ( H ( M ) , H ( N )) ⇒ H ( R Hom B ( M , N )) and analogous for Tor.

Applying to: Q k ) ∼ = ¯ S ⊗ ¯ G ∼ Λ ( k , M ⊗ L R Hom ¯ = R Hom R ( M , k ) S R Hom R ( M , k ) ∼ = ¯ Λ ⊗ ¯ F ∼ k ⊗ L = M ⊗ L Q k ¯ gives

Applying to: Q k ) ∼ = ¯ S ⊗ ¯ G ∼ Λ ( k , M ⊗ L R Hom ¯ = R Hom R ( M , k ) S R Hom R ( M , k ) ∼ = ¯ Λ ⊗ ¯ F ∼ k ⊗ L = M ⊗ L Q k ¯ gives E 2 = Ext ∗ Λ ( k , Tor Q ∗ ( M , k )) ⇒ Ext ∗ R ( M , k ) ¯

Applying to: Q k ) ∼ = ¯ S ⊗ ¯ G ∼ Λ ( k , M ⊗ L R Hom ¯ = R Hom R ( M , k ) S R Hom R ( M , k ) ∼ = ¯ Λ ⊗ ¯ F ∼ k ⊗ L = M ⊗ L Q k ¯ gives E 2 = Ext ∗ Λ ( k , Tor Q ∗ ( M , k )) ⇒ Ext ∗ R ( M , k ) ¯ ¯ S R ( M , k ) , k ) ⇒ Tor Q ∗ (Ext ∗ E 2 = Tor ∗ ( M , k )

These were previously known by Avramov-Buchweitz, and Avramov-Gasharov-Peeva, respectively. The second was inspired by spectral sequence of Benson-Carlson (TAMS ’94).

These were previously known by Avramov-Buchweitz, and Avramov-Gasharov-Peeva, respectively. The second was inspired by spectral sequence of Benson-Carlson (TAMS ’94). In particular, gives (from first page) well known inequalities: P Q M ( t ) P R M ( t ) ≤ (1 − t 2 ) c P Q M ( t ) ≤ P R M ( t )(1 + t ) c with equality if and only if the corresponding spectral sequences collapse on the first page if and only if higher homotopies (resp. operators) are minimal.

Putting these together: M ( t )(1 + t ) c ≤ P Q M ( t ) P Q M ( t ) ≤ P R (1 − t ) c so we see that both cannot collapse at once.

Putting these together: M ( t )(1 + t ) c ≤ P Q M ( t ) P Q M ( t ) ≤ P R (1 − t ) c so we see that both cannot collapse at once. What’s happening?

Analogy with equivariant cohomology

Analogy with equivariant cohomology X is a smooth manifold, T torus acting smoothly on X

� � � � Analogy with equivariant cohomology X is a smooth manifold, T torus acting smoothly on X Goresky, Kottwitz and MacPherson (GKM) show that there is a commutative diagram R � D f dg ( ¯ dg (¯ D f S ) Λ) ∼ = L ∼ ∼ = = D b T (pt) p ∗ D b T ( X ) T (pt) ∼ ¯ ¯ S = H ∗ = R [ T 1 , . . . , T c ] Λ = H ∗ ( T ) D b T ( X ) equivariant derived category of X .

So we have −⊗ R k D b T ( X ) ∼ = D f ( R ) → D b − − − − T (pt)