aa r X i v : . [ m a t h . R A ] A ug EXT-ALGEBRAS OF GRADED SKEW EXTENSIONS
Y. SHEN, X. WANG, AND G.-S. ZHOUA bstract . In this paper, we study the Ext-algebras of graded skew extensions. For a connected graded algebra A and a graded automorphism σ , we analyze the Yoneda product of the Ext-algebra of graded skew extension A [ z ; σ ], and prove this Ext-algebra is an R -smash product of the Ext-algebra of A and the one of polynomialalgebra k [ z ].
1. I ntroduction
The Ext-algebra Ext ∗ A ( k , k ) is a powerful tool to comprehend a connected graded algebra A . Equippedwith the Yoneda product, Ext-algebras are associative algebras. Such a structure has been discussed, espe-cially for Koszul type algebras (such as [4, 6, 7]). From the view of A ∞ -algebras, Ext-algebras carry enoughinformation to recover algebras ([8]).Graded Ore extension is a common constructive method for graded algebras. The usual associative Ext-algebras of Ore extensions have been studied in [10] for K algebras. In the study of Artin-Schelter regularalgebras and Calabi-Yau algebras, there are many examples coming from graded skew extensions, whichare graded Ore extensions with zero derivations. For Koszul algebras, He, Van Oystaeyen and Zhang provethat the Ext-algebras of graded skew extensions are trivial extensions in [7]. The aim of this paper is togive a straightforward description for the Yoneda products of Ext-algebras of graded skew extensions in ageneral setting.In fact, graded skew extensions are R -smash (twisted tensor) products, namely, tensor vector spacesof algebras with braiding structures. The initial purpose of R -smash product is to study factorization andproduct problems in noncommutative world. In this viewpoint, for a connected graded algebra A and agraded automorphism σ of A , the graded skew extension A [ z ; σ ] can be factorized into A and polynomialalgebra k [ z ]. Starting from the free resolutions of trivial modules of A and k [ z ], we analyze the structure ofExt-algebra of A [ z ; σ ] as an associative algebra. The result shows that the Ext-algebra is still an R -smashproduct. Theorem. (Theorem 3.7) Let A be a connected graded algebra, and σ be an automorphism of A. LetB = A [ z ; σ ] be a graded skew extension of A. Write E ( k [ z ]) , E ( A ) and E ( B ) to be Ext-algebras of k [ z ] , Aand B, respectively. Then there exists a linear isomorphism R E : E ( A ) ⊗ E ( k [ z ]) → E ( k [ z ]) ⊗ E ( A ) such thatas bigraded algebras E ( B ) (cid:27) E ( k [ z ]) R E E ( A ) (cid:27) E ( A ) R − E E ( k [ z ]) . Here is an outline of the paper. In Section 2, we fix some notations, review definitions of R -smashproducts, and prove a result that Ext-algebras provide a functor from the category of connected graded Mathematics Subject Classification.
Key words and phrases.
Skew extensions, Ext-algebras, R -smash products. algebras to the category of bigraded algebras. In Section 3, we focus on the Yoneda product of Ext-algebrasof graded skew extensions and give two applications.Throughout the paper, k is a field and all vector spaces and algebras are over k . Unadorned ⊗ means ⊗ k ,and (graded) k -dual functor is denoted by ( − ) ∗ .2. P reliminaries Notations.
We give a brief introduction of some notations used in the paper. We refer [1] for moreabout homological algebras.Let A = L i ∈ Z A i be a graded algebra. We say A is connected , if A i = i < A = k . In thiscase, we write ε A : A → k for the canonical augmentation of A . For a graded automorphism σ of A , thegraded skew extension A [ z ; σ ] is obtained by giving the graded polynomial ring A [ z ] a new multiplication,subject to za = σ ( a ) z for any a ∈ A . We are interested in the case that indeterminant z is of positive degree.This assumption ensures A [ z ; σ ] is connected provided so is A .We denote by Gr A the category of all graded left A -modules with A -module homomorphisms of degreezero. For an object M ∈ Gr A and an integer i , the i -th shift M ( i ) is also an object in Gr A by setting M ( i ) j = M i + j for any j . Let M , N be two graded left A -modules. For each morphism f : M → N in Gr A ,there is a morphism from M ( i ) to N ( i ) corresponding to f , which we still denote by f for simplicity. Writegraded vector spaces Hom A ( M , N ) = L i ∈ Z Hom Gr A ( M , N ( i )), and its derived one Ext ∗ A ( M , N ). If L is aright graded A -module, tensor product L ⊗ A M is defined to be a graded vector space, where the degree ofelement l ⊗ m is the sum of degrees of