aa r X i v : . [ m a t h . A C ] J un Cohomology of torsion and completion of N -complexes Xiaoyan YangDepartment of Mathematics, Northwest Normal University, Lanzhou 730070, ChinaE-mail: [email protected]
Abstract
We introduce the notions of Koszul N -complex, ˇCech N -complex and telescope N -complex, explicit derived torsion and derived completion functors in the de-rived category D N ( R ) of N -complexes using the ˇCech N -complex and the tele-scope N -complex. Moreover, we give an equivalence between the category ofcohomologically a -torsion N -complexes and the category of cohomologically a -adic complete N -complexes, and prove that over a commutative noetherian ring,via Koszul cohomology, via RHom cohomology (resp. ⊗ cohomology) and vialocal cohomology (resp. derived completion), all yield the same invariant. Key Words:
Koszul N -complex; telescope N -complex; torsion; completion Introduction and Preliminaries
The notion of N -complexes (graded objects with N -differentials d ) was introduced byMayer [15] in his study of simplicial complexes and its abstract framework of homologicaltheory was studied by Kapranov [14] and Dubois-Violette [4]. Since then the homologicalproperties of N -complexes have attracted many authors, for example [3, 6, 8, 9, 19, 20, 21].Iyama, Kato and Miyachi [12] studied the homotopy category K N ( B ) of N -complexes of anadditive category B as well as the derived category D N ( A ) of an abelian category A . Theyproved that both K N ( B ) and D N ( A ) are triangulated, and established a theory of projective(resp. injective) resolutions and derived functors. They also showed that the well knownequivalences between homotopy category of chain complexes and their derived categoriesalso generalize to the case of N -complexes.Let R be a commutative ring and a an ideal of R . Denote by Mod R the category of R -modules. There are two operations associated to this data: the a -torsion and the a -adic com-pletion. For an R -module M , the a -torsion elements form the a -torsion submodule Γ a ( M ) ∼ =lim −→ i> Hom R ( R/ a i , M ) of M . The a -adic completion of M is Λ a ( M ) := lim ←− i> ( R/ a i ⊗ R M ).Therefore, we have two additive functorsΓ a , Λ a : Mod R → Mod R .The derived category of Mod R is denoted by D ( R ). Then the derived functors Γ a , LΛ a : D ( R ) → D ( R )exist. The right derived functor RΓ a has been studied in great length already by Grothendieckand others in the context of local cohomology. The left derived functors LΛ a was studied byMatlis [15] and Greenlees-May [10].Let a be a weakly proregular ideal of R , this includes the noetherian case, but there areother interesting examples. Porta, Shaul and Yekutieli [17] extended earlier work by Alonso-Jeremias-Lipman [1], Schenzel [18] and Dwyer-Greenlees [5]. They proved that the derivedfunctors RΓ a and LΛ a can be computed by telescope complexes, and established the MGMequivalence, where the letters “MGM” stand for Matlis, Greenlees and May.The first aim of this paper is to extend works of Porta, Shaul and Yekutieli to the categoryof N -complexes. We introduce the definitions of Koszul N -complex, ˇCech N -complex andtelescope N -complex, and explicit derived torsion and derived completion functors in D N ( R )using these N -complexes. Theorem A.
Let x = x , · · · , x d be a weakly proregular sequence in R and a the idealgenerated by x . For any N -complex X , there are functorial quasi-isomorphisms RΓ a ( X ) ≃ → ˇ C ( x ; R ) ⊗ R X ∼ = Tel( x ; R ) ⊗ R X , Hom R (Tel( x ; R ) , X ) ≃ → LΛ a ( X ) . Denote by D N ( R ) a -tor and D N ( R ) a -com the full subcategories of D N ( R ) consisting of co-homologically a -torsion N -complexes and cohomologically a -adic complete N -complexes,respectively (see Definition 6.1). We show that Theorem B.
Let a be a weakly proregular ideal of R . Then the functors RΓ a : D N ( R ) a - com ⇄ D N ( R ) a - tor : LΛ a form an equivalence. Let a be an ideal in a commutative noetherian ring R and K the Koszul complex on afinite set of n generators for a . It is well known that the following numbers are equal when M is a finitely generated R -module: • n + inf { ℓ ∈ Z | H ℓ ( K ⊗ R M ) = 0 } ; • inf { ℓ ∈ Z | Ext ℓR ( R/ a , M ) = 0 } ; • inf { ℓ ∈ Z | H ℓ a ( M ) ∼ = 0 } , where H ∗ a ( M ) is the a -local cohomology of M .Each of the quantities displayed above is meaningful. These have proved to be of immenseutility even in dealing with problems concerning modules alone. Foxby and Iyengar [7]proved that the numbers obtained from the three formulas above coincide for any complex.It is natural to ask if these three approaches yield the same invariant for N -complexes. Thesecond aim of current paper is to answer the question for any N -complex and consider itsdual statement over commutative noetherian rings. . Preliminaries and basic facts
We assume throughout this paper that all rings are commutative.This section is devoted to recalling some notions and basic facts which we need in thelater sections. For terminology we shall follow [3, 12] and [20]. N -complexes. Fix an integer N >
2. An N -complex X is a sequence of R -modules · · · d n − −→ X n − d n − −→ X n d n −→ X n +1 d n +1 −→ · · · satisfying d N = 0. That is, composing any N -consecutive morphisms gives 0. A morphism f : X → Y of N -complexes is a collection of maps f n : X n → Y n making all the rectanglescommute. In this way we get a category of N -complexes, denoted by C N ( R ).For any R -module M , j ∈ Z and t = 1 , · · · , N , we define D jt ( M ) : · · · → X j − t +1 d j − t +1 −−−−→ · · · d j − −−→ X j − d j − −−→ X j → → · · · be an N -complex given by X n = M for all j − t + 1 n j and d n = 1 M for all j − t + 1 n j − X be an N -complex. For n ∈ Z , we defineZ nt ( X ) = Ker( d n + t − · · · d n ), B nt ( X ) = Im( d n − · · · d n − t ) for t = 0 , · · · , N ,C nt ( X ) = Coker( d n − · · · d n − t ), H nt ( X ) = Z nt ( X ) / B nN − t ( X ) for t = 1 , · · · , N − N -complex X is called N -acyclic if H nt ( X ) = 0 for all n and t . Proposition 1.1. ([12])
Let → X → Y → Z → be a short exact sequence in C N ( R ) .For n ∈ Z and t N − , there is a long exact sequence of cohomologies · · · → H n − ( N − t ) N − t ( Z ) → H nt ( X ) → H nt ( Y ) → H nt ( Z ) → H n + tN − t ( X ) → · · · . Let X be an N -complex. Define suspension functors Σ , Σ − : K N ( R ) → K N ( R ) as follows(Σ X ) n = X n +1 ⊕ · · · ⊕ X n + N − , d Σ X = ··· ··· ... ... ... ... ... ... ··· − d N − − d N − − d N − ··· − d − d ,(Σ − X ) n = X n − N +1 ⊕ · · · ⊕ X n − , d Σ − X = − d ··· − d ··· ... ... ... ... ... ... − d N − ··· − d N − ··· .Let f : X → Y be a morphism in C N ( R ). The mapping cone C ( f ) of f is defined as C ( f ) n = Y n ⊕ (Σ X ) n , d nC ( f ) = d f ··· ··· ... ... ... ... ... ... ··· − d N − − d N − ··· − d − d .Two morphisms f, g : X → Y of N -complexes are called homotopic if there exists { s n : X n → Y n − N +1 } such that n − f n = P N − i =0 d N − − i s n + i d i , ∀ n .We denote the homotopy category of N -complexes by K N ( R ). Then the category K N ( R ) istriangulated, and every exact triangle in K N ( R ) is isomorphic to the form X f −→ Y g −→ C ( f ) h −→ Σ X ,where X, Y ∈ K N ( R ) and g = " ... , h = " ··· ··· ... ... ... ... ... ... ··· .A morphism f : X → Y is called a quasi-isomorphism if the induced morphism H it ( f ) :H it ( X ) → H it ( Y ) is an isomorphism for any i and t = 1 , · · · , N −
1, or equivalently ifthe mapping cone C ( f ) belongs to K ac N ( R ) the full subcategory of K N ( R ) consisting of N -acyclic N -complexes. The derived category D N ( R ) of N -complexes is defined as the quotientcategory K N ( R ) / K ac N ( R ), which is also triangulated. Definition 1.2. ([14]) Let q be a primitive N -th root of 1 ( q N = 1), and let ( X, d X ),( Y, d Y )be two N -complexes of R -modules.(a) The q -Hom is the N -complex Hom R ( X, Y ) defined byHom R ( X, Y ) n = Q i ∈ Z Hom R ( X i , Y i + n ) with differential d n ( f i ) = d i + nY f i − q n f i +1 d iX .(b) The q -tensor product is the N -complex X ⊗ R Y defined by( X ⊗ R Y ) n = ` i ∈ Z ( X i ⊗ R Y n − i ) with differential d n ( x ⊗ y ) = d X ( x ) ⊗ y + q | x | x ⊗ d Y ( y ),where x, y are supposed to be homogeneous and | x | denotes the degree of x . Remark 1.3. (1) By the definition of q -Hom and q -tensor product of N -complexes and theisomorphism in [12, Theorem 2.4], one can check the following isomorphisms:Hom R ( X, Σ Y ) ∼ = ΣHom R ( X, Y ),Hom R (Σ X, Y ) ∼ = Hom R ( X, Σ − Y ) ∼ = Σ − Hom R ( X, Y ),Σ X ⊗ R Y ∼ = X ⊗ R Σ Y ∼ = Σ( X ⊗ R Y ).(2) It follows from [16, Corollary 4.5] that ( X ⊗ R − , Hom R ( X, − )) form a adjoint pair. Lemma 1.4.
