On graded Gorenstein injective dimension
aa r X i v : . [ m a t h . A C ] J un ON GRADED GORENSTEIN INJECTIVE DIMENSION
AFSANEH ESMAEELNEZHAD AND PARVIZ SAHANDI
Abstract.
There are nice relations between graded homological dimensionsand ordinary homological dimensions. We study the Gorenstein injective di-mension of a complex of graded modules denoted by ∗ Gid, and derive itsproperties. In particular we prove the Chouinard’s like formula for ∗ Gid, andcompare it with the usual Gorenstein injective dimension. Introduction
Let R be a Noetherian Z -graded ring. In [9] and [10], Fossum and Fossum-Foxbyhave studied the graded homological dimension of graded modules and comparethem with classical homological dimensions. They shown that for a graded R -module M , one has ∗ id R M ≤ id R M ≤ ∗ id R M + 1 where id R M (resp. ∗ id R M )denotes for the injective dimension of M in the category of R -modules (resp. cat-egory of graded R -modules). It is natural to ask how these inequalities hold forthe Gorenstein injective dimension Gid R M . In this paper we give an answer thisquestion. Section 2 of this paper is devoted to review some hyper-homologicalalgebra for the derived category of the graded ring R . In Section 3 we definethe ∗ injective dimension of complexes of graded modules and homogeneous homo-morphisms, and derived its properties. In the final section we define the gradedGorenstein injective dimension of complexes of graded modules and homogeneoushomomorphisms denoted by ∗ Gid. Among other results we show that if thegraded ring R admits a ∗ dualizing complex, or is a non-negatively graded ring,then ∗ Gid R X ≤ Gid R X ≤ ∗ Gid R X + 1, where Gid R X is the Gorenstein in-jective dimension of X over R (see Corollary 4.19). Also in this case we prove aChouinard’s like formula for ∗ Gid R X (see Theorem 4.18). Our source of gradedrings and modules are [3] and [10].Throughout this paper R is a commutative Noetherian Z -graded ring.2. Derived category of complexes of graded modules
We use the notation from the appendix of [5]. Let X be a complex of R -modulesand R -homomorphisms. The supremum and the infimum of a complex X , denotedby sup( X ) and inf( X ) are defined by the supremum and infimum of { i ∈ Z | H i ( X ) =0 } . If m is an integer and X is a complex, then Σ m X denotes the complex X shifted m degrees to the left; it is given by(Σ m X ) ℓ = X ℓ − m and ∂ Σ m Xℓ = ( − m ∂ Xℓ − m Mathematics Subject Classification.
Key words and phrases. graded rings, graded modules, Chouinard’s formula, injective dimen-sion, Gorenstein injective dimension.P. Sahandi was supported in part by a grant from IPM (No. 91130030). for ℓ ∈ Z .The symbol D ( R ) denotes the derived category of R -complexes. The full sub-categories D ⊏ ( R ), D ⊐ ( R ), D (cid:3) ( R ) and D ( R ) of D ( R ) consist of R -complexes X while H ℓ ( X ) = 0, for respectively ℓ ≫ ℓ ≪ | ℓ | ≫ ℓ = 0. Homologyisomorphisms are marked by the sign ≃ . The right derived functor of the homo-morphism functor of R -complexes and the left derived functor of the tensor productof R -complexes are denoted by R Hom R ( − , − ) and − ⊗ L R − , respectively.Let M = ⊕ n ∈ Z M n and N = ⊕ n ∈ Z N n be two graded R -modules. The ∗ Homfunctor is defined by ∗ Hom R ( M, N ) = L i ∈ Z Hom i ( M, N ), such that Hom i ( M, N )is a Z -submodule of Hom R ( M, N ) consisting of all ϕ : M → N such that ϕ ( M n ) ⊆ N n + i for all n ∈ Z . In general ∗ Hom R ( M, N ) = Hom R ( M, N ) but equality holdsif M is finitely generated, see [3, Exercise 1.5.19]. Also the tensor product M ⊗ R N of M and N is a graded module with ( M ⊗ R N ) n is generated (as a Z -module) byelements m ⊗ n with m ∈ M i and n ∈ N j where i + j = n .Let { M α } α ∈ I be a family of graded R -modules. Then L α M α becomes a graded R -module with ( L α M α ) n = L α ( M α ) n for all n ∈ Z , see [10, Page 289]. Alsorecall that the direct product exists in the category of graded modules. Then thedirect product is denoted by ∗ Q α M α and ( ∗ Q α M α ) n = Q α ( M α ) n for all n ∈ Z ,see [10, Page 289]. In this case there are the following bijections [10, Page 289] ∗ Hom R ( M α M α , − ) ∼ = −→ ∗ Y α ∗ Hom R ( M α , − ) , ∗ Hom R ( − , ∗ Y α M α ) ∼ = −→ ∗ Y α ∗ Hom R ( − , M α ) . Likewise direct and inverse limits are exists in the category of graded modules with( ∗ lim −→ M α ) n = lim −→ ( M α ) n , ( ∗ lim ←− M α ) n = lim ←− ( M α ) n , see [10, Page 289]. Let ( R, m ) be a ∗ local (Noetherian) ring R , that is, a gradedring with a unique homogeneous maximal ideal m . The m - ∗ adic completion of R is ∗ b R = ∗ lim ←− R/ m n , which is a Noetherian graded ring by [9, Corollary VIII.2]. It is known that the m - ∗ adic completion ∗ b R is a flat R -module [10, Corollary 3.3], and that if E := ∗ E R ( R/ m ) is the ∗ injective envelope of R/ m over R , then ∗ Hom R ( E, E ) ∼ = ∗ b R [9,Theorem VIII.3].The symbol ∗ C ( R ) denotes the category of complexes of graded R -modules andhomogeneous differentials. Note that the category of graded modules is an abeliancategory, hence ∗ C ( R ) has a derived category, (see [15]), which will be denotedby ∗ D ( R ). Analogously we have ∗ C ⊏ ( R ), ∗ C ⊐ ( R ), ∗ C (cid:3) ( R ) and ∗ C ( R ) (resp. ∗ D ⊏ ( R ), ∗ D ⊐ ( R ), ∗ D (cid:3) ( R ) and ∗ D ( R )) which are the full subcategories of ∗ C ( R )(resp. ∗ D ( R )). If we use the notation X ∈ ∗ C ( ♯ ) ( R ), we mean H( X ) ∈ ∗ C ♯ ( R ).For R -complexes X and Y of graded modules, with homogeneous differentials ∂ X and ∂ Y , we define the homomorphism complex ∗ Hom R ( X, Y ) as follows: ∗ Hom R ( X, Y ) ℓ = ∗ Y p ∈ Z ∗ Hom R ( X p , Y p + ℓ ) N GRADED GORENSTEIN INJECTIVE DIMENSION 3 and when ψ = ( ψ p ) p ∈ Z belongs to ∗ Hom R ( X, Y ) ℓ the family ∂ ∗ Hom R ( X,Y ) ℓ ( ψ ) in ∗ Hom R ( X, Y ) ℓ − has p -th component ∂ ∗ Hom R ( X,Y ) ℓ ( ψ ) p = ∂ Yp + ℓ ψ p − ( − ℓ ψ p − ∂ Xp . When X ∈ ∗ C f ⊐ ( R ) and Y ∈ ∗ C ⊏ ( R ) all the products ∗ Q p ∈ Z ∗ Hom R ( X p , Y p + ℓ )are finite. Thus using [3, Exercise 1.5.19], we have ∗ Y p ∈ Z ∗ Hom R ( X p , Y p + ℓ ) = M p ∈ Z Hom R ( X p , Y p + ℓ ) , for every ℓ ∈ Z . Therefore ∗ Hom R ( X, Y ) = Hom R ( X, Y ).We also define the tensor product complex X ⊗ R Y as follows:( X ⊗ R Y ) ℓ = M p ∈ Z ( X p ⊗ R Y ℓ − p )and the ℓ -th differential ∂ X ⊗ R Yℓ is given on a generator x p ⊗ y ℓ − p in ( X ⊗ R Y ) ℓ ,where x p and y ℓ − p are homogeneous elements, by ∂ X ⊗ R Yℓ ( x p ⊗ y ℓ − p ) = ∂ Xp ( x p ) ⊗ y ℓ − p + ( − p x p ⊗ ∂ Yℓ − p ( y ℓ − p ) . If X and Y are R -complexes of graded modules, then ∗ Hom R ( X, − ), ∗ Hom R ( − , Y ),and X ⊗ R − are functors on ∗ C ( R ).Note that any object of ∗ C ⊏ ( R ) has an ∗ injective resolution by [15, Page 47],and any object of ∗ C ⊐ ( R ) has an ∗ projective resolution by [15, Page 48]. The rightderived functor of the ∗ Hom functor in the category of graded complexes is denotedby R ∗ Hom R ( − , − ) and set ∗ Ext iR ( − , − ) = H − i ( R ∗ Hom R ( − , − )). It is easily seenthat if R is a Noetherian Z -graded ring and X a homologically finite complex ofgraded modules and Y ∈ ∗ C ( R ) then R ∗ Hom R ( X, Y ) = R Hom R ( X, Y ). Also theleft derived functor of − ⊗ R − in the category of graded complexes is denoted by − ⊗ L ∗ R − . Since ∗ projective graded R -modules coincide with projective R -modulesby [10, Proposition 3.1] we easily see that − ⊗ L ∗ R − coincides with the ordinary leftderived functor of − ⊗ R − in the category of complexes. So we use − ⊗ L R − insteadof − ⊗ L ∗ R − .For the homomorphism and the tensor product functors we have the followinguseful proposition, see [5, A.2.8, A.2.10, and A.2.11] for the ungraded case. Theproof is the same as the ungraded case so we omit it. Proposition 2.1.