homogeneous elements l and m .Let M be a graded left (resp. right) graded A -module, and ν be a graded automorphism of A . Then ν M (resp. M ν ) denotes the twisted graded left (resp. right) A -module, which equals M as abelian groups with A -action a ∗ m = ν ( a ) m (resp. m ∗ a = m ν ( a )) for all a ∈ A and m ∈ M .We denote by Ch(Gr A ) the category of cochain complexes of graded left A -modules with cochain com-plex morphisms. For an object ( X (cid:5) , d X ) ∈ Ch(Gr A ) and an integer i , the shift of cochain complex X [ i ] (cid:5) is anew complex with terms X [ i ] n = X n + i and di ff erential ( − i d X . The endomorphism space End A ( X (cid:5) ) is also acochain complex with n -th term Q i Hom A ( X i , X i + n ) and di ff erential d X f − ( − n f d X for any f ∈ End A ( X (cid:5) ) n .Endowed with the canonical multiplication, End A ( X (cid:5) ) is a di ff erential bigraded algebra. For any gradedright A -module L , we may consider it as a cochain complex concentrated in degree 0. Then L ⊗ A X (cid:5) is acochain complex with L ⊗ A X n as terms and di ff erential L ⊗ A d X .Let f : X (cid:5) → Y (cid:5) be a morphism of cochain complexes in Ch(Gr A ). The mapping cone of f is a cochaincomplex, denoted by Cone( f ), whose n -th term is X n + ⊕ Y n with n -th di ff erential d n Cone( f ) ( x , y ) = ( − d n + X ( x ) , f n ( x ) + d nY ( y )) , ∀ x ∈ X n + , y ∈ Y n . R -smash product. R -smash product is a kind of solutions for factorization problems, and a moregeneral construction than tensor products. We collect some results of graded version in [2]. Definition 2.1.
Let A and B be two (bi-)graded algebras, and R : B ⊗ A → A ⊗ B be a (bi-)graded k -linearmap. Define A R B to be A ⊗ B as a (bi-)graded vector space with multiplication m A R B = ( m A ⊗ m B )(id A ⊗ R ⊗ id B ) , XT-ALGEBRAS OF GRADED SKEW EXTENSIONS 3 where m A , m B are multiplications of A , B respectively. We say A R B is an R-smash product , if it is associa-tive and 1 A R B is the unit.By [2, Theorem 2.5], R -smash products always satisfy a normal condition, that is, R (1 B ⊗ a ) = a ⊗ B and R ( b ⊗ A ) = A ⊗ b for any a ∈ A , b ∈ B . Since the normal condition is a necessary condition for R -smashproducts, we always omit this part for the description of linear maps R in the sequel. Example 2.2.
Let A be a graded algebra, and σ be a graded automorphism of A . Define a graded linearmap R : k [ z ] ⊗ A → A ⊗ k [ z ] P i ≥ c i z i ⊗ a P i ≥ σ i ( a ) ⊗ c i z i . Then A R k [ z ] is an R -smash product, and the skew extension A [ z ; σ ] (cid:27) A R k [ z ].The following theorem is an important description of R -smash products. Theorem 2.3. [2, Theorem 2.10]
Let A , B and C be (bi-)graded algebras. Then the following two conditionsare equivalent (a)
As (bi-)graded algebras, C (cid:27) A R B for some (bi-)graded k-linear map R : B ⊗ A → A ⊗ B; (b) there exist (bi-)graded algebra homomorphisms f A : A → C and f B : B → C such thatm C ◦ ( f A ⊗ f B ) : A ⊗ B → Cis an isomorphism of (bi-)graded vector spaces, where m C is the multiplication of C.In this case, R = (cid:16) m C ◦ ( f A ⊗ f B ) (cid:17) − ◦ m C ◦ ( f B ⊗ f A ) . Ext-algebras.
Let A be a connected graded algebra. There exists a graded free resolution P (cid:5) of A k : P (cid:5) = ( · · · → A ⊗ V d − −−→ A ⊗ V d − −−→ A ⊗ V d − −−→ A → → · · · ) ε A −−→ A k , where V n is a graded vector spaces and write V = k , satisfying Im d − n ⊆ A ≥ ⊗ V n − for all n = , , · · · . Sucha complex P (cid:5) is called a minimal free resolution of A k , which is unique up to isomorphisms in Ch(Gr A ). Itis easy to know that Hom A ( P (cid:5) , k ) has zero di ff erential, so the graded vector space Ext nA ( k , k ) = Hom A ( A ⊗ V n , k ) (cid:27) V ∗ n for all n ≥
0. Equipped with the Yoneda product, the bigraded vector space E ( A ) : = M ( n , i ) ∈ Z Ext nA ( k , k ) i is a bigraded associative algebra, called the Ext-algebra of A .In other hand, the cohomology of di ff erential bigraded algebra End A ( P (cid:5) ) is isomorphic to the Ext-algebra E ( A ). With the help of quasi-isomorphism ε A : P (cid:5) → A k , we have the following quasi-isomorphismEnd A ( P (cid:5) ) Hom A ( P (cid:5) ,ε A ) −−−−−−−−−→ Hom A ( P (cid:5) , k ) . We close this section by a result saying that Ext-algebra arises a contravariant functor from the categoryof connected graded algebras to the category of bigraded algebras. It maybe a classical result, but we cannotfind an appropriate paper to cite. We devote a short space to give a proof.
