Change of grading, injective dimension and dualizing complexes
aa r X i v : . [ m a t h . K T ] J a n Change of grading, injective dimension and dualizingcomplexes
A. Solotar, P. Zadunaisky ∗ Abstract
Let G , H be groups, ϕ : G −→ H a group morphism, and A a G -gradedalgebra. The morphism ϕ induces an H -grading on A , and on any G -graded A -module, which thus becomes an H -graded A -module. Given an injective G -graded A -module, we give bounds for its injective dimension when seen as H -graded A -module. Following ideas by Van den Bergh, we give an application of our resultsto the stability of dualizing complexes through change of grading. MSC: D , E , E , W , G . Keywords: injective modules, change of grading, dualizing complexes. Introduction
Graded rings are ubiquitous in algebra. One of the main reasons is that the presenceof a grading simplifies proofs and allows to generalize many results (for example, thetheories of commutative and noncommutative graded algebras are easier to reconcilethan their ungraded counterparts). Furthermore, results can often be transfered fromthe graded to the ungraded context through standard techniques. In more categoricalterms, there is a natural forgetful functor from the category Gr G A of graded modulesover a G -graded algebra A , to the category Mod A of modules over A , and the chal-lenge is to find a way to transfer information in the opposite direction. When G = Z this is usually done through “filtered-and-graded” arguments and spectral sequences.In this article we exploit a different technique, namely the existence of three functors ϕ ! , ϕ ∗ , ϕ ∗ , where ϕ ! : Gr G A −→ Mod A is the usual forgetful functor (sometimes alsocalled the push-down functor), ϕ ∗ is its right adjoint, and ϕ ∗ is the right adjoint of ϕ ∗ . This technique has two advantages over the usual filtered-and-graded methods,namely that it does not depend on the choice of a non-canonical filtration, and thatthe group G is arbitrary. Its main drawback is that the functors in this triple do not ∗ This work has been supported by the projects UBACYT
BA,pip-conicet
CO, and MATHAMSUD-REPHOMOL. The first named author is a re-search member of CONICET (Argentina). The second named author is a FAPESP PostDoc Fellow, grant: - - S˜ao Paulo Research Foundation (FAPESP). reserve finite generation, noetherianity, or other “finiteness” properties unless furtherhypotheses are in place.The problem we consider is the following. Suppose you are given an injectiveobject I in the category Gr Z A . In general I is not injective as A -module, but if A is noetherian then its injective dimension is at most one. Now, what happens if weconsider gradings by more general groups? In general, given groups G , H and agroup morphism ϕ : G −→ H , any G -graded object can be seen as an H -gradedobject through ϕ , see paragraph . . In particular a G -graded algebra A inherits an H -grading, and there is a natural functor ϕ ! : Gr G A −→ Gr H A , between the categoriesof G -graded and H -graded modules. The question thus becomes: given an injectiveobject I in Gr G A , what is the injective dimension of ϕ ! ( I ) in Gr H A ?This question has been considered several times in the literature, but it has receivedno unified treatment. A classical result of R. Fossum and H.-B. Foxby [FF , Theorem . ] states that if A is Z -graded noetherian and commutative then a Z -graded-injectivemodule has injective dimension at most 1. M. Van den Bergh claims in the article[vdB , below Definition . ] that this result extends to the noncommutative case if thealgebra is N -graded and A is equal to the base field; a proof of this fact can be foundin the preprint [Yek ]. Other antecedents include [Eks ], where it is shown that if A is a noetherian Z -graded algebra then the injective dimension of A is finite if and onlyif its graded injective dimension is finite. Following the ideas of [Lev , section ],one can show that if A is N -graded and noetherian, and M is a Z -graded module suchthat M n = n ≪
0, then the graded injective dimension of M coincides with itsinjective dimension as A -module. Most of these results are obtained by the usual routeof going from ungraded to graded objects through filtrations and spectral sequences.The only result that we could find in the literature regarding injective modules gradedby groups other than Z states that if A is graded over a finite group then a gradedmodule is graded injective if and only if it is injective [NVO , . . ].In order to give a general answer to the question we work with the functors ϕ ! , ϕ ∗ , ϕ ∗ mentioned above, which were originally introduced by A. Polishchuk and L.Positselski in [PP ]. These functors, collectively called the change of grading functors ,turn out to be particularly well-adapted to the transfer of information of homologicalnature. Our main result, which includes most of the previous ones as special cases, isthe following. Theorem.