Let S be a graded ring which is an R -algebra. ∗ Adjointness.
Let
Z, Y ∈ ∗ C ( S ) and X ∈ ∗ C ( R ) . Then Z ⊗ S Y ∈ ∗ C ( S ) and ∗ Hom R ( Y, X ) ∈ ∗ C ( S ) , and there is an isomorphism of S -complexes ρ ZY X : ∗ Hom R ( Z ⊗ S Y, X ) ∼ = −→ ∗ Hom S ( Z, ∗ Hom R ( Y, X )) , which is natural in Z, Y and X . ∗ Tensor-Evaluation.
Let
Z, Y ∈ ∗ C ( S ) and X ∈ ∗ C ( R ) . Then ∗ Hom S ( Z, Y ) ∈ ∗ C ( R ) and Y ⊗ R X ∈ ∗ C ( S ) , and there is a natural morphism of S -complexes ω ZY X : ∗ Hom S ( Z, Y ) ⊗ R X −→ ∗ Hom S ( Z, Y ⊗ R X ) . The morphism is invertible under each of the next two extra conditions (i) Z ∈ ∗ C fp (cid:3) ( S ) , Y ∈ ∗ C ⊐ ( S ) , and X ∈ ∗ C ⊐ ( R ) ; or (ii) Z ∈ ∗ C fp (cid:3) ( S ) , Y ∈ ∗ C ⊏ ( S ) , and X ∈ ∗ C (cid:3) ( R ) . AFSANEH ESMAEELNEZHAD AND PARVIZ SAHANDI ∗ Hom-Evaluation.
Let
Z, Y ∈ ∗ C ( S ) and X ∈ ∗ C ( R ) . Then ∗ Hom S ( Z, Y ) ∈ ∗ C ( R ) and ∗ Hom R ( Y, X ) ∈ ∗ C ( S ) , and there is a natural morphism of S -complexes θ ZY X : Z ⊗ S ∗ Hom R ( Y, X ) −→ ∗ Hom R ( ∗ Hom S ( Z, Y ) , X ) . The morphism is invertible under each of the next two extra conditions (i) Z ∈ ∗ C fp (cid:3) ( S ) , Y ∈ ∗ C ⊐ ( S ) , and X ∈ ∗ C ⊏ ( R ) ; or (ii) Z ∈ ∗ C fp ⊐ ( S ) , Y ∈ ∗ C ⊏ ( S ) , and X ∈ ∗ C (cid:3) ( R ) .By Z ∈ ∗ C fp ( S ) we mean that Z consists of finitely generated projective S -modules. We recall the definition of the depth and width of complexes. Let a be an idealin a ring R and X a complex of graded R -modules. The a -depth and a -width of X over R are defined respectively bydepth( a , X ) := − sup R Hom R ( R/ a , X ) , width( a , X ) := inf( R/ a ⊗ L R X ) . If ( R, m ) is a local ring then set depth R X := depth( m , X ) and width R X :=width( m , X ).Now let ( R, m ) be a ∗ local graded ring and X be a complex of graded R -modules.By [3, Proposition 1.5.15(c)], − ⊗ R R m is a faithfully exact functor on the categoryof graded R -modules. Then we havewidth( m , X ) = inf { i | H i ( R/ m ⊗ L R X ) = 0 } = inf { i | H i ( R/ m ⊗ L R X ) ⊗ R R m = 0 } = inf { i | H i ( R m / m R m ⊗ L R m X m ) = 0 } = width( m R m , X m ) = width R m X m . Likewise we have depth( m , X ) = depth R m X m . Proposition 2.2.
Let ( R, m , k ) be a ∗ local ring and X ∈ C ⊐ ( R ) and set E := ∗ E R ( k ) . Then width( m , X ) = depth( m , R ∗ Hom R ( X, E )) . Proof.
We have the following computations:width( m , X ) = inf( k ⊗ L R X ) = − sup R ∗ Hom R ( k ⊗ L R X, E )= − sup R ∗ Hom R ( k, R ∗ Hom R ( X, E ))= depth( m , R ∗ Hom R ( X, E )) . The second equality hods since E is faithfully ∗ injective and the third one uses the ∗ adjointness isomorphism. (cid:3) The following lemma is the graded version of one of Foxby’s accounting principles[5, Lemma A.7.9].
Lemma 2.3.
Let ( R, m , k ) be a ∗ local ring. Then (a) If X ∈ ∗ C ( k ) , then there is a quasiisomorphism H ( X ) → X . (b) If X ∗ C ( ⊏ ) ( R ) and W ∈ ∗ C ( k ) then inf R ∗ Hom R ( W, X ) = inf R ∗ Hom R ( k, X ) − sup W. N GRADED GORENSTEIN INJECTIVE DIMENSION 5
Proof.
Part ( a ) is easy since very graded k -module is free by [3, Exercise 1.5.20].For part ( b ) note that W = k ⊗ L k W . Then using the ∗ adjointness isomorphism wehave R ∗ Hom R ( W, X ) = R ∗ Hom k ( W, R ∗ Hom R ( k, X )) = ∗ Hom k ( W, R ∗ Hom R ( k, X )) . Thus using part ( a ) we haveinf R ∗ Hom R ( W, X ) = inf { ℓ | R ∗ Hom R ( W, X ) ℓ = 0 } = inf { ℓ | ∗ Y i + j = ℓ ∗ Hom k ( W − j , R ∗ Hom R ( k, X ) i ) = 0 } = inf R ∗ Hom R ( k, X ) − sup W. This completes the proof. (cid:3)
The following equality is the graded version of [20, Lemma 2.6].
Proposition 2.4.
Let ( R, m , k ) be a ∗ local ring X, Y ∈ ∗ C (cid:3) ( R ) , then width( m , R ∗ Hom R ( X, Y )) = depth( m , X ) + width( m , Y ) − depth( m , R ) . Proof.