Theorem 2.4.
Taking Ext-algebra E ( − ) is a contravariant functor from the category of connected gradedalgebras to the category of bigraded algebras. Y. SHEN, X. WANG, AND G.-S. ZHOU
Proof.
Let A and B be two connected graded algebras, and f : A → B be a graded algebra homomorphism.The algebra homomorphism f makes any object in Gr B also an object in Gr A . Clearly, Hom B ( M , N ) ⊆ Hom A ( M , N ) for all M , N ∈ Gr B .Let P (cid:5) and Q (cid:5) be minimal free resolutions of A k and B k , respectively. And Q (cid:5) is also a cochain complexin Ch(Gr A ). By Comparison Theorem, there is a morphism of cochain complexes e f : P (cid:5) → Q (cid:5) in Ch(Gr A )lifting f such that ε B e f = ε A . Define a bigraded linear map E ( f ) : E ( B ) = Hom B ( Q (cid:5) , k ) ֒ → Hom A ( Q (cid:5) , k ) Hom A ( e f , k ) −−−−−−−−→ Hom A ( P (cid:5) , k ) = E ( A ) , that is, E ( f )( γ ) = γ ◦ e f for any γ ∈ E ( B ).Let α ∈ E i ( B ) s and β ∈ E j ( B ) t for any i , j ≥ s , t ≤
0. We also consider them as morphisms α : Q (cid:5) → B k ( s )[ i ] and β : Q (cid:5) → B k ( t )[ j ] be in Ch(Gr B ). We lift α to a morphism e α : Q (cid:5) → Q ( s )[ i ] (cid:5) inCh(Gr B ) such that ε B [ i ] ◦ e α = α . Write ϕ : = E ( f )( α ) : P (cid:5) → A k ( s )[ i ], and e ϕ : P (cid:5) → P ( s )[ i ] (cid:5) to be a lift of ϕ satisfying ε A [ i ] ◦ e ϕ = ϕ in Ch(Gr A ). Then we obtain the following commutative diagram in Ch(Gr A ). Q (cid:5) α % % ❏❏❏❏❏❏❏❏❏❏ e α (cid:15) (cid:15) P (cid:5) e f o o ϕ y y tttttttttt e ϕ (cid:15) (cid:15) A k ( s )[ i ] Q ( s )[ i ] (cid:5) ε B [ i ] ttttttttt P ( s )[ i ] (cid:5) ε A [ i ] e e ❏❏❏❏❏❏❏❏❏ e f [ i ] o o So ε B [ i ] ◦ ( e f [ i ] ◦ e ϕ − e α ◦ e f ) = . Notice that ε B [ i ] is a quasi-isomorphism and the functor Hom A ( P (cid:5) , − )preserves quasi-isomorphism, we have that e f [ i ] ◦ e ϕ − e α ◦ e f : P (cid:5) → Q ( s )[ i ] (cid:5) is null homotopic. By theminimality of P (cid:5) and Q (cid:5) , we have β [ i ] ◦ e f [ i ] ◦ e ϕ = β [ i ] ◦ e α ◦ e f , and E ( f )( β ) · E ( f )( α ) = ( β ◦ e f )[ i ] ◦ e ϕ = β [ i ] ◦ e f [ i ] ◦ e ϕ = β [ i ] ◦ e α ◦ e f = E ( f )( β · α ) . Hence, E ( f ) is a bigraded algebra homomorphism.It remains to show the given rule of associating morphisms for E ( − ) satisfies the requirement for con-travariant functors. It is obvious that E (id A ) = id E ( A ) . Let C be another connected graded algebra, and g : B → C be a graded algebra homomorphism. Let F (cid:5) be a minimal free resolution of C k . By ComparisonTheorem, there is a morphism of cochain complexes e g : Q (cid:5) → F (cid:5) lifting g , satisfying ε C e g = ε B . It is clear e g ◦ e f is a morphism of cochain complexes from P (cid:5) to F (cid:5) induced by g ◦ f , such that ε C ◦ ( e g ◦ e f ) = ε A .Therefore, E ( g ◦ f ) = E ( f ) ◦ E ( g ). (cid:3)
3. T he Y oneda products of E xt - algebras of graded skew extensions This section focuses on the Yoneda products of Ext-algebras of graded skew extensions. Let A be aconnected graded algebra and let σ be a fixed graded automorphism of A . Let B = A [ z ; σ ] be the gradedskew extension of A , where deg z = l ≥ ι A : A → B , ι z : k [ z ] → B , XT-ALGEBRAS OF GRADED SKEW EXTENSIONS 5 and two graded algebra epimorphisms: π A : B → A P mi = a i z i a , π z : B → k [ z ] P mi = a i z i P mi = ε A ( a i ) z i , satisfying π A ◦ ι A = id A , π z ◦ ι z = id k [ z ] . Corollary 3.1.
