Applications of the change-of-rings spectral sequence to the - PowerPoint PPT Presentation

Applications of the change-of-rings spectral sequence to the computation of Hochschild cohomology Mariano Suárez-Alvarez mariano@dm.uba.ar Mar del Plata, March 6–17, 2006

� � � Operations on cohomology Theorem Let A be an algebra, M, N ∈ A Mod and d ≥ 0 . Let O = ( O p ) p ≥ 0 be a sequence of natural transformations of functors of A-modules O p : Ext p A ( N , − ) → Ext p + d ( M , − ) . A Assume that, for each short exact sequence P ′ ֌ P ։ P ′′ , the following diagram commutes: ∂ Ext p A ( N , P ′′ ) Ext p + 1 ( N , P ′ ) A O p O p + 1 ∂ � Ext p + d + 1 Ext p + d ( M , P ′′ ) ( M , P ′ ) A A Then there exists exactly one Y ( O ) ∈ Ext d A ( M , N ) such that O p ( − ) = ( − ) ◦ Y ( O ) .

Operations on cohomology Corollary There is an isomorphism of bifunctors of A-modules Y : sOp • A ( − , − ) ∼ = Ext • A ( − , − ) . Corollary Let A be an algebra and d ≥ 0 . Let O = ( O p ) p ≥ 0 be a sequence of natural transformations of functors of A-bimodules O p : H p ( A , − ) → H p + d ( A , − ) which commutes with boundary maps. Then there exists exactly one Y ( O ) ∈ HH d ( A ) such that O p ( − ) = ( − ) ⌣ Y ( O ) .

Change of rings Corollary Let φ : A → B be a map of rings, and let M ∈ A Mod . For each q ≥ 1 there is a unique class ζ q ∈ Ext 2 B ( Tor A q − 1 ( B , M ) , Tor A q ( B , M )) such that the differential d p , q : Ext p q ( B , M ) , − ) → Ext p + 2 B ( Tor A ( Tor A q − 1 ( B , M ) , − ) 2 B of the spectral sequence E p , q = Ext p B ( Tor A q ( B , M ) , − ) ⇒ Ext • A ( M , − ) 2 is given on α ∈ Ext p B ( Tor A q ( B , M ) , − ) by d p , q ( α ) = α ◦ ζ q . 2

Change of rings Theorem Let φ : A → B be an epimorphism of algebras. There exists a spectral sequence, functorial on B-bimodules, E p , q ∼ = Ext p B e ( Tor A q ( B , B ) , − ) ⇒ H • ( A , − ) 2 which has E • , 0 ∼ = H • ( B , − ) . 2 For each q ≥ 1 there exists a unique class ζ q ∈ Ext 2 B e ( Tor A q − 1 ( B , B ) , Tor A q ( B , B )) such that d p , q ( − ) = ( − ) ◦ ζ q . 2 = I / I 2 and If φ is surjective and I = ker φ , Tor A 1 ( B , B ) ∼ ζ 1 ∈ Ext 2 B e ( B , I / I 2 ) = H 2 ( B , I / I 2 ) is the class of the infinitesimal extension � 0 � I / I 2 � A / I 2 � B 0

Monogenic algebras Theorem i = 0 a i X i ∈ k [ X ] . Let k be a field and fix a monic f = � N Let d = ( f , f ′ ) , pick q ∈ k [ X ] such that f = qd, and put N i ( i − 1 ) X i − 2 = q 2 ∆ 2 ( f ) . � u = q 2 a i 2 i = 0 Let A = k [ X ] / ( f ) . There is an isomorphism of graded commutative algebras k [ x 0 , τ 1 , ζ 2 ] HH • ( A ) ∼ ( f ( x ) , d ( x ) τ, f ′ ( x ) ζ, τ 2 − u ( x ) ζ ) . =

Monogenic algebras Proposition Let N � � a i ( s + 1 ) X s q X t . w = i = 0 s , t ≥ 0 s + t + 1 = i The Gerstenhaber Lie structure on HH • ( A ) is such that [ τ, x ] = q ( x ) , [ ζ, τ ] = w ( x ) ζ, [ x , x ] = [ τ, τ ] = [ τ, ζ ] = [ x , ζ ] = 0 .

Nice morphisms Theorem Let φ : A → B be an epimorphism of algebras. The following statements are equivalent: a) φ : A → B is a homological epimorphism; b) Tor A + ( B , M ) = 0 for all M ∈ A Mod ; c) Tor A + ( B , B ) = 0 ; d) φ e : A e → B e is a homological epimorphism. When they hold, there is an isomorphism of functors of B-bimodules ∼ H • ( B , − ) = → H • ( A , − ) . −

Nice ideals Corollary Let φ : A → B be a surjective homological epimorphism and let I = ker φ . There is a long exact sequence � Ext p � HH p ( A ) � HH p ( B ) � Ext p + 1 � · · · · · · A e ( A , I ) A e ( A , I )

� Nice ideals Proposition Let φ : A → B be a surjective homological epimorphism such that I = ker φ is A-flat on one side. Then H 0 ( B , − ) ∼ = H 0 ( A , − ) on B Mod B and there is a natural long exact sequence of functors of B-bimodules � H p ( B , − ) � H p ( A , − ) · · · ⌣ζ � Ext p − 1 � H p + 1 ( B , − ) � · · · A e ( I / I 2 , − ) with ζ ∈ H 2 ( B , I / I 2 ) the class of the infinitesimal extension � I / I 2 � A / I 2 � B � 0 0

Nice ideals Lemma Let φ : A → B be a surjective morphism of algebras and put I = ker φ . Then  B , if q = 0 ;    I / I 2 , if q = 1 ;   q ( B , B ) ∼ Tor A = � µ � ker I ⊗ A I − → I , if q = 2 ;    Tor A  q − 2 ( I , I ) , if q > 2 . 

� Nice ideals: an example Let A = kQ / J be an admissible quotient of the path algebra on a quiver Q and let e ∈ Q 0 . Assume ◮ Every minimal relation in J involving a path passing through e also involves a path not passing through e ; and ◮ e is on no oriented cycle of Q . Then I = AeA ⊳ A is homological and, if B = A / I , there is a long exact sequence � Ext p � HH p ( A ) · · · A ( D ( eA ) , Ae ) � HH p ( B ) � Ext p + 1 � · · · ( D ( eA ) , Ae ) A

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