Use Lemma 2.3 and the same argument as [20, Lemma 2.6]. (cid:3)
Let a ∈ R be homogeneous and set α = deg( a ). Then the complex 0 → R ( − α ) a → R → Koszul complex of a , anddenoted by K ( a ), where R ( − α ) denotes the graded R -module with grading givenby R ( − α ) n = R n − α . Note that K ( a ) ∈ ∗ C ( R ). Now let a be a homogeneous idealof R and a , · · · , a n be a set of generators of a by homogeneous elements. TheKoszul complex of a , denoted by K := K ( a ), and define as K ( a ) ⊗ R · · · ⊗ R K ( a n ).It is shown in [12] that width( a , X ) = inf( K ⊗ R X ).Let α : X → Y be a homogeneous morphism between R -complexes of graded R -modules. The mapping cone of α is a complex of graded R -modules given by M ( α ) ℓ : Y ℓ ⊕ X ℓ − and δ M ( α ) ℓ ( y ℓ , x ℓ − ) = ( δ Yℓ ( y ℓ ) + α ℓ − ( x ℓ − ) , − δ Xℓ − ( x ℓ − )) . Note that δ M ( α ) ℓ is a homogeneous differentiation from Y ℓ ⊕ X ℓ − to Y ℓ − ⊕ X ℓ − .It is easy to see that the morphism α : X → Y between complexes of graded R -modules is quasi-isomorphism if and only if the mapping cone of α is homologicallytrivial (see [5, Lemma A.1.19]). Also for the covariant and contravariant functors ∗ Hom R ( V, − ) and ∗ Hom R ( − , W ) we have the following: M ( ∗ Hom R ( V, α )) = ∗ Hom R ( V, M ( α )) , M ( ∗ Hom R ( α, W )) = Σ ∗ Hom R ( M ( α ) , W ) . See [5, A2.1.2 and A.2.1.4].The ungraded version of the following result contained in [7, Proposition 2.7].
Proposition 2.5.
Let B be a class of graded R -modules, and α : X → Y be aquasiisomorphism between complexes of graded R -modules such that ∗ Hom R ( α, V ) : ∗ Hom R ( Y, V ) ≃ −→ ∗ Hom R ( X, V ) AFSANEH ESMAEELNEZHAD AND PARVIZ SAHANDI is quasiisomorphism for every module V ∈ B . Let e V ∈ ∗ C ( R ) be a complex con-sisting of modules from B . Then the induced morphism, ∗ Hom R ( α, e V ) : ∗ Hom R ( Y, e V ) −→ ∗ Hom R ( X, e V ) , is a quasiisomorphism, provided that either (a) e V ∈ ∗ C ⊏ ( R ) , or (b) X, Y ∈ ∗ C ⊏ ( R ) .Proof. In either cases it is enough to show that ∗ Hom R ( M ( α ) , e V ) is homologi-cally trivial. On the other hand by graded version of [7, Lemma 2.5] the resultholds if ∗ Hom R ( M ( α ) , e V ℓ ) is homologically trivial for each ℓ ∈ Z which this is ourassumption. (cid:3) ∗ injective dimension The injective dimension of a complex X is defined and studied in [1], denotedby id R X . A graded module J is called ∗ injective if it is an injective object inthe category of graded modules. The injective dimension of a graded module M in the category of graded modules, is denoted by ∗ id R M (cf. [10, 15, 3]). The ∗ injective dimension of a complex of graded modules X is studied in [15, Page 83].Let n ∈ Z . A homologically left bounded complex of graded modules X , is saidto have ∗ injective dimension at most n , denoted by ∗ id R X ≤ n , if there exists an ∗ injective resolution X → I , such that I i = 0 for i < − n . If ∗ id R X ≤ n holds,but ∗ id R X ≤ n − ∗ id R X = n . If ∗ id R X ≤ n for all n ∈ Z we write ∗ id R X = −∞ . If ∗ id R X ≤ n for no n ∈ Z we write ∗ id R X = ∞ . Thefollowing theorem inspired by [1, Theorem 2.4.I and Corollary 2.5.I]. Theorem 3.1.
For X ∈ ∗ D ⊏ ( R ) and n ∈ Z the following are equivalent: (1) ∗ id R X ≤ n. (2) n ≥ − sup U − inf( R ∗ Hom R ( U, X )) for all U ∈ ∗ D (cid:3) ( R ) and H ( U ) = 0 . (3) n ≥ − inf X and ∗ Ext n +1 R ( R/J, X ) = 0 for every homogeneous ideal J of R . (4) n ≥ − inf X and for any (resp. some) ∗ injective resolution I of X , thegraded R -module Ker( ∂ − n : I − n → I − n − ) is ∗ injective.Moreover the following hold: ∗ id R X = sup { j ∈ Z | ∗ Ext jR ( R/J, X ) = 0 for some homogeneous ideal J } = sup {− sup( U ) − inf( R ∗ Hom R ( U, X )) | U ≇ in ∗ D (cid:3) ( R ) } . Proof. (1) ⇒ (2) Let t := sup U and I be an ∗ injective resolution of X , such that,for all i < − n , I i = 0. Then we have ∗ Ext iR ( U, X ) ∼ = H − i ( ∗ Hom R ( U, I )) . Since ∗ Hom R ( U, I ) − i = 0 for − i < − n − t , the assertion follows.(2) ⇒ (3) It is trivial that ∗ Ext n +1 R ( R/J, X ) = 0 for every homogeneous ideal J of R . For the second assertion let U = R in (2). So that Ext iR ( R, X ) = ∗ Ext iR ( R, X ) = 0 for i > n . Now by [1, Lemma 1.9(b)], we have H − i ( X ) = 0for − i < − n . This means that n ≥ − inf X .(3) ⇒ (4) By hypothesis of (4) H i ( I ) = 0 for i < − n . Thus the complex · · · → → → I − n → I − n − → · · · → I i → I i − → · · · , N GRADED GORENSTEIN INJECTIVE DIMENSION 7 gives an ∗ injective resolution of Ker ∂ − n . In particular ∗ Ext R ( R/J,
Ker ∂ − n ) = H − n − ∗ Hom R ( R/J, I ) = ∗ Ext n +1 R ( R/J, X ) = 0for every homogeneous ideal J of R . Thus Ker ∂ − n is ∗ injective by [10, Corollary4.3].(4) ⇒ (1) Let I be any ∗ injective resolution of X . By (5) we have Ker ∂ − n is ∗ injective. Thus ∗ id R X < − n by definition.The next two equalities are trivial. (cid:3) For a local ring ( R, m , k ) and for an R -complex X and i ∈ Z the i th Bass numberand Betti number of X are defined respectively by µ iR ( X ) := dim k H − i ( R Hom R ( k, X ))and β Ri ( X ) := dim k H i ( k ⊗ L R X ) . It is well-known that for X ∈ D ⊏ ( R ) one has (cf.[1, Proposition 5.3.I])id R X = sup { m ∈ Z |∃ p ∈ Spec( R ); µ mR p ( X p ) = 0 } . As a graded analogue we have:
Proposition 3.2.
For X ∈ ∗ D ⊏ ( R ) we have the following equality ∗ id R X = sup { m ∈ Z |∃ p ∈ ∗ Spec( R ); µ mR p ( X p ) = 0 } . Proof.