The bigraded algebra homomorphisms E ( π A ) : E ( A ) → E ( B ) and E ( π z ) : E ( k [ z ]) → E ( B ) are both injective.Proof. By Theorem 2.4, E ( − ) is a contravariant functor. Then E ( ι A ) ◦ E ( π A ) = id E ( A ) and E ( ι B ) ◦ E ( π B ) = id E ( B ) . Therefore E ( π A ) and E ( π B ) are both injective. (cid:3) In the viewpoint of R -smash products, graded skew extension B can be factored into two subalgebras A and k [ z ]. Following Corollary 3.1, E ( A ) and E ( k [ z ]) are both bigraded subalgebras of E ( B ). A naturalquestion is whether E ( B ) is a smash product of E ( A ) and E ( k [ z ]). We devote the rest of this section to givea positive answer.There is a classical free resolution of trivial module B k constructed in [5, 10]. We make a little modifica-tion to the free resolution for the graded skew extension. Let P (cid:5) be a minimal free resolution of A k : P (cid:5) = ( · · · −−→ A ⊗ V m d − m −−−→ A ⊗ V m − d − m + −−−−→ · · · −−→ A ⊗ V d − −−→ A → → · · · ) ε A −−→ A k , where each V i is a graded vector space.We have a graded ( B , A )-bimodule homomorphism ρ z : B σ ( − l ) −· z −−→ B . Define a cochain complex inCh(Gr B ) F (cid:5) : = Cone ( ρ z ⊗ A P (cid:5) : B σ ( − l ) ⊗ A P (cid:5) → B ⊗ A P (cid:5) ) . Lemma 3.2.
The morphism of cochain complexes ε B : F (cid:5) → B k is a quasi-isomorphism. As a consequence,F (cid:5) is a minimal free resolution of B k.Proof. One obtains the morphism π A provides a quasi-isomorphism from the mapping cone Cone( ρ z ) of B σ ( − l ) ρ z −−→ B to A as cochain complexes of graded ( B , A )-bimodules. Then π A ⊗ A k : Cone( ρ z ⊗ A k ) = Cone( ρ z ) ⊗ A k → A ⊗ A k (cid:27) B k is still a quasi-isomorphism. Now we have the following commutativediagram: B σ ( − l ) ⊗ A P (cid:5) ρ z ⊗ A P (cid:5) / / B σ ( − l ) ⊗ A ε A ≃ (cid:15) (cid:15) B ⊗ A P (cid:5) / / B ⊗ A ε A ≃ (cid:15) (cid:15) F (cid:5) ϕ (cid:15) (cid:15) ✤✤✤ ε B ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ B σ ( − l ) ⊗ A k ρ z ⊗ A k / / B ⊗ A k / / Cone( ρ z ⊗ A k ) π A ⊗ A k / / B k , where ϕ : = ( B σ ( − l ) ⊗ A ε A ) [1] ⊕ ( B ⊗ A ε A ) is also a quasi-isomorphism. So ε B is a quasi-isomorphism. Theminimality of F (cid:5) comes from the minimality of P (cid:5) and the action of ρ z . (cid:3) In the sequel, we use the minimal free resolution F (cid:5) of B k constructed above to bridge the relationsamong Ext-algebras E ( k [ z ]) , E ( A ) and E ( B ). Recall that E ( A ) (cid:27) M i ≥ Hom A ( P − i , k ) (cid:27) M i ≥ V ∗ i , Y. SHEN, X. WANG, AND G.-S. ZHOU where V = k . By Lemma 3.2, we have E ( B ) = V ∗ , and E j ( B ) (cid:27) Hom B ( F − j , k ) (cid:27) Hom B ( B σ ( − l ) ⊗ A P − j + , k ) ⊕ Hom B ( B ⊗ A P − j , k ) (cid:27) V ∗ j − ( l ) ⊕ V ∗ j , for j ≥ E i ( A ) with V ∗ i and E j ( B ) with V ∗ j − ( l ) ⊕ V ∗ j for each i ≥ , j ≥ k [ z ], there is a canonical minimal free resolution of k [ z ] k :Cone( ρ ′ z ) : = (0 → k [ z ]( − l ) ρ ′ z −−→ k [ z ] → ε k [ z ] −−−→ k [ z ] k , where ρ ′ z is the k [ z ]-module homomorphism of multiplication by z . Then E ( k [ z ]) (cid:27) Hom k [ z ] ( k [ z ]( − l ) , k ) = Hom Gr k [ z ] ( k [ z ]( − l ) , k ( − l )) (cid:27) k ∗ ( l )is a one dimensional graded vector space concentrated in degree − l , and identified with k ∗ ( l ). We fix ξ to bethe canonical basis of E ( k [ z ]) corresponding to the identity map in k ∗ ( l ). Note that V = k . Remark 3.3.