For any morphism f : X → Y in C N ( R ) and any N -complex Z , one has (1) Hom R ( Z, C ( f )) ∼ = C (Hom R ( Z, f )) . (2) Hom R ( C ( f ) , Z ) ∼ = Σ − C (Hom R ( f, Z )) . (3) C ( f ) ⊗ R Z ∼ = C ( f ⊗ Z ) and Z ⊗ R C ( f ) ∼ = C ( Z ⊗ f ) .Proof. (1) For any h α i β i i ∈ Hom R ( Z, C ( f )) n ∼ = Hom R ( Z, Y ) n ⊕ Hom R ( Z, Σ X ) n , we have n Hom R ( Z,C ( f )) h α i β i i = d n + iC ( f ) h α i β i i − q n h α i +1 β i +1 i d iZ = h d n + iY f n + i +1 d n + i Σ X i h α i β i i − q n h α i +1 d iZ β i +1 d iZ i = h d n + iY α i + f n + i +1 β i − q n α i +1 d iZ d n + i Σ X β i − q n β i +1 d iZ i = (cid:20) d n Hom R ( Z,Y ) Hom R ( Z,f ) n +1 d n Hom R ( Z, Σ X ) (cid:21) h α i β i i , which implies that d n Hom R ( Z,C ( f )) = (cid:20) d n Hom R ( Z,Y ) Hom R ( Z,f ) n +1 d n ΣHom R ( Z,X ) (cid:21) , as desired.(2) For any h α i β i i ∈ Hom R ( C ( f ) , Z ) n ∼ = Hom R ( Y, Z ) n ⊕ Hom R (Σ X, Z ) n , since h α i β i i corre-sponds to a morphism [ α i β i ] : Y i ⊕ (Σ X ) i → Z n + i we have d n Hom R ( C ( f ) ,Z ) h α i β i i = d n + iZ [ α i β i ] − q n [ α i +1 β i +1 ] d iC ( f ) = d n + iZ [ α i β i ] − q n [ α i +1 β i +1 ] h d iY f i +1 d i Σ X i = [ d n + iZ α i − q n α i +1 d iY d n + iZ β i − q n α i +1 f i +1 − q n β i +1 d i Σ X ]= (cid:20) d n Hom R ( Y,Z ) − q n Hom R ( f,Z ) i +1 d n Hom R (Σ X,Z ) (cid:21) h α i β i i , which implies that d n Hom R ( C ( f ) ,Z ) = (cid:20) d n Hom R ( Z,Y ) − q n Hom R ( f,Z ) i +1 d n ΣHom R ( Z,X ) (cid:21) , as desired.(3) These follow from d C ( f ) ⊗ R Z = h d Y ⊗ RZ f ⊗ Z d Σ X ⊗ RZ i and d Z ⊗ R C ( f ) = h d Z ⊗ RY Z ⊗ f d Z ⊗ R Σ X i . (cid:3) Let P be the class of projective R -modules. An N -complex P is called semi-projective if P n ∈ P for all n , and every f : P → E is null homotopic whenever E ∈ K ac N ( R ). Let I bethe class of injective R -modules. An N -complex I is called semi-injective if I n ∈ I for all n , and every f : E → I is null homotopic whenever E ∈ K ac N ( R ).Let X be an N -complex. By Lemma 1.4, we have four triangle functors Hom R ( X, − ),Hom R ( − , X ) : K N ( R ) → K N ( Z ) and − ⊗ R X, X ⊗ R − : K N ( R ) → K N ( Z ). Then [12,Corollary 3.29] yields the following derived functorsRHom R ( X, − ) , RHom R ( − , X ) : D N ( R ) → D N ( Z ), − ⊗ L R X, X ⊗ L R − : D N ( R ) → D N ( Z ).They can be computed via semi-projective and semi-injective resolution of the N -complexesby [12, Theorem 3.27], respectively.Let X be an N -complex. The stupid truncation τ >i ( X ) : 0 → X i +1 → X i +2 → · · · and τ i ( X ) : · · · → X i − → X i →
0. For i, j ∈ Z let C [ i,j ] N ( R ) be the full subcategory of C N ( R ) hose objects are the N -complexes concentrated in the degree range [ i, j ] := { i, · · · , j } .Here is a useful criterion for quasi-isomorphisms. Lemma 1.5.
Let R and R ′ be two rings and F, G : Mod R → C N ( R ′ ) two additive functors,and let η : F → G be a natural transformation. Consider the extensions F, G : C N ( R ) → C N ( R ′ ) . Suppose X ∈ C N ( R ) satisfies the following conditions: (1) There are j , j ∈ Z such that F ( X i ) , G ( X i ) ∈ C [ j ,j ] N ( R ′ ) for every i ∈ Z . (2) The homomorphism η X i : F ( X i ) → G ( X i ) is a quasi-isomorphism for every i ∈ Z .Then η X : F ( X ) → G ( X ) is a quasi-isomorphism.Proof. Assume that X is bounded. If X ≃ D i ( M ) for some R -module M and i ∈ Z thenthis is given. Otherwise the inductive step is done using the short exact sequence0 → τ >i ( X ) → X → τ i ( X ) → N -complexes. Now assume X is arbitrary. We prove that H it ( η X ) : H it ( F ( X )) → H it ( G ( X ))is an isomorphism for every i ∈ Z and a fixed t . For any i j set τ [ i,j ] := τ j ◦ τ >i . Givenan integer i , the morphism H it ( η X ) only depends on the morphism τ [ i − N + t,i + t ] ( η X ) : τ [ i − N + t,i + t ] ( F ( X )) → τ [ i − N + t,i + t ] ( G ( X ))of N -complexes. Thus we can replace η X with η X ′ , where X ′ = τ [ j + i − N + t,j + i + t ] ( X ). But X ′ is bounded, so the morphism η X ′ is a quasi-isomorphism. (cid:3) The Koszul N -complex In this section, we give a construction of Koszul N -complexes and compute the cohomologyof a few concrete Koszul N -complexes. Definition 2.1.
Let x be an element in R . The Koszul N -complex on x , denoted by K • ( x ; R ), is the mapping cone C ( x ) of x : D ( R ) → D ( R )0 → R → R → · · · → R x → R → R in degrees − N +1 , · · · ,
0. Suppose we are given a sequence x = x , · · · , x d of elementsin R . By induction, the Koszul N -complex on x , denoted by K • ( x ; R ), is the mapping cone C ( x d ) of x d : K • ( x , · · · , x d − ; R ) → K • ( x , · · · , x d − ; R ).One can check that K • ( x ; R ) ∼ = K • ( x ; R ) ⊗ R · · · ⊗ R K • ( x d ; R ). Example 2.2.