The argument is the same as proof of [1, Proposition 5.3.I] with somechanges. Denote the supremum by i . By Theorem 3.1, we have ∗ id R X ≥ i . Hencethe equality holds if i = ∞ . Thus assume that i is finite. By Theorem 3.1 we haveto show that if ∗ Ext jR ( M, X ) = 0 for some finitely generated graded R -module M ,then j ≤ i ; this implies that ∗ id R X ≤ i . The elements of Ass ( M ) are homogeneousprime ideals. Thus we have a filtration 0 = M ⊂ M ⊂ · · · ⊂ M t = M of gradedsubmodules of M such that for each i we have M i /M i − ∼ = R/ p i with p i ∈ Supp M and is homogeneous. From the long exact sequence of ∗ Ext jR ( − , X ) = 0 we havethe set { q ∈ Spec( R ) | there is an h ≥ j such that ∗ Ext hR ( R/ q , X ) = 0 } , is not empty. Let p maximal in this set, and for a homogeneous x ∈ R \ p considerthe exact sequence 0 → R/ p x → R/ p → R/ ( p + Rx ) → . It induces an exact sequence ∗ Ext hR ( R/ ( p + Rx ) , X ) → ∗ Ext hR ( R/ p , X ) x → ∗ Ext hR ( R/ p , X ) → ∗ Ext h +1 R ( R/ ( p + Rx ) , X )in which the left-hand term is trivial because of the maximality of p . Thus ∗ Ext hR ( R/ p , X ) x → ∗ Ext hR ( R/ p , X ) is injective for all homogeneous elements x ∈ R \ p , hence so isthe homogeneous localization homomorphism ∗ Ext hR ( R/ p , X ) → ∗ Ext hR ( R/ p , X ) ( p ) .Thus the free R ( p ) / p R ( p ) -module ∗ Ext hR ( R/ p , X ) ( p ) is nonzero. Consequently( ∗ Ext hR ( R/ p , X ) ( p ) ) p R ( p ) ∼ = ∗ Ext hR p ( R p / p R p , X p )is nonzero. This implies that j ≤ h ≤ i . (cid:3) Remark 3.3. (1) A graded module is called ∗ projective if it is a projective object inthe category of graded modules. By [10, Proposition 3.1] the ∗ projective graded R -modules coincide with projective R -modules. The projective dimension of a gradedmodule M in the category of graded modules, is denoted by ∗ pd R M (cf. [10] ). Let n ∈ Z . A homologically right bounded complex of graded modules X , is said to AFSANEH ESMAEELNEZHAD AND PARVIZ SAHANDI have ∗ projective dimension at most n , denoted by ∗ pd R X ≤ n , if there exists a ∗ projective resolution P → X , such that P i = 0 for i > n . If ∗ pd R X ≤ n holds,but ∗ pd R X ≤ n − does not, we write ∗ pd R X = n . If ∗ pd R X ≤ n for all n ∈ Z we write ∗ pd R X = −∞ . If ∗ pd R X ≤ n for no n ∈ Z we write ∗ pd R X = ∞ .(2) For X ∈ ∗ D ⊐ ( R ) by the same method as in [1, Theorem 2.4.P and Corollary2.5.P] we have ∗ pd R X = sup { j ∈ Z | ∗ Ext jR ( X, N ) = 0 for some graded R -module N } = sup { inf( U ) − inf( R ∗ Hom R ( X, U )) | U ≇ in ∗ D (cid:3) ( R ) } . (3) It is easy to see that for X ∈ ∗ D ⊐ ( R ) , we have ∗ pd R X ≤ pd R X . The proof of the following proposition is easy so we omit it (see [3, Theorem1.5.9]). Let J be an ideal of the graded ring R . Then the graded ideal J ∗ isdenoted to the ideal generated by all homogeneous elements of J . It is well-knownthat if p is a prime ideal of R , then p ∗ is a homogeneous prime ideal of R by [3,Lemma 1.5.6]. Proposition 3.4.
Assume that X ∈ ∗ D (cid:3) ( R ) and p is a non homogeneous primeideal in R . Then µ i +1 R p ( X p ) = µ iR p ∗ ( X p ∗ ) and β R p i ( X p ) = β R p ∗ i ( X p ∗ ) for any integer i ≥ . Corollary 3.5.
Let X ∈ ∗ D (cid:3) ( R ) and p be a non-homogeneous prime ideal in R .Then depth X p = depth X p ∗ + 1 . Proof.
Using Proposition 3.4, we can assume that both depth X p and depth X p ∗ are finite. So the equality follows from the fact that over a local ring ( R, m , k ) wehave depth R X = inf { i ∈ Z | µ iR ( X ) = 0 } . (cid:3) Foxby in [11] defined the small support of a homologically right bounded complex X over a Noetherian ring R , denoted by supp R X , assupp R X = { p ∈ Spec R |∃ m ∈ Z : β R p m ( X p ) = 0 } . Let ∗ supp R X be a subset of supp R X consisting of homogeneous prime idealsof supp R X . Then from Proposition 3.4 we see that p ∈ supp R X if and only if p ∗ ∈ ∗ supp R X . Also using [11, Proposition 2.8] and Corollary 3.5 (or directlyfrom Proposition 3.4) we havewidth R p X p < ∞ ⇔ width R p ∗ X p ∗ < ∞ . Proposition 3.6.
Assume that X ∈ ∗ D (cid:3) ( R ) , and p is a non homogeneous primeideal in R . Then width R p X p = width R p ∗ X p ∗ Proof.
We can assume that both width X p and width X p ∗ are finite numbers. Andthe argument is dual to the proof of [3, Theorem 1.5.9]. (cid:3) The ungraded version of the following theorem was proved for modules byChouinard [4, Corollary 3.1] and extended to complexes by Yassemi [20, Theorem2.10].
Theorem 3.7.
Let X ∈ ∗ D (cid:3) ( R ) . If ∗ id R X < ∞ then ∗ id R X = sup { depth R p − width X p | p ∈ ∗ Spec( R ) } . N GRADED GORENSTEIN INJECTIVE DIMENSION 9
Proof.
We have the following computations ∗ id R X = sup { m ∈ Z |∃ p ∈ ∗ Spec( R ) : µ mR p ( M p ) = 0 } = sup { m ∈ Z |∃ p ∈ ∗ Spec( R ) : H m ( R Hom R p ( κ ( p ) , M p )) = 0 } = sup {− inf R Hom R p ( κ ( p ) , M p ) | p ∈ ∗ Spec( R ) } = sup { depth R p − width R p M p | p ∈ ∗ Spec( R ) } . The first equality is by Proposition 3.2 and the last one is by [20, Lemma 2.6(a)]. (cid:3)
The following corollary was already known for graded modules in [10, Corollary4.12].
Corollary 3.8.
For every X ∈ ∗ D (cid:3) ( R ) , we have ∗ id R X ≤ id R X ≤ ∗ id R X + 1 . Proof.
First of all note that by proposition 3.4, id R X < ∞ if and only if ∗ id R X < ∞ . The first inequality is clear by Theorem 3.7 and [20, Theorem 2.10]. For thesecond one let p ∈ Spec R be such that id R X = depth R p − width R p M p by [20,Theorem 2.10]. By Corollary 3.5 and Proposition 3.6 we havedepth R p − width R p M p ≤ depth R p ∗ − width R p ∗ M p ∗ + 1 ≤ ∗ id R X + 1 , where the second inequality holds by Theorem 3.7. (cid:3) Here we define the ∗ dualizing complex for a graded ring and prove some relatedresults that we need in the next section. Definition 3.9. A ∗ dualizing complex for a graded ring R is a homologically finiteand bounded complex D , of graded R -modules, such that ∗ id R D < ∞ and thehomothety morphism ψ : R → R ∗ Hom R ( D, D ) is invertible in ∗ D ( R ) . Corollary 3.10.
Any ∗ dualizing complex for R is a dualizing complex for R . The proof of the following lemma is the same as [15, Chapter V, Proposition3.4].
Lemma 3.11.