In order to make notations uniform, we consider Ext-algebras E ( k [ z ]), E ( A ) and E ( B ) asgraded k -dual spaces of some graded vector spaces. However, we need to keep in mind that each ele-ment in such identifications for Ext-algebras also has a corresponding representation through minimal freeresolutions.For any homogeneous element f ∈ ( V ∗ i ) − t = E i ( A ) − t , there is a corresponding morphism α : = P (cid:5) → k ( − t )[ i ] in Ch(Gr A ) induced by ε A ⊗ f : A ⊗ V i → A k ( − t ).In other side, for any g ∈ ( V ∗ i − ( l )) − t − l ⊆ E i ( B ) − t − l , there is a corresponding morphism β : = F (cid:5) → k ( − t − l )[ i ] in Ch(Gr B ) induced by B σ ( − l ) ⊗ A ( A ⊗ V i − ) π z ⊗ A ( A ⊗ g ) −−−−−−−−→ k [ z ]( − l − t ) ε k [ z ] −−−→ B k ( − t − l ). Suchrepresentations are useful to prove results below. Lemma 3.4. E ( π z )( ξ ) is the identity map of V ∗ ( l ) ⊆ E ( B ) .Proof. The composition of graded algebra homomorphisms A ι A −−→ B π z −−→ k [ z ] makes k [ z ] a graded ( B , A )-module, and k [ z ] σ (cid:27) k [ z ] as graded ( B , A )-modules. Let pr : k [ z ] ⊗ A P (cid:5) → k [ z ] ⊗ A A (cid:27) k [ z ] be theprojection to the 0-th term of the cochain complex k [ z ] ⊗ A P (cid:5) , which is a morphism in Ch(Gr B ). Then wehave a commutative diagram in Ch(Gr B ) as below B σ ( − l ) ⊗ A P (cid:5) ρ z ⊗ P (cid:5) / / π z ⊗ A P (cid:5) (cid:15) (cid:15) B ⊗ A P (cid:5) π z ⊗ A P (cid:5) (cid:15) (cid:15) / / F (cid:5) e π z (cid:15) (cid:15) ✤✤✤ ε B (cid:28) (cid:28) ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ k [ z ]( − l ) ⊗ A P (cid:5) ρ z ⊗ A P (cid:5) / / pr (cid:15) (cid:15) k [ z ] ⊗ A P (cid:5) / / pr (cid:15) (cid:15) Cone( ρ z ⊗ A P (cid:5) ) ϕ (cid:15) (cid:15) ✤✤✤ k [ z ]( − l ) ρ ′ z / / k [ z ] / / Cone( ρ ′ z ) ε k [ z ] / / B k , where e π z = ( π z ⊗ A P (cid:5) )[1] ⊕ ( π z ⊗ A P (cid:5) ), and ϕ = pr [1] ⊕ pr .The basis ξ of E ( k [ z ]) is also related to the morphism β : Cone( ρ ′ z ) → k ( − l )[1] whose ( − ε k [ z ] : k [ z ]( − l ) → k ( − l ), by Remark 3.3. By Theorem 2.4, one has E ( π z )( β ) = β ◦ ϕ ◦ e π z . XT-ALGEBRAS OF GRADED SKEW EXTENSIONS 7
However, the nonzero values of ( − ϕ all takes from k [ z ]( − l ) ⊗ A A . Hence, the onlynonzero part of E ( π z )( β ) equals the composition of following morphisms φ : B σ ( − l ) ⊗ A A π z ⊗ A A −−−−−→ k [ z ]( − l ) ⊗ A A (cid:27) k [ z ]( − l ) ε k [ z ] −−−→ k ( − l ) . When identifying E ( B ) as V ∗ ( l ) ⊗ V ∗ , φ corresponds to the identity map of V ∗ ( l ). The result follows. (cid:3) Lemma 3.5.
For any f ∈ V ∗ i = E i ( A ) , then E ( π A )( f ) = f ∈ V ∗ i ⊆ E i ( B ) .Proof. Consider the following commutative diagram in Ch(Gr B ): B σ ( − l ) ⊗ A P (cid:5) ρ z ⊗ A P (cid:5) / / (cid:15) (cid:15) B ⊗ A P (cid:5) / / π A ⊗ A P (cid:5) (cid:15) (cid:15) F (cid:5) (0 ,π A ⊗ A P (cid:5) ) (cid:15) (cid:15) ✤✤✤ ε B ❆❆❆❆❆❆❆❆ / / P (cid:5) / / P (cid:5) ε A / / B k . Similar to the proof of Lemma 3.4, the result follows. (cid:3)
It is the turn to see the Yoneda product of images of E ( A ) and E ( k [ z ]) in E ( B ). Lemma 3.6.