Let ϕ : G −→ H be a group morphism, let L = ker ϕ and let d be the projectivedimension of the trivial L-module k . Let A be a G-graded noetherian algebra, and let I be aninjective object of Gr G A. Then the injective dimension of ϕ ! ( I ) is at most d. The proof depends on two facts. First, that if I is G -graded injective then ϕ ! ( I ) isan injective object in the additive subcategory generated by all modules of the form ϕ ! ( M ) with M a G -graded A -module; in other words, modules in the image of ϕ ! are Hom HA ( − , ϕ ! ( I )) -acyclic and hence can be used to build acyclic resolutions, seeLemma . . The second is a result of independent interest, stating that given an H -graded A -module N we can obtain a resolution of N by objects in the additive categorygenerated by ϕ ! ( ϕ ∗ ( N )) , see Proposition . ; this resolution can be used to calculate he H -graded extension modules between N and ϕ ! ( I ) , which gives the desired bound.The article is structured as follows. In Section we review some basic facts onthe category of graded modules and recall some general properties of the change ofgrading functors established in the article [RZ ]. In Section we prove our mainresults on how regrading affects injective dimension. Finally in Section we givesimilar results at the derived level and use them to study the behavior of dualizingcomplexes with respect to regradings, a question originally raised by Van den Berghin [vdB ].Throughout the article k is a commutative ring, and unadorned hom spaces andtensor products are always over k . Also all modules over rings are left modules unlessotherwise stated. The letters G , H will always denote groups, and ϕ : G −→ H will bea group morphism. Acknowledgements:
The authors would like to thank Mariano Su´arez- ´Alvarez fora careful reading of a previous version of this article. The change of grading functors . . A G -graded k -module is a k -module V with a fixed decomposition V = L g ∈ G V g ;we say that v ∈ V is homogeneous of degree g if v ∈ V g , and V g is called the g -homogeneous component of V . We usually say graded instead of G -graded if G isclear from the context.Given two G -graded modules V and W , their tensor product is also a G -gradedmodule, where for each g ∈ G ( V ⊗ W ) g = M g ′ ∈ G V g ′ ⊗ W ( g ′ ) − g A map between graded k -modules f : V −→ W is said to be G-homogeneous , or simplyhomogeneous, if f ( V g ) ⊂ W g for all g ∈ G . By definition, a homogeneous map f : V −→ W induces maps f g : V g −→ W g for each g ∈ G , and f = L g ∈ G f g ; werefer to f g as the homogeneous component of degree g of f . The support of a G -graded k -module V is supp V = { g ∈ G | V g = } .The category Gr G k has G -graded modules as objects and homogeneous k -linearmaps as morphisms. Kernels and cokernels of homogeneous maps between graded k -modules are graded in a natural way, so a complex0 −→ V ′ −→ V −→ V ′′ −→ Gr G k is a short exact sequence if and only if it is a short exact sequence of k -modules, or equivalently if for each g ∈ G the sequence formed by taking g -homogeneouscomponents is exact.Given an object V in Gr G k and g ∈ G , we denote by V [ g ] the G -graded k -modulewhose homogeneous component of degree g ′ is V [ g ] g ′ = V g ′ g . This gives a naturalautoequivalence of Gr G k . . . We now recall the general definitions regarding G -graded k -algebras. The readeris referred to [NVO , Chapter ] for proofs and details.A G -graded k -algebra is a G -graded k -module A which is also a k -algebra, suchthat for all g , g ′ ∈ G and all a ∈ A g , a ′ ∈ A g ′ we have aa ′ ∈ A gg ′ . If A is a G -gradedalgebra then its structural map ρ : A −→ A ⊗ k [ G ] is defined as a ∈ A g a ⊗ g ∈ A g ⊗ k [ G ] g for each g ∈ G ; the fact that A is a G -graded algebra implies that this is amorphism of algebras.A G -graded A -module is an A -module M which is also a G -graded k -module suchthat for each g , g ′ ∈ G and all a ∈ A g , m ∈ M g ′ it happens that am ∈ M gg ′ . Once again,we usually say graded instead of G -graded. We say that A is graded left noetherianif every graded A -submodule of a finitely generated graded A -module is also finitelygenerated. If G is a polycyclic-by-finite group then A is graded noetherian if and onlyif it is noetherian [CQ , Theorem . ].We denote by Gr G A the category whose objects are G -graded A -modules andwhose morphisms are G -homogeneous A -linear maps. Notice that if M is a graded A -module then the graded k -module M [ g ] is also a graded A -module, with the sameunderlying A -module structure, so shifting also induces an autoequivalence of Gr G A .The category Gr G A has arbitrary direct sums and products. The direct