Let ( R, m , k ) be a ∗ local ring and that D is a ∗ dualizing complexfor R . Then there exists an integer t such that H t ( R ∗ Hom R ( k, D )) ∼ = k andH i ( R ∗ Hom R ( k, D )) = 0 for i = t . Assume that ( R, m ) is a ∗ local ring. A ∗ dualizing complex D is said to be normalized ∗ dualizing complex , if t = 0 in the lemma. It is easy to see that a suitableshift of any ∗ dualizing complex is a normalized one. Also using [15, Chapter V,Proposition 3.4] we see that if D is a normalized ∗ dualizing complex for ( R, m ),then D m is a normalized dualizing complex for R m . Lemma 3.12.
Let ( R, m , k ) be a ∗ local ring and that D is a normalized ∗ dualizingcomplex for R . Then there exists a natural functorial isomorphism on the categoryof graded modules of finite length to itself φ : H ( R ∗ Hom R ( − , D )) → ∗ Hom R ( − , ∗ E R ( k )) , where ∗ E R ( k ) is the ∗ injective envelope of k over R . Proof.
Since D is a normalized ∗ dualizing complex for R , T := H ( R ∗ Hom R ( − , D ))is an additive contravariant exact functor from the category of graded modules offinite length to itself. Let M be a graded R -module and m ∈ M is homoge-neous element of degree α . Then ǫ m : R ( − α ) → M is a homogeneous morphismwhich sends 1 into m . Thus we have a homogeneous morphism φ ( M ) : T ( M ) → ∗ Hom R ( M, T ( R )) which sends a homogeneous element x ∈ T ( M ) to a morphism f x ∈ ∗ Hom R ( M, T ( R )) such that f x ( m ) = T ( ǫ m )( x ) for every homogeneous ele-ment m ∈ M . It is easy to see that it is functorial on M . Thus we showed thatthere is a natural functorial morphism φ : T → ∗ Hom R ( − , T ( R )). Therefore bythe same method of [14, Lemma 4.4 and Propositions 4.5], there is a functorialisomorphism φ : H ( R ∗ Hom R ( − , D )) → ∗ Hom R ( − , ∗ lim −→ T ( R/ m n )) , from the category of graded modules of finite length to itself. Using the techniqueof proof of [14, Proposition 4.7] in conjunction with [10, Corollary 4.3], we seethat ∗ lim −→ T ( R/ m n ) is an ∗ injective R -module. Since D is a normalized ∗ dualizingcomplex for R we have ∗ Hom R ( k, ∗ lim −→ T ( R/ m n )) ∼ = H ( R ∗ Hom R ( k, D )) ∼ = k. Thus in particular we can embed k to ∗ lim −→ T ( R/ m n ). To show that ∗ lim −→ T ( R/ m n )is an ∗ essential extension of k , let Q be a graded submodule of ∗ lim −→ T ( R/ m n ) suchthat k ∩ Q = 0. Then ∗ Hom R ( k, Q ) can be embed in ∗ Hom R ( k, ∗ lim −→ T ( R/ m n )) ∼ = k. Therefore ∗ Hom R ( k, Q ) = 0. On the other hand Ass( T ( R/ m n )) is in V ( m ) foreach n ∈ N . Now by [19, Proposition 2.1], the fact that each prime ideal ofAss( ∗ lim −→ T ( R/ m n )) is the annihilator of a homogeneous element [3, Lemma 1.5.6],and the definition of ∗ lim −→ , we haveAss( ∗ lim −→ T ( R/ m n )) ⊆ [ n ∈ N Ass( T ( R/ m n )) ⊆ V ( m ) . Consequently Q has support in V ( m ), so that Q = 0. Therefore ∗ lim −→ T ( R/ m n ) ∼ = ∗ E R ( k ). (cid:3) Let a be an ideal of R . The right derived local cohomology functor with supportin a is denoted by R Γ a ( − ). Its right adjoint, L Λ a ( − ), is the left derived localhomology functor with support in a (see [13] for detail).Now we have the following proposition, which its proof use Lemma 3.12, and theargument is the same as [15, Chapter V, Proposition 6.1]. Proposition 3.13.
Let ( R, m , k ) be a ∗ local ring and that D be a normalized ∗ dualizing complex for R . Then R Γ m ( D ) ≃ ∗ E R ( k ) . ∗ Gorenstein injective dimension
In this section we introduce the concept of ∗ Gorenstein injective dimension ofcomplexes and we derive its main properties. In particular we prove a Chouinard’slike formula for this dimension, and compare it with the usual Gorenstein injectivedimension.
N GRADED GORENSTEIN INJECTIVE DIMENSION 11
Definition 4.1.
A graded R -module N is called ∗ Gorenstein injective, if there ex-ists an acyclic complex I of ∗ injective R -modules and homogeneous homomorphismssuch that M ∼ = Ker( I → I ) and for every ∗ injective module E , the complex ∗ Hom R ( E, I ) is exact. It is clear that every ∗ injective R -module is ∗ Gorenstein injective. So that every Y ∈ ∗ D ⊏ ( R ) has a ∗ Gorenstein injective resolution. The ∗ Gorenstein injectivedimension of Y ∈ ∗ D ⊏ ( R ) denoted by ∗ Gid R Y , is define as: ∗ Gid R Y := inf (cid:26) sup { ℓ ∈ Z | B − ℓ = 0 } (cid:12)(cid:12)(cid:12)(cid:12) B ℓ is ∗ Gorenstein injective and B ∈ ∗ D ⊏ ( R ) is isomorphic to Y (cid:27) . By a careful revision of the proof of dual version of [16, Theorems 2.5 and 2.20]we have the following two results.
Theorem 4.2.
The class of ∗ Gorenstein injective R -modules is ∗ injectively re-solving, that is for any short exact sequence → X ′ → X → X ′′ → with X ′∗ Gorenstein injective R -module, X ′′ is ∗ Gorenstein injective if and only if X is ∗ Gorenstein injective.
Proposition 4.3.
Let N be a graded R -module with finite ∗ Gorenstein injectivedimension, and let n be an integer. Then the following conditions are equivalent: (1) ∗ Gid R N ≤ n . (2) ∗ Ext iR ( L, N ) = 0 for all i > n , and all R -modules L with finite ∗ id R L . (3) ∗ Ext iR ( I, N ) = 0 for all i > n , and all ∗ injective R -modules I . (4) For every exact sequence → N → H → · · · → H n − → C n → where H , · · · , H n − are ∗ Gorenstein injectives, then C n is also ∗ Gorenstein in-jective.Consequently, the ∗ Gorenstein injective dimension of M is determined by theformulas: ∗ Gid R N = sup { i ∈ N | ∗ Ext iR ( L, N ) = 0 for some graded R -module L with finite ∗ id R L } = sup { i ∈ N | ∗ Ext iR ( I, N ) = 0 for some graded ∗ injective module I } . The ungraded version of the following theorem is in [7, Theorem 2.8]. The proofuses Proposition 2.5 and the same technique of proof of [7, Theorem 2.8].
Theorem 4.4.
Let V ≃ −→ W be a quasiisomorphism between complexes of graded R -modules, where each module in V and W has finite ∗ projective dimension orfinite ∗ injective dimension. If B ∈ ∗ C ⊏ ( R ) is a complex of ∗ Gorenstein injectivemodules, then the induced morphism ∗ Hom R ( W, B ) → ∗ Hom R ( V, B ) is a quasiisomorphism under each of the next two condition: (a) V, W ∈ ∗ C ⊐ ( R ) , or (b) V, W ∈ ∗ C ⊏ ( R ) . Corollary 4.5.
Assume that Y ≃ B where B ∈ ∗ C ⊏ ( R ) is a complex of ∗ Gorensteininjective modules. If U ≃ V , where V ∈ ∗ C ⊐ ( R ) is a complex in which each modulehas finite ∗ projective dimension or finite ∗ injective dimension, then R ∗ Hom R ( U, Y ) ≃ ∗ Hom R ( V, B ) . Proof.