Let f ∈ ( V ∗ i ) − t = E i ( A ) − t for i , t ≥ . (a) Then E ( π z )( ξ ) · E ( π A )( f ) = ( − i f ∈ (cid:0) V ∗ i ( l ) (cid:1) − t − l ⊆ E i + ( B ) − t − l . (b) There exists a bigraded algebra automorphism τ of E ( A ) such thatE ( π A )( f ) · E ( π z )( ξ ) = τ ( f ) ∈ (cid:0) V ∗ i ( l ) (cid:1) − t − l ⊆ E i + ( B ) − t − l . Proof.
First of all, we fix corresponding representations along Remark 3.3. Write α : P (cid:5) → A k ( − t )[ i ] as thecorresponding element of f and β : Cone( ρ ′ z ) → k [ z ] k ( − l )[1] as the one of ξ . By Lemma 3.5, the image E ( π A )( f ) in E i ( B ) is still an element f in ( V ∗ i ) − t . So the morphism E ( π A )( α ) : F (cid:5) → B k ( − t )[ i ] in Ch(Gr B )induced by π A ⊗ A ( ε A ⊗ f ) : B ⊗ A ( A ⊗ V i ) → B k ( − t ) . By the proof of Lemma 3.4, we have the morphism E ( π z )( β ) : F (cid:5) → B k ( − l )[1] in Ch(Gr B ) is induced by B σ ( − l ) ⊗ A A π z ⊗ A A −−−−−→ k [ z ]( − l ) ε k [ z ] −−−→ B k ( − l ) . (a) By Comparison Theorem and the minimality of P (cid:5) , there exists a morphism e α : P (cid:5) → P (cid:5) ( − t )[ i ] suchthat ε A [ i ] ◦ e α = α . Note that the ( − i )-th component of e α : A ⊗ V i → A ( − t ) is just A ⊗ f .The Yoneda product of E ( π z )( ξ ) and E ( π A )( f ) can be read o ff from the following diagram B σ ( − l ) ⊗ A P (cid:5) ρ z ⊗ A P (cid:5) / / ( − i B σ ( − l ) ⊗ A e α (cid:15) (cid:15) B ⊗ A P (cid:5) / / B ⊗ A e α (cid:15) (cid:15) F (cid:5) φ (cid:15) (cid:15) ✤✤✤ E ( π A )( α ) ' ' PPPPPPPPPPPPP ( B σ ( − l ) ⊗ A P (cid:5) ( − t ))[ i ] ψ / / ( B ⊗ A P (cid:5) ( − t ))[ i ] / / F (cid:5) ( − t )[ i ] E ( π z )( β )[ i ] (cid:15) (cid:15) / / B k ( − t )[ i ] B k ( − t − l )[ i + , Y. SHEN, X. WANG, AND G.-S. ZHOU where ψ : = ( − i ( ρ z ⊗ A P (cid:5) ( − t ))[ i ] and φ : = ( − i ( B σ ( − l ) ⊗ A e α )[1] ⊕ ( B ⊗ A e α ). Then E ( π z )( β ) · E ( π A )( α ) = E ( π z )( β )[ i ] ◦ φ, and the nonezero part equals the following composition B σ ( − l ) ⊗ A ( A ⊗ V i ) ( − i B σ ( − l ) ⊗ A ( A ⊗ f ) −−−−−−−−−−−−−−→ B σ ( − l ) ⊗ A A ( − t ) π z ⊗ A A −−−−−→ k [ z ]( − t − l ) ε k [ z ] −−−→ B k ( − t − l ) . Thus, E ( π z )( ξ ) · E ( π A ) ( f ) = ( − i f ∈ (cid:16) V ∗ i ( l ) (cid:17) − t − l ⊂ E i + ( B ) − t − l by Remark 3.3.(b) By Comparison Theorem and the minimality of P (cid:5) , there exists a cochain complex isomorphism e σ : σ − P (cid:5) → P (cid:5) in Ch(Gr A ). Then e σ induces a di ff erential bigraded algebra isomorphism Ψ : End( P (cid:5) ) → End( σ − P (cid:5) ), and H Ψ is a bigraded algebra automorphism of E ( A ). Since we identify E ( A ) with ⊕ i ≥ V ∗ i , wewrite τ to be the automorphism of ⊕ i ≥ V ∗ i .To be specific, one has the following commutative diagramEnd A ( P (cid:5) ) Hom A ( P (cid:5) ,ε A ) / / Ψ (cid:15) (cid:15) Hom A ( P (cid:5) , k ) Hom A ( e σ, k ) (cid:15) (cid:15) End