Let x, y, z be three elements in R .(1) For N = 3, the Koszul 3-complex on x is K • ( x ; R ) : 0 → R → R x → R → R in degrees − , − ,
0. The Koszul 3-complex on x, y is • ( x, y ; R ) : 0 → R [ − ] −−−→ R (cid:20) y
00 1 − x − x (cid:21) −−−−−→ R h y
00 0 1 i −−−−→ R x y ] −−−→ R −→ − , · · · ,
0. The Koszul 3-complex on x, y, z is K • ( x, y, z ; R ) : 0 → R (cid:20) − (cid:21) −−−→ R z − y − y
01 0 − x x −−−−−−−−→ R z − z − y − y − − y x x − −−−−−−−−−−−−−−→ R y z z − x − x z − x − xy − y − x − y −−−−−−−−−−−−−−−−−→ R (cid:20) y z z
00 0 0 0 0 1 (cid:21) −−−−−−−−→ R x y z ] −−−−→ R −→ − , · · · , N = 4, the Koszul 4-complex on x is K • ( x ; R ) : 0 → R → R → R x → R → R in degrees − , − , − ,
0. The Koszul 4-complex on x, y is K • ( x, y ; R ) : 0 → R [ − ] −−−→ R (cid:20) − − (cid:21) −−−−−→ R y − x − x − x −−−−−−−−→ R (cid:20) y (cid:21) −−−−−−→ R h y
00 0 1 i −−−−→ R x y ] −−−→ R −→ − , · · · ,
0. The Koszul 4-complex on x, y, z is K • ( x, y, z ; R ) : 0 → R (cid:20) − (cid:21) −−−→ R − − −
10 1 1 −−−−−−−−→ R z − y − y − y − − −
10 0 0 x x x −−−−−−−−−−−−−−→ R z − z − y − y − y − y − − y − − x x x − −−−−−−−−−−−−−−−−−−−−−−−→ R z z − − z − y − y − y − − y − y − − y
00 0 x x x − − −−−−−−−−−−−−−−−−−−−−−−−−−−−→ R y z z z − x − x − x z − x − xy − xy − y − x − xy − y − x − y −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ R y z z z −−−−−−−−−−−−−→ R (cid:20) y z z
00 0 0 0 0 1 (cid:21) −−−−−−−−→ R x y z ] −−−−→ R −→ ith the ten nonzero modules in degrees − , · · · , N -complex on N -complexes and Koszul cohomology. Definition 2.3.
Let x = x , · · · , x d be a sequence of elements in R and X an N -complex.The Koszul N -complex of x on X is the N -complex K • ( x ; X ) := K • ( x ; R ) ⊗ R X .For t = 1 , · · · , N −
1, the Koszul cohomology of x on X isH jt ( x ; X ) = H jt ( K • ( x ; X )) for j ∈ Z . Example 2.4.
Let x, y be two elements in R and M an R -module.(1) For N = 3, the Koszul 3-complex of x on M is K • ( x ; M ) : 0 → M → M x → M → M in degrees − , − ,
0. Therefore, one has thatH − ( x ; M ) = H − ( x ; M ) = 0,H − ( x ; M ) = H − ( x ; M ) = (0 : M x ),H ( x ; M ) = H ( x ; M ) = M/xM .The Koszul 3-complex of x, y on M is K • ( x, y ; M ) : 0 → M [ − ] −−−→ M (cid:20) y
00 1 − x − x (cid:21) −−−−−→ M h y
00 0 1 i −−−−→ M x y ] −−−→ M −→ − ( x, y ; M ) = H − ( x, y ; M ) = 0,H − ( x, y ; M ) = H − ( x, y ; M ) = (0 : M ( x, y )),H ( x, y ; M ) = H ( x, y ; M ) = M/ ( x, y ) M .(2) For N = 4, the Koszul 4-complex of x on M is K • ( x ; M ) : 0 → M → M → M x → M → M in degrees − , − , − ,
0. Therefore, one has thatH − ( x ; M ) = H − ( x ; M ) = H − ( x ; M ) = H − ( x ; M ) = H − ( x ; M ) = H − ( x ; M ) = 0,H − ( x ; M ) = H − ( x ; M ) = H − ( x ; M ) = (0 : M x ),H ( x ; M ) = H ( x ; M ) = H ( x ; M ) = M/xM .The Koszul 4-complex of x, y on M is K • ( x, y ; M ) : 0 → M [ − ] −−−→ M (cid:20) − − (cid:21) −−−−−→ M y − x − x − x −−−−−−−−→ M (cid:20) y (cid:21) −−−−−−→ M h y
00 0 1 i −−−−→ M x y ] −−−→ M −→ − t ( x, y ; M ) = H − t ( x, y ; M ) = 0 for t = 1 , , − ( x, y ; M ) = H − ( x, y ; M ) = H − ( x, y ; M ) = (0 : M ( x, y )),H ( x, y ; M ) = H ( x, y ; M ) = H ( x, y ; M ) = M/ ( x, y ) M .(3) Let x = x , ..., x d be a sequence in R . The Koszul N -complex on x is K • ( x ; M ) : 0 → M → M → · · · → M x → M → M in degrees − N + 1 , · · · ,
0. Therefore, one has thatH − tt ( x ; M ) = (0 : M x ) for t = 1 , · · · , N − t ( x ; M ) = M/x M for t = 1 , · · · , N − − Nt ( x ; M ) = 0 for t = 1 , · · · , N − N -complexes0 → K • ( x ; M ) → K • ( x , x ; M ) → Σ K • ( x ; M ) → − Nt ( K • ( x , x ; M )) = (0 : M ( x , x )) for t = 1 , · · · , N − t ( K • ( x , x ; M )) = M/ ( x , x ) M for t = 1 , · · · , N − − N − tt ( K • ( x , x ; M )) = 0 for t = 1 , · · · , N − (cid:26) H − kN − tt ( x ; M ) = 0 d = 2 k H − kNt ( x ; M ) = 0 d = 2 k − t = 1 , · · · , N − (cid:26) H − kNt ( x ; M ) = (0 : M x ) d = 2 k H − kN − tt ( x ; M ) = (0 : M x ) d = 2 k + 1 for t = 1 , · · · , N − t ( x ; M ) = M/ x M for t = 1 , · · · , N − Proposition 2.5.
Given a sequence of elements x = ( x , · · · , x d ) in R , one has an isomor-phism in K N ( R ) K • ( x ; R ) ∼ = Σ d Hom R ( K • ( x ; R ) , R ) .Proof. For x there exists an exact triangle R x −→ R −→ K • ( x ; R ) → Σ R in K N ( R ). Applyingthe functor RHom R ( − , R ) to this triangle, one gets an exact triangleΣ − R → Hom R ( K • ( x ; R ) , R ) → R x −→ R .Thus K • ( x ; R ) ∼ = ΣHom R ( K • ( x ; R ) , R ) in K N ( R ). For x there exists an exact triangle K • ( x ; R ) x −→ K • ( x ; R ) −→ K • ( x , x ; R ) → Σ K • ( x ; R ) in K N ( R ). Applying the functorRHom R ( − , R ) to this triangle, one gets an exact triangleΣ − Hom R ( K • ( x ; R ) , R ) → Hom R ( K • ( x , x ; R ) , R ) → Hom R ( K • ( x ; R ) , R ) → Hom R ( K • ( x ; R ) , R ), hich implies that K • ( x , x ; R ) ∼ = Σ Hom R ( K • ( x , x ; R ) , R ) in K N ( R ). Continuing thisprocess, we obtain the isomorphism we seek. (cid:3) The ˇC ech N -complex For x ∈ R , the localization R x is obtained by inverting the multiplicatively closed set { , x, x , · · · } . Let ι : R → R x be the canonical map sending each r ∈ R to the class of thefraction r/ ∈ R x . This section gives a construction of ˇCech N -complexes. Construction 3.1.