Is the same as [7, Corollary 2.10] using Theorem 4.4(a). (cid:3)
The ungraded version of the following theorem is contained in [7, Theorem 3.3],and its proof is dual of [7, Theorem 3.1]. We present the proof of (3) ⇒ (4) for lateruse. Theorem 4.6.
Let Y ∈ ∗ D ⊏ ( R ) be a complex of finite ∗ Gorenstein injective di-mension. For n ∈ Z the following are equivalent: (1) ∗ Gid R Y ≤ n. (2) n ≥ − sup U − inf R ∗ Hom R ( U, Y ) for all U ∈ ∗ D (cid:3) ( R ) of finite ∗ projectiveor finite ∗ injective dimension with H ( U ) = 0 . (3) n ≥ − inf R ∗ Hom R ( J, Y ) for all ∗ injective R -modules J . (4) n ≥ − inf Y and for any left-bounded complex B ≃ Y of ∗ Gorenstein injec-tive modules, the
Ker( B − n → B − ( n +1) ) is a ∗ Gorenstein injective module.Moreover the following hold: ∗ Gid R Y = sup {− sup( U ) − inf R ∗ Hom R ( U, Y ) | ∗ id R ( U ) < ∞ and H ( U ) = 0 } = sup {− inf R ∗ Hom R ( J, Y ) | J is ∗ injective } . Proof. (2) ⇒ (3) and (4) ⇒ (1) are clear. (1) ⇒ (2) is dual of (1) ⇒ (2) in [7, Theorem3.1].(3) ⇒ (4): To establish n ≥ − inf Y , it is sufficient to show thatsup {− inf R ∗ Hom R ( J, Y ) | J is ∗ injective } ≥ − inf Y. ( ∗ )By assumption g := ∗ Gid R ( Y ) is finite, so Y ≃ B for some complex of ∗ Gorensteininjective modules: B = 0 → B s → B s − → · · · → B − g +1 → B − g → . Now it is clear that − g ≤ inf Y . By Lemma 4.5, for any ∗ injective module J , thecomplex ∗ Hom R ( J, B ) is isomorphic to R ∗ Hom R ( J, Y ) in ∗ D ( R ). If g = − inf Y then the differential ∂ − g +1 : B − g +1 → B − g is not surjective. Now by the definitionof ∗ Gorenstein injective modules there exists an ∗ injective module J such that J → B − g is surjective. Notice that the differential ∗ Hom R ( J, ∂ − g +1 ) in the complex ∗ Hom R ( J, B ) is not surjective, for otherwise, for any φ ∈ ∗ Hom R ( J, B − g ) thereexists a ψ ∈ ∗ Hom R ( J, B − g +1 ) such that φ = ∂ − g +1 ψ . This implies that ∂ − g +1 issurjective which is a contradiction. Therefore ∗ Hom R ( J, B ) has nonzero homologyin degree − g = inf Y . This gives ( ∗ ). Next assume that g > − inf Y = − t andconsider the exact sequence B : 0 → Z Bt → B t → B t − → · · · → B − g +1 → B − g → . ( ∗∗ )It shows that ∗ Gid R Z Bt ≤ t + g and it is not difficult to see that the equality musthold. For otherwise ∗ Gid R Y < g and by Proposition 4.3, H − g ( R ∗ Hom R ( J, Y )) ∼ = ∗ Ext t + g ( J, Z Bt ) = 0 for some ∗ injective module J . Therefore ( ∗ ) follows, whichgives the inequality n ≥ − inf Y .To prove the second part of (4) let B be a left-bounded complex of ∗ Gorensteininjective modules such that B ≃ Y . By assumption ∗ Gid R Y is finite, so thereexists a bounded complex ˜ B of ∗ Gorenstein injective modules such that ˜ B ≃ Y .Since n ≥ − inf Y = − inf ˜ B , the kernel Z ˜ Bn fits in an exact sequence0 → Z ˜ B − n → ˜ B − n → ˜ B − n − → · · · → ˜ B t → . By Proposition 4.3 and the isomorphism ∗ Ext i ( J, Z ˜ B − n ) ≃ H − ( n + i ) ( R ∗ Hom R ( J, Y )) =0 for i > Z ˜ B − n is ∗ Gorenstein injective.
N GRADED GORENSTEIN INJECTIVE DIMENSION 13
Now it is enough to prove that if I and B are left bounded complexes of respec-tively ∗ injective and ∗ Gorenstein injective modules and B ≃ Y ≃ I then the kernel Z I − n is ∗ Gorenstein injective if and only if Z B − n is so. Let B and I be such com-plexes. As I consists of ∗ injectives by [15, Chapter I, Lemma 4.5] there is a quasiisomorphism π : B → I which induces a quasi isomorphism between the complexes π ⊃ n : B ⊃ n → I ⊃ n . The mapping cone M ( π ⊃ n ) = · · · → B n − ⊕ I n → B n ⊕ I n +1 → B n +1 ⊕ Z Bn → Z In → ∗ Gorensteininjective modules. It follows by Theorem 4.2 that Z In is ∗ Gorenstein injective if andonly if B n +1 ⊕ Z Bn is so, which is tantamount to Z Bn being ∗ Gorenstein injective.The two equalities are immediate consequences of the equivalence of (1)-(4). (cid:3)
Corollary 4.7.
Let Y ∈ ∗ D ⊏ ( R ) . Then ∗ Gid R Y ≤ ∗ id R Y, with equality if ∗ id R Y is finite.Proof. Is the same as [5, Proposition 6.2.6] using Theorem 4.6 and Corollary 3.8. (cid:3)
Recall the finitistic injective dimension of R which defined asFID( R ) := sup { id R M | M is an R -module with id R M < ∞} . The finitistic Gorenstein injective dimension
FGID( R ), finitistic ∗ injective dimen-sion ∗ FID( R ) and finitistic ∗ Gorenstein injective dimension ∗ FGID( R ) are definesimilarly. In [16, Theorem 2.29], Holm proved that FGID( R ) = FID( R ). Also it isknown that over a commutative Noetherian ring R we have FID( R ) ≤ dim R by [2,Corollary 5.5] and [17, II. Theorem 3.2.6].The following theorem is the graded version of Holm’s result [16, Theorem 2.29].Its proof is dual of [16, Theorem 2.28]. We give the sketch of proof. Theorem 4.8. If R is any graded ring ∗ FGID( R ) = ∗ FID( R ) .Proof. Using Corollary 4.7 we have ∗ FID( R ) ≤ ∗ FGID( R ). If N is a graded R -module with 0 < ∗ Gid R N < ∞ , then the graded version of [16, Theorem 2.15] givesa graded R -module C such that ∗ id R C = ∗ Gid R N −
1. Therefore ∗ FGID( R ) ≤ ∗ FID( R ) + 1. Now for the reverse inequality ∗ FGID( R ) ≤ ∗ FID( R ), assume that0 < ∗ FGID( R ) = m < ∞ . Pick a module N with ∗ Gid R N = m . Hence the sametechnique of [18, Lemma 2.2], gives a graded module T such that ∗ id R T = m . (cid:3) Corollary 4.9.
Let Y ∈ ∗ D ⊏ ( R ) be a complex of finite ∗ Gorenstein injective di-mension. Then ∗ Gid R Y ≤ FID( R ) − inf Y. Proof.
Note that by ( ∗∗ ) in the proof of Theorem 4.6 above we have ∗ Gid R Y = ∗ Gid R M − inf Y for some graded R -module M . Thus ∗ Gid R Y ≤ ∗ FGID( R ) − inf Y = ∗ FID( R ) − inf Y ≤ FID( R ) − inf Y . (cid:3) Let D be a ∗ dualizing complex of R . Then D is a dualizing complex of R , so wehave the Bass category B ( R ) with respect to D (cf. [7, Page 237]). It is knownthat for Y ∈ D ⊏ ( R ), we have Y ∈ B ( R ) if and only if Gid R Y < ∞ , [7, Theorem4.4]. Definition 4.10.