A ( σ − P (cid:5) ) Hom A ( σ − P (cid:5) ,ε A ) / / Hom A ( σ − P (cid:5) , k ) . So for any h ∈ ( V ∗ j ) − s , τ ( h ) corresponds to a morphism σ − P (cid:5) → k [ j ]( − s ) formed by σ − ( A ⊗ V j ) e σ − j −−−→ A ⊗ V j ε A ⊗ h −−−−→ k ( − s ) . Using di ff erent constructions of mapping cones, we have0 / / (cid:15) (cid:15) B ⊗ A P (cid:5) / / B ⊗ A P (cid:5) ι (cid:15) (cid:15) B σ ( − l ) ⊗ A P (cid:5) / / B ⊗ A P (cid:5) / / (cid:15) (cid:15) F (cid:5) π (cid:15) (cid:15) B σ ( − l ) ⊗ A P (cid:5) / / / / ( B σ ( − l ) ⊗ A P (cid:5) )[1] . Define a cochain morphism in Ch(Gr B ) as follows θ : F (cid:5) π −−→ ( B σ ( − l ) ⊗ A P (cid:5) )[1] (cid:27) −−→ ( B ( − l ) ⊗ A σ − P (cid:5) )[1] B ( − l ) ⊗ A e σ −−−−−−−→ ( B ( − l ) ⊗ A P (cid:5) )[1] ι −−→ F (cid:5) ( − l )[1] , and it satisfies E ( π z )( β ) = ε k [ z ] [1] ◦ θ . Then E ( π A )( α ) · E ( π z )( β ) = E ( π A )( α )[1] ◦ θ , and the nonzero part is B σ ( − l ) ⊗ A ( A ⊗ V i ) (cid:27) B ( − l ) ⊗ A σ − ( A ⊗ V i ) B ( − l ) ⊗ A e σ − i −−−−−−−−→ B ( − l ) ⊗ A ( A ⊗ V i ) π A ⊗ A ( ε A ⊗ f ) −−−−−−−−−→ B k ( − t − l ) . So E ( π A )( f ) · E ( π z )( ξ ) = τ ( f ) ∈ (cid:16) V ∗ i ( l ) (cid:17) − t − l ⊆ E i + ( B ) − t − l by Remark 3.3 and the correspondence of τ . (cid:3) Now we can prove the main result.
Theorem 3.7.
Let A be a connected graded algebra, and σ be an automorphism of A. Let B = A [ z ; σ ] be agraded skew extension of A. Write E ( A ) , E ( k [ z ]) and E ( B ) to be Ext-algebras of A, k [ z ] and B, respectively.Then as bigraded algebras E ( B ) (cid:27) E ( k [ z ]) R E E ( A ) (cid:27) E ( A ) R − E E ( k [ z ]) , for some k-linear normal isomorphism R E : E ( A ) ⊗ E ( k [ z ]) → E ( k [ z ]) ⊗ E ( A ) satisfyingR E ( f ⊗ g ) = ( − i g ⊗ τ ( f ) . XT-ALGEBRAS OF GRADED SKEW EXTENSIONS 9 where f ∈ E i ( A ) , g ∈ E ( k [ z ]) , and τ is the bigraded automorphism of E ( A ) in Lemma 3.6(b).Proof. By Corollary 3.1, we have two injections E ( π z ) : E ( k [ z ]) ֒ → E ( B ) , E ( π A ) : E ( A ) ֒ → E ( B ) . Write m : = m E ( B ) ( E ( π z ) ⊗ E ( π A )) : E ( k [ z ]) ⊗ E ( A ) → E ( B ) , m : = m E ( B ) ( E ( π A ) ⊗ E ( π z )) : E ( A ) ⊗ E ( k [ z ]) → E ( B ) , where m E ( B ) is the Yoneda product of E ( B ). Clearly, for any f ∈ E ( A ), m (1 ⊗ f ) = f and m ( f ⊗ = f byLemma 3.5. So m and m are both isomorphisms of bigraded vector spaces following from Lemma 3.6.Using Theorem 2.3, there exist k -linear isomorphisms of bigraded vector spaces: E ( A ) ⊗ E ( k [ z ]) E ( k [ z ]) ⊗ E ( A ) ( R E ) − : = m − ◦ m o o R E : = m − ◦ m / / such that E ( B ) (cid:27) E ( k [ z ]) R E E ( A ) (cid:27) E ( A ) R − E E ( k [ z ]) as bigraded algebras.For any f ∈ E i ( A ) and g ∈ E ( k [ z ]), by Lemma 3.6, we have m ( f ⊗ g ) = ( − i m ( g ⊗ τ ( f )) . Hence, R E ( f ⊗ g ) = ( − i g ⊗ τ ( f ). (cid:3) Example 3.8.