Let x be an element in R . We have a commutative diagram:... (cid:15) (cid:15) ... x (cid:15) (cid:15) ... x (cid:15) (cid:15) ... x (cid:15) (cid:15) ... K • ( x ; R ) : 0 (cid:15) (cid:15) / / R x (cid:15) (cid:15) R x (cid:15) (cid:15) · · · R x / / x (cid:15) (cid:15) R / / K • ( x ; R ) : 0 (cid:15) (cid:15) / / R x (cid:15) (cid:15) R x (cid:15) (cid:15) · · · R x / / x (cid:15) (cid:15) R / / K • ( x ; R ) : 0 / / R R · · · R x / / R / / R ( − , R ) to this diagram, we have a commutative diagram:0 / / R x / / R x (cid:15) (cid:15) · · · R x (cid:15) (cid:15) R / / x (cid:15) (cid:15) / / R x / / R x (cid:15) (cid:15) · · · R x (cid:15) (cid:15) R / / x (cid:15) (cid:15) / / R x / / R x (cid:15) (cid:15) · · · R x (cid:15) (cid:15) R / / x (cid:15) (cid:15) N -complex0 → R ι −→ R x −→ R x −→ · · · −→ R x −→ R in degree 0 and R x in degrees 1 , · · · , N −
1, which is called the ˇCech N -complex on x , denoted by ˇ C • ( x ; R ). We also have an exact triangle in K N ( R )Σ − R x → ˇ C • ( x ; R ) → R ι −→ R x .Let x, y be two elements in R . Then the morphisms K • ( x s ; R ) : 0 u (cid:15) (cid:15) / / R xy (cid:15) (cid:15) R xy (cid:15) (cid:15) · · · R x s / / xy (cid:15) (cid:15) R / / y (cid:15) (cid:15) K • ( x s − ; R ) : 0 / / R R · · · R x s − / / R / / • ( x s ; R ) : 0 v (cid:15) (cid:15) / / R x (cid:15) (cid:15) R x (cid:15) (cid:15) · · · R x s / / x (cid:15) (cid:15) R / / K • ( x s − ; R ) : 0 / / R R · · · R x s − / / R / / C N ( R ):0 / / K • ( x s ; R ) / / v (cid:15) (cid:15) K • ( x s , y s ; R ) / / (cid:15) (cid:15) Σ K • ( x s ; R ) Σ u (cid:15) (cid:15) / / / / K • ( x s − ; R ) / / K • ( x s − , y s − ; R ) / / Σ K • ( x s − ; R ) / / R ( − , R ) to the diagram, one obtains a direct systemHom R ( K • ( x, y ; R ) , R ) → Hom R ( K • ( x , y ; R ) , R ) → Hom R ( K • ( x , y ; R ) , R ) → · · · .In the limit we get an N -complex lim −→ Hom R ( K • ( x s , y s ; R ) , R ), which is called the ˇCech N -complex on x, y , denoted by ˇ C • ( x, y ; R ). We also have an exact triangle in K N ( R )Σ − ˇ C • ( x ; R ) y → ˇ C • ( x, y ; R ) → ˇ C • ( x ; R ) ι −→ ˇ C • ( x ; R ) y .For a sequence x = ( x , · · · , x d ) of elements in R , set x s = x s , · · · , x sd and y = ( x , · · · , x d − ).By induction, the ˇCech N -complex ˇ C • ( x ; R ) on x is lim −→ Hom R ( K • ( x s ; R ) , R ) and we have anexact triangle in K N ( R )Σ − ˇ C • ( y ; R ) x d → ˇ C • ( x ; R ) → ˇ C • ( y ; R ) ι −→ ˇ C • ( y ; R ) x d .In fact, by induction, one can obtain the following isomorphismˇ C • ( x ; R ) ∼ = ˇ C • ( x ; R ) ⊗ R · · · ⊗ R ˇ C • ( x d ; R ). Example 3.2.
Let x, y be two elements in R .For N = 3, the ˇCech 3-complex on x isˇ C • ( x ; R ) : 0 → R ι x → R x → R x → x, y isˇ C • ( x, y ; R ) : 0 → R [ ι x ι y ] −−→ R x ⊕ R y (cid:20) ι y
00 1 (cid:21) −−−−→ R x ⊕ R xy ⊕ R y h ι y − ι x − ι x i −−−−−−→ R xy ⊕ R xy [ 1 − −−−−→ R xy −→ , , , , N = 4, the ˇCech 4-complex on x isˇ C • ( x ; R ) : 0 → R ι x → R x → R x → R x → x, y isˇ C • ( x, y ; R ) : 0 → R [ ι x ι y ] −−→ R x ⊕ R y (cid:20) ι y
00 1 (cid:21) −−−−→ R x ⊕ R xy ⊕ R y ι y −−−−−→ R x ⊕ R xy ⊕ R xy ⊕ R y " ι y − ι x − ι x − ι x −−−−−−−−→ R xy ⊕ R xy ⊕ R xy h −
10 1 − i −−−−−→ R xy ⊕ R xy [ 1 − −−−−→ R xy −→ here the seven nonzero modules are in degrees 0 , , , , , , Lemma 3.3.
For a sequence x = x , · · · , x d in R , the natural morphism e : ˇ C ( x ; R ) → R ( R is viewed as the N -complex D ( R )) induces a quasi-isomorphism e ⊗ , ⊗ e : ˇ C ( x ; R ) ⊗ R ˇ C ( x ; R ) ≃ → ˇ C ( x ; R ) .Proof. By symmetry it is enough to look only at1 ⊗ e : ˇ C ( x ; R ) ⊗ R ˇ C ( x ; R ) → ˇ C ( x ; R ).Since the N -complexes ˇ C ( x i ; R ) are semi-flat, it is enough to consider the case d = 1 and x = x . We have the following commutative diagram:0 (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / Σ − R x (cid:15) (cid:15) / / Σ − ˇ C ( x ; R ) x (cid:15) (cid:15) / / Σ − R x (cid:15) (cid:15) / / / / Σ − ˇ C ( x ; R ) x (cid:15) (cid:15) / / ˇ C ( x ; R ) ⊗ R ˇ C ( x ; R ) (cid:15) (cid:15) ⊗ e / / ˇ C ( x ; R ) (cid:15) (cid:15) / / / / Σ − R x / / (cid:15) (cid:15) ˇ C ( x ; R ) / / (cid:15) (cid:15) R (cid:15) (cid:15) / /
00 0 0Note that x : R x → R x is an isomorphism in D N ( R ), it follows that Σ − ˇ C ( x ; R ) x is acyclic.This completes the proof. (cid:3) Given an N -complex X , set ˇ C • ( x ; X ) := ˇ C • ( x ; R ) ⊗ R X . The R -moduleˇH jt ( x ; X ) = H jt ( ˇ C ( x ; X )) for t = 1 , · · · , N − j -th ˇCech cohomology of x on X . Example 3.4. (1) Let x be an element in R . ThenˇH t ( x ; R ) = { r ∈ R | r/ R x } = { r ∈ R | x s r = 0 for some s > } = [ s > (0 : R x s ) ∼ = lim −→ Hom R ( R/ ( x s ) , R ) , ˇH tN − t ( x ; R ) = R x /R for t = 1 , · · · , N − x = x , · · · , x d be a sequence of element in R and M an R -module. The ˇCech N -complex of x on M isˇ C ( x ; M ) : 0 → M ι −→ M x −→ M x −→ · · · −→ M x −→ ith modules M in degree 0 and M x in degrees 1 , · · · , N −
1, Therefore, one has thatˇH t ( x ; M ) ∼ = lim −→ Hom R ( R/ ( x s ) , M ) for t = 1 , · · · , N − tN − t ( x ; M ) = M x /M for t = 1 , · · · , N − Nt ( x ; M ) = 0 for t = 1 , · · · , N − N -complexes0 → Σ − ˇ C ( x ; M ) x → ˇ C ( x , x ; M ) → ˇ C ( x ; M ) → t ( x , x ; M ) ∼ = lim −→ Hom R ( R/ ( x s , x s ) , M ) for t = 1 , · · · , N − Nt ( x ; M ) = M x ,x / (Im M x + Im M x ) for t = 1 , · · · , N − N + tN − t ( x , x ; M ) = 0 for t = 1 , · · · , N − (cid:26) ˇH jt ( x ; M ) = 0 for j > kN d = 2 k − jN − t ( x ; M ) = 0 for j > kN + t d = 2 k for 1 t N − (cid:26) ˇH ( k − N + tN − t ( x ; M ) = M x ··· x d / Σ ci =1 image M x ··· x i − x i +1 ··· x d d = 2 k − kNt ( x ; M ) = M x ··· x d / Σ ci =1 image M x ··· x i − x i +1 ··· x d d = 2 k for 1 t N − t ( x ; M ) ∼ = lim −→ Hom R ( R/ ( x s ) , M ) for 1 t N − Derived torsion of N -complexes In this section, we explicit derived torsion functors in D N ( R ) using the ˇCech N -complex.Let a be an ideal of R . For each R -module M , setΓ a ( M ) = { m ∈ M | a n m = 0 for some integer n } .There is a functorial homomorphism σ M : Γ a ( M ) → M which is just the inclusion. Whenthey coincide, M is said to be a -torsion. The association M → Γ a ( M ) extends to define aleft exact additive functor on C N ( R ), it is called the a -torsion functor. By [12, Corollary3.29], the functor Γ a has a right derived functorRΓ a : D N ( R ) → D N ( R ) , ξ : Γ a → RΓ a constructed using semi-injective resolutions. Proposition 4.1.