Let D be a ∗ dualizing complex of R . The ∗ Bass category ∗ B ( R ) with respect to D is define as: ∗ B ( R ) := (cid:26) Y ∈ ∗ D (cid:3) ( R ) (cid:12)(cid:12)(cid:12)(cid:12) ǫ Y : D ⊗ L R R ∗ Hom R ( D, Y ) → Y is an iso-morphism and R ∗ Hom R ( D, Y ) ∈ ∗ D (cid:3) ( R ) (cid:27) . It is easily seen that a complex Y ∈ ∗ D (cid:3) ( R ) is in B ( R ) if and only if is in ∗ B ( R ). Theorem 4.11.
Assume that R admits a ∗ dualizing complex D . Then For Y ∈ ∗ D ⊏ ( R ) the following are equivalent: (1) Y ∈ ∗ B ( R ) . (2) ∗ Gid R Y < ∞ .Proof. It is dual to the proof of [7, Theorem 4.1]. In fact the proof uses that if N isa graded R -module satisfying both N ∈ ∗ B ( R ) and ∗ Ext mR ( J, N ) = 0 for all integer m > ∗ injective R -module J , then N is ∗ Gorenstein injective, which itsproof is dual to [7, Lemma 4.6]. (cid:3)
Theorem 4.12.
Let ( R, m , k ) be a ∗ local ring that admits a ∗ dualizing complex D .For a complex Y ∈ ∗ D (cid:3) ( R ) of finite ∗ Gorenstein injective dimension, we have width( m , Y ) = depth( m , R ) + inf R ∗ Hom R ( ∗ E R ( k ) , Y ) . Proof.
We follow the method of [7, Theorem 6.5]. By Theorem 4.11, Y ∈ ∗ B ( R );in particular Y ≃ D ⊗ L R R ∗ Hom R ( D, Y ). Furthermore, we can assume that D isa normalized ∗ dualizing complex, so that by Proposition 3.13 we have R Γ m ( D ) ∼ = ∗ E R ( k ) and that D m is a normalized dualizing complex for R m . We compute asfollows: width( m , Y ) = width( m , D ⊗ L R R ∗ Hom R ( D, Y ))= width( m , D ) + width( m , R ∗ Hom R ( D, Y ))= inf D m + inf L Λ m R ∗ Hom R ( D, Y )= depth R m + inf R ∗ Hom R ( R Γ m ( D ) , Y )= depth( m , R ) + inf R ∗ Hom R ( ∗ E R ( k ) , Y ) . The second equality is by [20, Theorem 2.4(b)], the third one by the fact that D m is homologically finite and by [13, Theorem 2.11], and the forth one by [13, 2.6],and the fact that R ∗ Hom R ( D, Y ) = R Hom R ( D, Y ). (cid:3) The ungraded version of the following result is in [7, Proposition 5.5].
Proposition 4.13.
Assume that R admits a ∗ dualizing complex and let Y ∈ ∗ D (cid:3) ( R ) .Then for any homogeneous prime ideal p ∈ R there is an inequality ∗ Gid R ( p ) Y ( p ) ≤ ∗ Gid R Y. Proof.
It is enough to show that if N is a ∗ Gorenstein injective, then N ( p ) is ∗ Gorenstein injective over R ( p ) . This is similar to the proof of [5, Theorem 6.2.13]using Corollary 4.9. (cid:3) The following proposition is the graded version of [8, Lemma 2.1]. For part ( b )we follow the technique of [8, Lemma 2.1]. We present the proof to give some hintsfor the graded analogues. Before doing that we need a lemma. N GRADED GORENSTEIN INJECTIVE DIMENSION 15
Lemma 4.14.
Let ( R, m ) be a ∗ local non-negatively graded ring. Then the m - ∗ adiccompletion ∗ b R of R , is a ∗ faithfully flat R -module, that is ∗ b R is R -flat and for anygraded R -module M , M = 0 if and only if M ⊗ R ∗ b R = 0 .Proof. It is well known that ∗ b R is a flat R -module by [10, Corollary 3.3]. Now let M be a graded R -module such that M ⊗ R ∗ b R = 0. Note that m = m ⊕ R ⊕ R ⊕ · · · ,where m is the unique maximal ideal of R . So that ( ∗ b R ) = lim ←− ( R/ m n ) =lim ←− R / m n = c R , where c R is the m -adic completion of R . Let t be an integerand set M ( t ) := M t ⊕ M t +1 ⊕ M t +2 ⊕ · · · , which is a graded submodule of M .Hence M ( t ) ⊗ R ∗ b R = 0. Therefore ( M ( t ) ⊗ R ∗ b R ) t = 0. Since ( M ( t ) ⊗ R ∗ b R ) t is generated as a Z -module by x ⊗ r for x ∈ M t and r ∈ ( ∗ b R ) = c R and t isarbitrary, we see that M ⊗ R c R = 0. On the other hand we have R ⊗ R c R = R ⊗ R ( R ⊗ R c R ) = ( R ⊗ R R ) ⊗ R c R = ( R ⊗ R R ) ⊗ R c R = R ⊗ R c R = c R .Hence 0 = M ⊗ R c R = ( M ⊗ R R ) ⊗ R c R = M ⊗ R ( R ⊗ R c R ) = M ⊗ R c R .Since c R is a faithfully flat R -module, we get that M = 0. This completes theproof. (cid:3) Proposition 4.15.