We compute the Ext-algebra of quantum plane k p [ x , y ] as an easy example, where p ∈ k × .Write A = k [ x ] and k p [ x , y ] (cid:27) A [ y ; σ ] where σ is an automorphism of A by sending x to px . The Ext-algebraof A is isomorphic to k h u i / ( u ) and E ( k [ y ]) (cid:27) k h v i / ( v ). Choose P (cid:5) : 0 → A ( − −· x −−→ A → A k . Then we have a commutative diagram0 / / σ − A ( − −· x / / p · σ (cid:15) (cid:15) σ − A / / σ (cid:15) (cid:15) / / A ( − −· x / / A / / . It induces an algebra automorphism of E ( A ) by sending u to pu . By Theorem 3.7, the Ext-algebra ofquantum plane is E ( k q [ x , y ]) (cid:27) (cid:16) k h u i / ( u ) (cid:17) R E (cid:16) k h v i / ( v ) (cid:17) (cid:27) k h u , v i / ( u , v , vu + puv ) , where R E ( v ⊗ u ) = − pu ⊗ v .Finally, we give two applications of Theorem 3.7.For a connected graded algebra A generated in degree 1, it is called a K p algebra if E ( A ) is generatedby { E i ( A ) } pi = as an associative algebra. In particular, K algebras are the class of algebras defined in [3],which is considered as a natural generalization of Koszul algebras. For such a class of algebras, we have animmediate result from Theorem 3.7, which contains a special case of [10, Theorem 1.2]. Corollary 3.9.
Let A be a connected graded algebra generated in degree , and σ be an automorphism ofA. Let B = A [ z ; σ ] be a graded skew extension of A. Then A is a K p algebra if and only if B is a K p algebra. The other application is about the Frobenius property and Artin-Schelter regular algebras. We say afinite dimensional algebra E is a Frobenius algebra , if there exists a nondegenerate associative bilinearform h− , −i : E × E → k . We refer [11] for more details. Corollary 3.10.
Let A be a connected graded algebra and σ be an automorphism of A. Let B = A [ z ; σ ] be a graded skew extension of A. If the Ext-algebra E ( A ) is a Frobenius algebra, then Ext-algebra E ( B ) isalso a Frobenius algebra.Proof. By Theorem 3.7, E ( B ) (cid:27) E ( k [ z ]) R E E ( A ) for some graded linear map R E : E ( A ) ⊗ E ( k [ z ]) → E ( k [ z ]) ⊗ E ( A ). Let τ be the bigraded automorphism of E ( A ) in Lemma 3.6.Clearly E ( k [ z ]) is a Frobenius algebra. Write the nondegenerate associative bilinear forms for E ( A ) and E ( k [ z ]) are h− , −i A and h− , −i z respectively. Define a bilinear form h− , −i : E ( B ) × E ( B ) → k satisfying h g R E f , g R E f i = h g , g i z h f , f i A , if g ∈ E ( k [ z ]) , ( − i h g , g i z h τ ( f ) , f i A , if g ∈ E ( k [ z ]) , where f ∈ E i ( A ) , f ∈ E ( A ) and g ∈ E ( k [ z ]) for i >
0. By straightforward computation, the bilinear form h− , −i is nondegenerate associative. So E ( B ) is a Frobenius algebra. (cid:3) We say a connected graded algebra A is Artin-Schelter regular if the global dimension of A equals d < ∞ ,Ext iA ( k , A ) = i , d and Ext dA ( k , A ) is 1-dimensional. By the approach of A ∞ -algebras, we know that A is Artin-Schelter regular if and only if the Ext-algebra E ( A ) is Frobenius [9, Corollary D]. Now we have anew way to prove graded Ore extension preserving Artin-Schelter regularity. Corollary 3.11.
Let A be an Artin-Schelter regular algebra. Let σ be a graded automorphism of A and δ be a σ -derivation of A. Then graded Ore extension A [ z ; σ, δ ] is Artin-Schelter regular.Proof. We consider the case of graded skew extension B = A [ z ; σ ] firstly. By [9, Corollary D], one obtainsthe Ext-algebra E ( A ) is Frobenius, which implies the Ext-algebra E ( B ) is also Frobenius by Corollary 3.10.Hence B is a Artin-Scehlter regular algebra.There is a canonical filtration on A [ z ; σ, δ ] such that the associated graded algebra is the graded skewextension B . Then the result follows from [12, Theorem 3.6] (cid:3) Acknowledgments.
Y. Shen is supported by the NSFC (Grant Nos. 11626215, 11701515) and ScienceFoundation of Zhejiang Sci-Tech University (Grant No. 16062066-Y); X. Wang is supported by the NSFC(Grant No. 11671351); G.-S. Zhou is supported by the NSFC (Grant No. 11601480).R eferences [1] L. L. Avramov, H. B. Foxby, S. Halperin, Di ff erential graded homological algebra , preprint.[2] S. Caenepeel, B. Ion, G. Militaru and S. Zhu, The factorization problem and the smash biproduct of algebras and coalgebras ,Algebr. Represent. Theory, (2000) 19–42.[3] T. Cassidy and B. Shelton,
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