There is a functorial morphism σ R X : RΓ a ( X ) → X , such that σ X = σ R X ◦ ξ X as morphisms Γ a ( X ) → X in D N ( R ) .Proof. Let X α → I be a semi-injective resolution, and define σ R X = α − ◦ σ I ◦ ξ − I ◦ RΓ a ( α ).This is independent of the resolution. (cid:3) For each N -complex X and i ∈ Z , the i th local cohomology of X with support in a isH it, a ( X ) = H it (RΓ a ( X )) for t = 1 , · · · , N − xample 4.2. Let R = Z and p be a prime number, and let M be an indecomposable R -module. By the fundamental theorem of Abelian groups, M is isomorphic to Z /d Z whereeither d = 0 or d is a prime power. In either case the 3-complex0 → Q /d Z → Q / Z → Q / Z → Z /d Z . In what follows, Z p denotes Z with p inverted. Case 1. If M = Z /p e Z for some integer e >
1, then applying Γ ( p ) ( − ) to the resolutionabove yields the 3-complex 0 → Z p /p e Z → Z p / Z → Z p / Z → , ( p ) ( M ) = H , ( p ) ( M ) = Z /p e Z = M ,H , ( p ) ( M ) = H , ( p ) ( M ) = 0 = H , ( p ) ( M ) = H , ( p ) ( M ). Case 2. If M = Z /d Z with d nonzero and relatively prime to p , then applying Γ ( p ) ( − ) tothe resolution above yields the 3-complex0 → d Z p /d Z → Z p / Z → Z p / Z → , ( p ) ( M ) = Z p / ( d Z p + Z ) = H , ( p ) ( M ),H , ( p ) ( M ) = H , ( p ) ( M ) = 0 = H , ( p ) ( M ) = H , ( p ) ( M ). Case 3. If M = Z , then applying Γ ( p ) ( − ) to the resolution above yields the 3-complex0 → → Z p / Z → Z p / Z → , ( p ) ( M ) = Z p / Z = H , ( p ) ( M ),H , ( p ) ( M ) = H , ( p ) ( M ) = 0 = H , ( p ) ( M ) = H , ( p ) ( M ).Following [18], an inverse system { M i } i ∈ N of abelian groups, with transition maps p j,i : M j → M i , is called pro-zero if for every i there exists j > i such that p j,i is zero. Definition 4.3. (1) Let x = x , · · · , x d be a sequence of elements in R . The sequence x is called a weakly proregular sequence if for every i < t = 1 , · · · , N − { H it ( K • ( x s ; R )) } s ∈ N is pro-zero.(2) An ideal a of R is called a weakly proregular ideal if it is generated by some weaklyproregular sequence.Let x be an element in R . For any s >
0, we have a morphism of N -complexes K • ( x s ; R ) : 0 γ s (cid:15) (cid:15) / / R (cid:15) (cid:15) R (cid:15) (cid:15) · · · R x s / / (cid:15) (cid:15) R / / π s (cid:15) (cid:15) D ( R/ ( x s )) : 0 / / / / / / · · · / / / / R/ ( x s ) / / hich induces the following morphism of inverse systems · · · / / K • ( x ; R ) γ (cid:15) (cid:15) / / K • ( x ; R ) γ (cid:15) (cid:15) / / K • ( x ; R ) γ (cid:15) (cid:15) · · · / / R/ ( x ) / / R/ ( x ) / / R/ ( x )(4.1)where R/ ( x s ) is viewed as the N -complex D ( R/ ( x s )). Let X be an N -complex. (4.1) yieldsa morphism of direct systems:Hom R ( R/ ( x ) , X ) (cid:15) (cid:15) / / Hom R ( R/ ( x ) , X ) (cid:15) (cid:15) / / Hom R ( R/ ( x ) , X ) (cid:15) (cid:15) / / · · · Hom R ( K • ( x ; R ) , X ) / / Hom R ( K • ( x ; R ) , X ) / / Hom R ( K • ( x ; R ) , X ) / / · · · This gives rise to a functorial morphism of N -complexes δ x,X : lim −→ s> Hom R ( R/ ( x s ) , X ) → lim −→ s> Hom R ( K • ( x s ; R ) , X ) ∼ = ˇ C ( x ; R ) ⊗ R X .Let x = x , · · · , x d be elements in R . The Koszul N -complex on x s is the N -complex K • ( x s ; R ). This is equipped with a morphism of N -complexes of θ s : K • ( x s ; R ) → R/ ( x s ).Therefore, we obtain an inverse system of N -complexes · · · → K • ( x s +1 ; R ) → K • ( x s ; R ) → · · · → K • ( x ; R ),compatible with the morphisms θ s and natural maps R/ ( x s +1 ) → R/ ( x s ).The next result provide an explicit formula for computing RΓ a . Theorem 4.4.
Let x = x , · · · , x d be a weakly proregular sequence in R and a the idealgenerated by x . For any N -complex X , there is a functorial quasi-isomorphism δ R x ,X : RΓ a ( X ) → ˇ C ( x ; R ) ⊗ R X .Proof. If d = 1, then by construction of γ s , we have a quasi-isomorphism of N -complexes H s : 0 / / (0 : R x s ) (cid:127) _ (cid:15) (cid:15) (0 : R x s ) (cid:127) _ (cid:15) (cid:15) · · · (0 : R x s ) / / (cid:127) _ (cid:15) (cid:15) / / (cid:15) (cid:15) γ s : 0 / / R R · · · R x s / / ( x s ) / / F ( Y ) := Γ ( x ) ( Y ) and G ( Y ) := lim −→ s> Hom R ( K • ( x s ; R ) , Y ) for any N -complex Y .Let I be a semi-injective N -complex. It is enough to show that δ x ,I : F ( I ) → G ( I ) is aquasi-isomorphism. By Lemma 1.5 we may assume that I is a single injective module. Foreach j ∈ Z and t = 1 , · · · , N −
1, one has thatH jt (lim −→ s> Hom R (Ker γ s , I )) ∼ = lim −→ s> H jt (Hom R ( H s , I )) ∼ = lim −→ s> Hom R (H − jN − t ( H s ) , I ) ∼ = lim −→ s> Hom R (H − jN − t ( K • ( x ; R )) , I ) . ince x is weakly proresular, it follows that lim −→ s> Hom R (H − jN − t ( K • ( x ; R )) , I ) = 0 for j > t = 1 , · · · , N −
1. Hence lim −→ s> Hom R (Ker γ s , I ) is acyclic, and so F ( I ) ∼ = G ( I ) in D N ( R ).Now assume d > d . Let X ≃ −→ I be a semi-injective resolutionand set y = x , · · · , x d − . One has the following isomorphismsRΓ a ( X ) ∼ = lim −→ s> Hom R ( R/ ( x s ) , I ) ∼ = lim −→ s> Hom R ( R/ ( y s ) ⊗ R R/ ( x sd ) , I ) ∼ = lim −→ s> Hom R ( R/ ( y s ) , lim −→ s> Hom R ( R/ ( x sd ) , I )) ∼ = lim −→ s> Hom R ( K • ( y s ; R ) , lim −→ s> Hom R ( K • ( x sd ; R ) , I )) ∼ = lim −→ s> Hom R ( K • ( y s ; R ) ⊗ R K • ( x sd ; R ) , I ) ∼ = lim −→ s> Hom R ( K • ( x s ; R ) , I ) ∼ = ˇ C ( x ; R ) ⊗ R X, where the second one holds as R/ ( x s ) ∼ = R/ ( y s ) ⊗ R R/ ( x sd ), the third one is Hom-tensoradjointness, the fourth one is by induction, as claimed. (cid:3) Corollary 4.5.