Let N be a ∗ Gorenstein injective R -module. Then under eachof the following conditions (a) R admits a ∗ dualizing complex; or (b) R is a non-negatively graded ring,one has depth R p width R p N p for every p in ∗ Spec R , and equality holds if p is a maximal element in ∗ supp R N .Proof. For part ( a ) let p be a homogenous prime ideal. Consider the ∗ local ring( S, n ) := ( R ( p ) , p R ( p ) ). By Proposition 4.13, B := N ( p ) is a ∗ Gorenstein injective S -module. Thus ∗ Ext iS ( ∗ E S ( S/ n ) , B ) = 0for all i >
0, that is inf R ∗ Hom S ( ∗ E S ( S/ n ) , B ) ≥
0. On the other hand since B n ∼ = N p and S n ∼ = R p , by Theorem 4.12 we havewidth N p = depth R p + inf R ∗ Hom S ( ∗ E S ( S/ n ) , B ) . Thus depth R p width R p N p .Now if width R p N p is finite, we have inf R ∗ Hom S ( ∗ E S ( S/ n ) , B ) < ∞ . UsingLemma 4.5 we have R ∗ Hom S ( ∗ E S ( S/ n ) , B ) ≃ ∗ Hom S ( ∗ E S ( S/ n ) , B ) . Therefore the infimum must be zero. This proves the second statement.For ( b ) assume that p is a homogeneous prime ideal and T is a graded R ( p ) -module with ∗ pd R ( p ) T < ∞ . A standard dimension shifting argument shows that ∗ Ext iR ( p ) ( T, N ( p ) ) = 0 for all i >
0. Set d = depth( p R ( p ) , R ( p ) ) and choose ahomogeneous maximal R ( p ) -regular sequence x in p R ( p ) by [3, Proposition 1.5.11]. Since the ∗ projective dimension of R ( p ) / ( x ) is finite we have0 ≤ inf R ∗ Hom R ( p ) ( R ( p ) / ( x ) , N ( p ) ) ≤ width( p R ( p ) , R ∗ Hom R ( p ) ( R ( p ) / ( x ) , N ( p ) ))= width( p R ( p ) , N ( p ) ) − d = width R p N p − d, where the second inequality holds by [6, 4.3], and the first equality follows fromProposition 2.4.Now let p be a maximal element in ∗ supp R N . Set S = R ( p ) which is a ∗ local ringwith depth d , homogeneous maximal ideal n = p R ( p ) , B = N ( p ) , and l = R ( p ) / p R ( p ) .One has n ∈ supp S B and by exactly the same method of proof of [8, Lemma 2.1]we have ∗ Ext iS ( T, B ) = 0 = ∗ Ext iS ( E, B ) for all i > S -module T with ∗ pd S T finite and E := ∗ E S ( l ).Let K denote the Koszul complex on a homogeneous system of generators for n .Since H i ( K ⊗ S E ) are Artinian, and by [3, Corollary 1.6.13], we have n H i ( K ⊗ S E ) = 0, we see that H i ( K ⊗ S E ) are finitely generated. So there is a resolution L ≃ −→ K ⊗ S E by finitely generated free S-modules. Using the ∗ Hom-evaluation wehave K ⊗ S ( E ⊗ L S ∗ Hom S ( E, B )) ≃ L ⊗ S ∗ Hom S ( E, B ) ∼ = ∗ Hom S ( ∗ Hom S ( L, E ) , B ) . The free resolution above induces a quasiisomorphism α from ∗ Hom S ( K ⊗ S E, E ) ∼ = ∗ Hom S ( K, ∗ b S ) to ∗ Hom S ( L, E ). The mapping cone C = M ( α ) is a boundedcomplex of direct sums of ∗ b S and E . Thus ∗ Ext iS ( C j , B ) = 0 for all i > j . Hence, an application of ∗ Hom S ( − , B ) yields a quasiisomorphism ∗ Hom S ( α, B ) : ∗ Hom S ( ∗ Hom S ( L, E ) , B ) ≃ −→ ∗ Hom S ( ∗ Hom S ( K, ∗ b S ) , B ) . The modules in the complex ∗ Hom S ( K, ∗ b S ) are Ext-orthogonal to the modules inthe mapping cone of an injective resolution B ≃ −→ H . Therefore, one has ∗ Hom S ( ∗ Hom S ( K, ∗ b S ) , B ) ≃ ∗ Hom S ( ∗ Hom S ( K, ∗ b S ) , H )by the graded analogue of [7, Lemma 2.4]. Now piece together the last threequasiisomorphisms, and use ∗ Hom-evaluation to obtain K ⊗ S ( E ⊗ L S ∗ Hom S ( E, B )) ≃ K ⊗ S R ∗ Hom S ( ∗ b S, B ) . Therefore by [6, (4.2) and (4.11)], the complexes E ⊗ L S ∗ Hom S ( E, B ) and R ∗ Hom S ( ∗ b S, B )have the same width. From [20, Theorem 2.4(b)] and Proposition 2.4 we havewidth( n , E ) + width( n , ∗ Hom S ( E, B )) = width( n , B ) . Now we can see that width( n , ∗ Hom S ( E, B )) = 0 (see proof of [8, Lemma 2.1]).Consequently width R p N p = width( n , B ) = width( n , E ) = depth( n , ∗ b S ) using Propo-sition 2.2. Now since ∗ b S is a flat S -module we haveExt iS ( S/ n , ∗ b S ) ∼ = Ext iS ( S/ n , S ) ⊗ S ∗ b S. Keep in mind that Ext iS ( S/ n , S ) = ∗ Ext iS ( S/ n , S ) is a graded S -module. Henceusing Lemma 4.14 we have Ext iS ( S/ n , ∗ b S ) = 0 if and only if Ext iS ( S/ n , S ) = 0.Therefore depth( n , ∗ b S ) = depth( n , S ) = d . (cid:3) N GRADED GORENSTEIN INJECTIVE DIMENSION 17
Lemma 4.16.
Let X be a complex of graded R -modules. For any homogeneoussurjective homomorphism i : Y n −→ X n of graded R -modules there is a commutativediagram Y = · · · / / X n +2 γ / / = (cid:15) (cid:15) Y n +1 β / / i ′ (cid:15) (cid:15) Y n α / / i (cid:15) (cid:15) X n − / / = (cid:15) (cid:15) · · · X = · · · / / X n +2 γ ′ / / X n +1 β ′ / / X n α ′ / / X n − / / · · · such that Y is a complex of graded R -modules, that i ′ is surjective and Ker i ∼ = Ker i ′ and the induced map H ( Y ) −→ H ( X ) is an isomorphism.Proof. Let α = α ′ i and β : Y n +1 −→ Y n be the pullback of β ′ along i thus Y n +1 = { ( x, y ) | x ∈ X n +1 , y ∈ Y n and β ′ ( x ) = i ( y ) } . It is clear that Y n +1 is a graded R -module. Let i ′ : Y n +1 → X n +1 be a mapdefined by i ′ ( x, y ) = x which is seen to be surjective. Define γ : X n +2 → Y n +1 by γ ( x ) = ( γ ′ ( x ) ,
0) for every x ∈ X n +2 . It is clear that Y is a complex andKer i ∼ = Ker i ′ . Therefore the induced map H( Y ) −→ H( X ) is an isomorphism. (cid:3) Corollary 4.17.
Let Y ∈ ∗ D (cid:3) ( R ) such that ∗ Gid R Y = g > . Then there is anexact triangle G → I → Y → Σ G such that G is a ∗ Gorenstein injective moduleand I is a complex of graded R -modules such that ∗ id R I = g .Proof. Without loss of generality we can assume that Y has the form0 → I → I − → · · · → I − g +1 → B − g → , where the I j s are ∗ injective and B − g is ∗ Gorenstein injective. By definition B − g ishomomorphic image of some ∗ injective module J . Thus by Lemma 4.16 we have acommutative diagram0 / / I / / (cid:15) (cid:15) I − / / (cid:15) (cid:15) · · · / / I − g +2 / / (cid:15) (cid:15) G − g +1 / / i ′ (cid:15) (cid:15) J / / i (cid:15) (cid:15) / / I / / I − / / · · · / / I − g +2 / / I − g +1 / / (cid:15) (cid:15) B − g / / (cid:15) (cid:15) , i ∼ = Ker i ′ . Since Ker i is a ∗ Gorenstein injective module, usingTheorem 4.2, G − g +1 is also ∗ Gorenstein injective. By repeating this argument g times we get the desired exact triangle G → I → Y → Σ G . (cid:3) The following equality extends Chouinard’s formula [4] and [8, Theorem 2.2] tothe ∗ Gorenstein injective dimension.
Theorem 4.18.
Assume that R admits a ∗ dualizing complex or R is a non-negativelygraded ring. Let Y ∈ ∗ D (cid:3) ( R ) of finite ∗ Gorenstein injective dimension. Then thereis an equality ∗ Gid R Y = sup { depth R p − width R p Y p | p ∈ ∗ Spec R } . Proof.
The argument is similar to [8, Theorem 2.2], just use Proposition 4.15 andCorollary 4.17. (cid:3)
In the following corollary we compare the ∗ Gorenstein injective dimension withthe usual Gorenstein injective dimension.
Corollary 4.19.
Let Y ∈ ∗ D (cid:3) ( R ) .Then under each of the following conditions (a) R admits a ∗ dualizing complex; or (b) R is a non-negatively graded ring, and ∗ Gid R Y and Gid R Y are finite,one has ∗ Gid R Y ≤ Gid R Y ≤ ∗ Gid R Y + 1 . Proof. If R admits a ∗ dualizing complex, then Theorem 4.11, implies that ∗ Gid R Y < ∞ , if and only if Gid R Y < ∞ . In each both cases the inequalities follow from The-orem 4.18 and [8, Theorem 2.2]. (cid:3) References
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