Let x = x , · · · , x d be a set of generators for a weakly proregular ideal a ,and let X be an N -complex. (1) The morphism σ RRΓ a ( X ) : RΓ a (RΓ a ( X )) → RΓ a ( X ) is an isomorphism. Thus thefunctor RΓ a : D N ( R ) → D N ( R ) is idempotent. (2) There is a natural isomorphism H it, a ( X ) ∼ = ˇH it ( x ; X ) . The Telescope N -complexes and derived completion This section introduces the notion of Telescope N -complexes and explicits the derivedcompletion functor in D N ( R ) using it. Definition 5.1.
Let { e i | i > } be the basis of the countably generated free R -module L ∞ i =0 R . Given an element x ∈ R , define the morphism v : D ( L ∞ i =0 R ) → D ( L ∞ i =0 R ) of N -complexes by v ( e i ) = (cid:26) e if i = 0 e i − − xe i if i > . The telescope N -complex Tel( x ; R ) is the N -complex Σ − C ( v )0 → L ∞ i =0 R v → L ∞ i =0 R → L ∞ i =0 R → · · · → L ∞ i =0 R → , , · · · , N −
1. Given a sequence x = x , · · · , x d in R , we defineTel( x ; R ) := Tel( x ; R ) ⊗ R · · · ⊗ R Tel( x d ; R ).Then Tel( x ; R ) is an N -complex of free R -modules. Lemma 5.2.
Let x = x , · · · , x d be a sequence in R . One has a quasi-isomorphism x : Tel( x ; R ) ≃ → ˇ C ( x ; R ) .Proof. For any x j , by [17, Lemma 5.7], one can define a quasi-isomorphism of N -complexesTel( x j ; R ) : 0 w xj (cid:15) (cid:15) / / L ∞ i =0 R w xj (cid:15) (cid:15) v / / L ∞ i =0 R w xj (cid:15) (cid:15) · · · L ∞ i =0 R w N − xj (cid:15) (cid:15) / / C ( x j ; R ) : 0 / / R ι / / R x j · · · R x j / / w x j = · · · = w N − x j . Therefore, we haveTel( x ; R ) ∼ = Tel( x ; R ) ⊗ R · · · ⊗ R Tel( x d ; R ) ≃ → ˇ C ( x ; R ) ⊗ R · · · ⊗ R ˇ C ( x d ; R ) ∼ = ˇ C ( x ; R ).This shows the quasi-isomorphism we seek. (cid:3) Corollary 5.3.
Let x and y be two finite sequences in R and let a = ( x ) and b = ( y ) . If √ a = √ b , then Tel( x ; R ) and Tel( y ; R ) are homotopy equivalent.Proof. By Lemma 5.2, we have two quasi-isomorphisms Tel( x ; R ) ≃ → ˇ C ( x ; R ) and Tel( y ; R ) ≃ → ˇ C ( y ; R ). But ˇ C ( x ; R ) ∼ = ˇ C ( y ; R ) in D N ( R ), it follows that Tel( x ; R ) ∼ = Tel( y ; R ) in D N ( R ).Consequently, Tel( x ; R ) and Tel( y ; R ) are homotopy equivalent. (cid:3) Corollary 5.4.
Let x = x , · · · , x d be a weakly proregular sequence in R and a the idealgenerated by x . For any N -complex X , there is a functorial quasi-isomorphism RΓ a ( X ) → Tel( x ; R ) ⊗ R X . Let a be an ideal of R . We have an inverse system · · · ։ R/ a ։ R/ a ։ R/ a .Following [10], for an R -module M we writeΛ a ( M ) := lim ←− s> ( R/ a s ⊗ R M )for the a -adic completion of M . We get an additive functor Λ a : Mod R → Mod R and thereis a functorial morphism τ M : M → Λ a ( M ) for any M ∈ Mod R . By [12, Corollary 3.29], thefunctor Λ a has a left derived functorLΛ a ( − ) : D N ( R ) → D N ( R ) , ξ ′ : LΛ a → Λ a constructed using semi-projective resolutions. For any N -complex X ∈ D N ( R ), by analogywith Proposition 4.1, there is a functorial morphism τ L X : X → LΛ a ( X ) in D N ( R ), such that ξ ′ X ◦ τ L X = τ X as morphism X → Λ a ( X ).Let x be an element of R and X an N -complex. Then the diagram (4.1) yields a morphismof inverse systems: · · / / Hom R (Hom R ( K • ( x ; R ) , R ) , X ) (cid:15) (cid:15) / / Hom R (Hom R ( K • ( x ; R ) , R ) , X ) (cid:15) (cid:15) · · · / / R/ ( x ) ⊗ R X / / R/ ( x ) ⊗ R X This gives rise to a functorial morphism of N -complexes λ X : Hom R (Tel( x ; R ) , X ) ≃ Hom R ( ˇ C ( x ; R ) , X ) → lim ←− s> ( R/ ( x s ) ⊗ R Y ) = Λ ( x ) ( X ).The next results provide an explicit formula for computing LΛ a . Theorem 5.5.
Let x = x , · · · , x d be a weakly proregular sequence in R and a the idealgenerated by x . For any N -complex X , there is a functorial quasi-isomorphism Hom R (Tel( x ; R ) , X ) ≃ → LΛ a ( X ) .Proof. It is enough to consider a semi-projective N -complex X = P . By Lemma 1.5 wereduce to the case of a single projective module P . By [17, Theorem 5.21], one can obtain aquasi-isomorphism Hom R (Tel( x ; R ) , P ) ≃ → Λ ( x ) ( P ) in D N ( R ), where Λ ( x ) ( P ) is viewed asthe N -complex D (Λ ( x ) ( P )). By induction, we obtain the quasi-isomorphism we seek. (cid:3) Corollary 5.6.
Let a be a weakly proregular ideal of R . (1) For any N -complex X , there exists a functorial quasi-isomorphism RHom R (RΓ a ( R ) , X ) ≃ → LΛ a ( X ) . (2) The morphism τ LLΛ a ( X ) : LΛ a ( X ) → LΛ a (LΛ a ( X )) is an isomorphism. Thus the functor LΛ a : D N ( R ) → D N ( R ) is idempotent. MGM equivalence of N -complexes The task of this section to prove the MGM equivalence in D N ( R ), i.e., we show an equiv-alence between the category of cohomologically a -torsion N -complexes and the category ofcohomologically a -adic complete N -complexes. Definition 6.1. (1) An N -complex X ∈ D N ( R ) is called cohomologically a -torsion if themorphism σ R X : RΓ a ( X ) → X is an isomorphism. The full subcategory of D N ( R ) consistingof cohomologically a -torsion N -complexes is denoted by D N ( R ) a -tor .(2) An N -complex Y ∈ D N ( R ) is called cohomologically a -adic complete if the morphism τ L Y : Y → LΛ a ( Y ) is an isomorphism. The full subcategory of D N ( R ) consisting of cohomo-logically a -adic complete N -complexes is denoted by D N ( R ) a -com .We first show that the functor RΓ a is right adjoint to the inclusion D N ( R ) a -tor ֒ → D N ( R )and the functor LΛ a is left adjoint to the inclusion D N ( R ) a -com ֒ → D N ( R ). Proposition 6.2. (1)
The morphism σ R Y : RΓ a ( Y ) → Y induces an isomorphism Hom D N ( R ) a - tor ( X, RΓ a ( Y )) ≃ −→ Hom D N ( R ) ( X, Y ) , ∀ X ∈ D N ( R ) a - tor , Y ∈ D N ( R ) . The morphism τ L X : X → LΛ a ( X ) induces an isomorphism Hom D N ( R ) a - com (LΛ a ( X ) , Y ) ≃ −→ Hom D N ( R ) ( X, Y ) , ∀ X ∈ D N ( R ) , Y ∈ D N ( R ) a - com .Proof. We just prove (1) since (2) follows by duality.We need to show that ̺ X,Y : Hom D N ( R ) a -tor ( X, RΓ a ( Y )) = Hom D N ( R ) ( X, RΓ a ( Y )) → Hom D N ( R ) ( X, Y ) is an isomorphism. Referring then to the diagramHom D N ( R ) ( X, Y ) ν −→ Hom D N ( R ) (RΓ a ( X ) , RΓ a ( Y )) ρ ←− Hom D N ( R ) ( X, RΓ a ( Y )),where ν is the natural morphism and ρ is induced by the isomorphism σ R X : RΓ a ( X ) → X . Next we show that ρ − ν is inverse to ̺ X,Y . That ̺ X,Y ρ − ν ( α ) = α for any α ∈ Hom D N ( R ) ( X, Y ) amounts to the (obvious) commutativity of the diagramRΓ a ( X ) σ R X ≃ (cid:15) (cid:15) RΓ a ( α ) / / RΓ a ( Y ) σ R Y (cid:15) (cid:15) X α / / Y. That ρ − ν̺ X,Y ( β ) = β for β ∈ Hom D N ( R ) ( X, RΓ a ( Y )) amounts to commutativity ofRΓ a ( X ) σ R X ≃ (cid:15) (cid:15) RΓ a ( β ) / / RΓ a (RΓ a ( Y )) ≃ (cid:15) (cid:15) X β / / RΓ a ( Y ) . This shows the isomorphism we seek. (cid:3)
Here is the main result of our paper, similar results can be found in [17, Section 7].
Theorem 6.3.
Let a be a weakly proregular ideal of R . (1) For any X ∈ D N ( R ) , the morphism LΛ a ( σ R X ) : LΛ a (RΓ a ( X )) → LΛ a ( X ) is an iso-morphism. (2) For any X ∈ D N ( R ) , the morphism RΓ a ( τ L X ) : RΓ a ( X ) → RΓ a (LΛ a ( X )) is an isomor-phism. (3) For any X ∈ D N ( R ) , one has RΓ a ( X ) ∈ D N ( R ) a - tor and LΛ a ( X ) ∈ D N ( R ) a - com . (4) The functors RΓ a : D N ( R ) a - com ⇄ D N ( R ) a - tor : LΛ a form an equivalence.Proof. (1) By Corollary 5.4 and Theorem 5.5, we have a commutative diagramLΛ a (RΓ a ( X )) ≃ (cid:15) (cid:15) LΛ a ( σ R X ) / / LΛ a ( X ) ≃ (cid:15) (cid:15) RHom R (RΓ a ( R ) , RΓ a ( X )) RHom R (RΓ a ( R ) ,σ R X ) / / RHom R (RΓ a ( R ) , X ) . ince RΓ a ( R ) ∈ D N ( R ) a -tor , it follows from Proposition 6.2(1) that RHom R (RΓ a ( R ) , σ R X ) isan isomorphism, so is LΛ a ( σ R X ).(2) By Corollary 5.4 and Theorem 5.5, one has a commutative diagramRΓ a ( R ) ⊗ L R X ≃ (cid:15) (cid:15) RΓ a ( R ) ⊗ L R τ L X / / RΓ a ( R ) ⊗ L R LΛ a ( X ) ≃ (cid:15) (cid:15) RΓ a ( X ) RΓ a ( τ L X ) / / RΓ a (LΛ a ( X )) . Let E be a faithful injective R -module. Applying the functor RHom R ( − , E ) to RΓ a ( R ) ⊗ L R τ L X ,we have a commutative diagram:RHom R (RΓ a ( R ) ⊗ L R LΛ a ( X ) , E ) ≃ (cid:15) (cid:15) RHom R (RΓ a ( R ) ⊗ L R τ L X ,E ) / / RHom R (RΓ a ( R ) ⊗ L R X, E ) ≃ (cid:15) (cid:15) RHom R (LΛ a ( X ) , LΛ a ( E )) RHom R ( τ L X , LΛ a ( E )) / / RHom R ( X, LΛ a ( E )) . Since LΛ a ( X ) ∈ D N ( R ) a -com , it follows from Proposition 6.2(2) that RHom R ( τ L X , LΛ a ( E ))is an isomorphism, so is RHom R (RΓ a ( R ) ⊗ L R τ L X , E ) = Hom R (RΓ a ( R ) ⊗ L R τ L X , E ). But E isfaithful injective, so RΓ a ( R ) ⊗ L R τ L X is an isomorphism, and hence RΓ a ( τ L X ) is an isomorphism.(3) This is immediate from the idempotence of the functors RΓ a and LΛ a .(4) By (1), there are functorial isomorphisms X ∼ = LΛ a ( X ) ∼ = LΛ a (RΓ a ( X )) for X ∈ D N ( R ) a -com .By (2), there are functorial isomorphisms X ∼ = RΓ a ( X ) ∼ = RΓ a (LΛ a ( X )) for X ∈ D N ( R ) ( a )-tor .These isomorphisms yield the desired equivalence. (cid:3) Remark 6.4.
Let a be a weakly proregular ideal of R . For any X, Y ∈ D N ( R ), one hasthat the morphismsRHom R (RΓ a ( X ) , RΓ a ( Y )) RHom R (RΓ a ( τ L X ) , ←−−−−−−−−−−− RHom R (RΓ a (LΛ a ( X )) , RΓ a ( Y )) adjunction −−−−−−→ RHom R (LΛ a ( X ) , LΛ a (RΓ a ( Y ))) RHom R (1 , LΛ a ( σ R Y )) −−−−−−−−−−−→ RHom R (LΛ a ( X ) , LΛ a ( Y ))is an isomorphism in D N ( R ). . Invariant
In this section, we prove that over a commutative noetherian ring, via Koszul cohomology,via RHom cohomology (resp. ⊗ cohomology) and via local cohomology (resp. derivedcompletion), all yield the same invariant. Lemma 7.1.
Let x = x , ..., x d be a sequence of R and X an N -complex. For t = 1 , · · · , N − , one has ( x )H t ( x ; X ) = 0 = ( x )H t (Hom R ( K • ( x ; R ); X )) .Proof. For each x i , the morphism of N -complexes K • ( x i ; R ) → K • ( x i ; R ) given by multipli-cation by x i can be factored as follows: K • ( x i ; R ) : 0 / / R x i (cid:15) (cid:15) R x i (cid:15) (cid:15) · · · R x i (cid:15) (cid:15) R x i / / x i (cid:15) (cid:15) R / / D N ( R ) : 0 / / R R · · ·
R M M / / x i (cid:15) (cid:15) K • ( x i ; R ) : 0 / / R R · · ·
R R x i / / R / / x i is null-homotopic on K • ( x i ; R ), so the same hold for K • ( x ; R ) ⊗ R X and Hom R ( K • ( x ; R ); X ). Therefore, x i H t ( x ; X ) = 0 = x i H t (Hom R ( K • ( x ; R ); X ) for each x i ,and hence for x = x , ..., x d , as desired. (cid:3) Lemma 7.2.
Let x = x , ..., x d be a sequence of R and a the ideal generated by x , and let X be an N -complex. For t = 1 , · · · , N − , one has Hom R ( K • ( x ; R ) , X ) ∼ = RΓ a (Hom R ( K • ( x ; R ) , X )) .Proof. By the definition of a -torsion functor, we have the following isomorphismsRΓ a (Hom R ( K • ( x ; R ) , X )) ∼ = lim −→ RHom R ( R/ a s , Hom R ( K • ( x ; R ) , X )) ∼ = lim −→ RHom R ( R/ a s ⊗ R K • ( x ; R ) , X ) ∼ = Hom R ( K • ( x ; R ) , lim −→ RHom R ( R/ a s , X )) ∼ = Hom R ( K • ( x ; R ) , RΓ a ( X )) ∼ = Hom R ( K • ( x ; R ) , X ) , where the fifth one is by K • ( x ; R ) ∈ D N ( R ) a -tor and Proposition 6.2(1). (cid:3) Lemma 7.3.
Let x be an element in R . For i ∈ Z and a fixed t , one has H it (Hom R ( K ( x ; R ) , X )) = 0 implies H it (Hom R ( K ( x s ) , X )) = 0 for s > .Proof. By Octahedral axiom, we have a commutative diagram in K N ( R ) (cid:15) (cid:15) / / K ( x ; R ) (cid:15) (cid:15) K ( x ; R ) (cid:15) (cid:15) / / (cid:15) (cid:15) R / / K ( x ; R ) (cid:15) (cid:15) / / Σ R x (cid:15) (cid:15) x / / Σ RR (cid:15) (cid:15) / / K ( x ; R ) (cid:15) (cid:15) / / Σ R (cid:15) (cid:15) x / / Σ R (cid:15) (cid:15) / / Σ K ( x ; R ) Σ K ( x ; R ) / / K N ( R ). Applying the functor RHom R ( − , X )to the second column, one gets an exact triangleΣ − Hom R ( K ( x ; R ) , X ) → Hom R ( K ( x ; R ) , X ) → Hom R ( K ( x ; R ) , X ) → Hom R ( K ( x ; R ) , X ),which implies that H it (Hom R ( K ( x ; R ) , X )) = 0 whenever H it (Hom R ( K ( x ; R ) , X )) = 0. Byrepeating this process we get the claim. (cid:3) For an N -complex X , set σ n X : · · · d n − N −−−→ X n − N +1 d N − N +1 −−−−−→ Z n − N +2 N − ( X ) d n − N +2 −−−−→ · · · d n +1 −−−→ Z n ( X ) → σ > n X : 0 → C nN − ( X ) ¯ d n −→ · · · ¯ d N + N − −−−−−→ C n ( X ) ¯ d N + N − −−−−−→ X n + N − d N + N − −−−−−→ · · · .The next result shows that Koszul cohomology, RHom cohomology and local cohomologyyield the same invariant, which was proved by Foxby and Iyengar [7] for N = 2 (see [7,Theorem 2.1]). Theorem 7.4.
Let a be an ideal of a noetherian ring R and K the Koszul N -complex on asequence of n generators for a . For any X ∈ D N ( R ) and a fixed t , one has inf { ℓ ∈ Z | H ℓt (RHom R ( R/ a , X )) = 0 } = inf { ℓ ∈ Z | H ℓt, a ( X ) = 0 } = inf { ℓ ∈ Z | H ℓt (Hom R ( K, X )) = 0 } . Proof.
Denote the three numbers in question a, b, c , respectively.For an R/ a -module T , one can set P ν → T be a semi-projective resolution such that P i = 0for all i > R ( T, X ) ∼ = Hom R ( P, X ) ∼ = Hom R/ a ( P, RHom R ( R/ a , X )) ∼ = Hom R/ a ( P, σ > a (RHom R ( R/ a , X ))) , here the last isomorphism is by the dual of [12, Lemma 3.9]. For n < a and i ∈ Z , one ofthe inequalities i > n + i < a holds, so Hom R ( P, X ) n = Q i ∈ Z Hom R ( P i , X n + i ) = 0. SoH ℓt (RHom R ( T, X )) = 0 for ℓ < a. ( † )Apply the functor RHom R ( − , X ) to the exact triangle a s / a s +1 → R/ a s +1 → R/ a s → Σ a s / a s +1 in D N ( A ) yields the long exact sequence · · · → H ℓt (RHom R ( R/ a s , X )) → H ℓt (RHom R ( R/ a s +1 , X )) → H ℓt (RHom R ( a s / a s +1 , X )) → · · · .Then ( † ) implies that H ℓt (RHom R ( a s / a s +1 , X )) = 0 for ℓ < a . By the induction hypothesisand the long exact sequence above, we get thatH ℓt (RHom R ( R/ a s +1 , X )) = 0 for ℓ < a and s > ℓt (RΓ a ( X )) = 0 for ℓ < a . On the other hand, let X ≃ → I be a semi-injectiveresolution. Then the inverse system of epimorphisms · · · ։ R/ a ։ R/ a ։ R/ a induces adirect system of monomorphismsHom R ( R/ a , I ) Hom R ( R/ a , I ) Hom R ( R/ a , I ) · · · So H ℓt (RHom R ( R/ a , X )) = 0 for ℓ < b . This shows that a = b .By Lemma 7.2 and construction of K , one has that Hom R ( K, X ) ≃ RΓ a (Hom R ( K, X )) ≃ Hom R ( K, RΓ a ( X )). Hence we get H ℓt (Hom R ( K, X )) = 0 for ℓ < b . On the other hand, onehas that H ℓt, a ( X ) = 0 for ℓ < c by Lemma 7.3. This shows the equality b = c . (cid:3) The next result shows that Koszul cohomology, ⊗ cohomology and derived completionyield the same invariant, which was proved by Foxby and Iyengar [7] for N = 2 (see [7,Theorem 4.1]). Theorem 7.5.
Let a be an ideal of a noetherian ring R and K the Koszul N -complex on asequence of n generators for a . For any X ∈ D N ( R ) and a fixed t , one has sup { ℓ | H ℓt ( R/ a ⊗ L R X ) = 0 } = sup { ℓ | H ℓt (LΛ a ( X )) = 0 } = sup { ℓ | H ℓt ( K ⊗ R X ) = 0 } . Proof.
Denote the three numbers in question a, b, c , respectively.Let P ≃ → X be a semi-projective resolution. Then the inverse system of epimorphisms · · · ։ R/ a ։ R/ a ։ R/ a induces an inverse system of epimorphisms · · · ։ R/ a ⊗ R P ։ R/ a ⊗ R P ։ R/ a ⊗ R P .So H ℓt ( R/ a ⊗ L R X ) = 0 for ℓ > b . To the opposite inequality, note that T ⊗ L R X ∼ = T ⊗ L R/ a R/ a ⊗ L R X for any R/ a -module T . By analogy with the proof of Theorem 7.4, one hasH ℓt ( T ⊗ L R X ) = 0 for ℓ > a. ( ‡ ) pply − ⊗ L R X to the exact triangle a s / a s +1 → R/ a s +1 → R/ a s → Σ a s / a s +1 in D N ( R ) yieldsthe following long exact sequence · · · → H ℓt ( a s / a s +1 ⊗ L R X ) → H ℓt ( R/ a s +1 ⊗ L R X ) → H ℓt ( R/ a s ⊗ L R X ) → · · · .Then ( ‡ ) yields that H ℓt ( a s / a s +1 ⊗ L R X ) = 0 for ℓ > a . By the induction hypothesis and thelong exact sequence above, we get thatH ℓt ( R/ a s +1 ⊗ L R X ) = 0 for ℓ > a and s > a = b .Let E be a faithful injective R -module. We have the following equivalencesH ℓt ( K ⊗ R X ) = 0 for ℓ > a ⇐⇒ H − ℓN − t (Hom R ( K, Hom R ( X, E ))) = 0 for ℓ > a ⇐⇒ H − ℓN − t (RHom R ( R/ a , Hom R ( X, E ))) = 0 for ℓ > a ⇐⇒ H ℓt ( R/ a ⊗ R X ) = 0 for ℓ > a, where the third equivalence is by Theorem 7.4. This shows the equality a = c . (cid:3) ACKNOWLEDGEMENTS
This research was partially supported by National Natural Science Foundation of China(11761060).
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