sum ofgraded modules is again graded in an obvious way, but this is not the case for directproducts. Given a collection of graded A -modules { V i | i ∈ I } , their direct product isthe graded A -module whose homogeneous decomposition is given by M g ∈ G ∏ i ∈ I V ig .In other words, the forgetful functor O : Gr G A −→ Mod A preserves direct sums, butnot direct products.The category Gr G A is a Grothendieck category with enough projective and injectiveobjects. Given an object M of Gr G A , we will denote by pdim GA M and injdim GA M itsprojective and injective dimensions, respectively. Given two graded A -modules M , N we denote by Hom GA ( M , N ) the k -module of all G -homogeneous A -linear morphismsfrom M to N . Since Gr G A has enough injectives, we can define for each i ≥ i -thright derived functor of Hom GA , which we denote by R i Hom GA .There is also an enriched homomorphism functor Hom GA , given by Hom GA ( M , N ) = M g ∈ G Hom GA ( M , N [ g ]) ,which is a G -graded k -submodule of Hom k ( M , N ) . We denote its right derived func-tors by R i Hom GA . . . Let A be a G -graded k -algebra. As shown in [RZ , Section . ], a group homo-morphism ϕ : G −→ H induces functors ϕ ! , ϕ ∗ : Gr G A −→ Gr H A and ϕ ∗ : Gr H A −→ Gr G A . We quickly review the construction for completeness. et V be a G -graded k -module. We define ϕ ! ( V ) to be the H -graded k -modulewhose homogeneous component of degree h ∈ H is given by ϕ ! ( V ) h = M { g ∈ G | ϕ ( g )= h } V g .Analogously given a map f : V −→ W between G -graded k -modules, we define ϕ ! ( f ) to be the k -linear map whose homogeneous component of degree h ∈ H is given by ϕ ! ( f ) h = M { g ∈ G | ϕ ( f )= h } f g .Notice that ϕ ! ( V ) has the same underlying k -module as V . In particular, ϕ ! ( A ) isan H -graded k -algebra which is equal to A as k -algebra, and if V is a G -graded A -module then ϕ ! ( V ) is an H -graded ϕ ! ( A ) -module with the same underlying A -modulestructure. Since the action of A remains unchanged, if f is A -linear then so is ϕ ! ( f ) .This defines the functor ϕ ! : Gr G A −→ Gr H ϕ ! ( A ) . From now on we usually write A instead of ϕ ! ( A ) to lighten up the notation, since the context will make it clear whetherwe are considering it as a G -graded or as an H -graded algebra.We define ϕ ∗ ( V ) and ϕ ∗ ( f ) , to be the H -graded k -module, and H -homogeneousmap whose homogeneous components of degree h ∈ H are given by ϕ ∗ ( V ) h = ∏ { g ∈ G | ϕ ( g )= h } V g , ϕ ∗ ( f ) h = ∏ { g ∈ G | ϕ ( f )= h } f g ,respectively. If V is also an A -module, we define the action of a homogeneous element a ∈ A g ′ with g ′ ∈ G over an element ( v g ) g ∈ ϕ − ( h ) ∈ ϕ ∗ ( V ) h as a ( v g ) = ( av g ) . Withthis action ϕ ∗ ( V ) becomes an H -graded A -module, and we have defined the functor ϕ ∗ : Gr G A −→ Gr H A .Now let V ′ , W ′ be H -graded k -modules and let f ′ : V ′ −→ W ′ be a homogeneousmap. We set ϕ ∗ ( V ′ ) ⊂ V ′ ⊗ k [ G ] to be the subspace generated by all elements of theform v ⊗ g with v ∈ V ′ homogeneous of degree ϕ ( g ) , and ϕ ∗ ( f )( v ⊗ g ) = f ( v ) ⊗ g .In other words, for each g ∈ G the homogeneous components of ϕ ∗ ( V ′ ) and ϕ ( f ′ ) ofdegree g are given by ϕ ∗ ( V ′ ) g = V ′ ϕ ( g ) ⊗ k g , f g = f ϕ ( g ) ⊗ Id .If V ′ is an H -graded A -module, then V ′ ⊗ k [ G ] is an A ⊗ k [ G ] -module, and it is aninduced A -module through the structure map ρ : A −→ A ⊗ k [ G ] ; it is immediateto check that with this action it becomes a G -graded A -module with ( V ′ ⊗ k [ G ]) g = V ′ ⊗ k g for each g ∈ G , and that ϕ ∗ ( V ′ ) ⊂ V ′ ⊗ k [ G ] is a G -graded A -submodule. It isalso easy to check that if f ′ is homogeneous and A -linear then so is ϕ ∗ ( f ′ ) . Thus wehave defined a functor ϕ ∗ : Gr H A −→ Gr G A . . . We refer to ϕ ! , ϕ ∗ and ϕ ∗ collectively as the change of grading functors . It is clearfrom the definitions that the change of grading functors are exact, and that ϕ ! , ϕ ∗ reflect exactness, i.e. a complex is exact if and only if its image by any of them is alsoexact. The functor ϕ ∗ reflects exactness if and only if ϕ is surjective. As mentionedbefore, we have some adjointness relations between these functors. Proposition ([RZ , Proposition . . ]) . The functor ϕ ∗ is right adjoint to ϕ ! and left adjointto ϕ ∗ .Proof. Let M be an object of Gr G A and N an object of Gr H A . We define maps Hom HA ( ϕ ! ( M ) , N ) α . . Hom GA ( M , ϕ ∗ ( N )) β n n as follows. Given f : ϕ ! ( M ) −→ N , for each g ∈ G and each m ∈ M g set α ( f )( m ) = f ( m ) ⊗ g . Conversely, given f : M −→ ϕ ∗ ( N ) , let ǫ : k [ G ] −→ k be the counit of k [ G ] , i.e. the algebra map defined by setting ǫ ( g ) =
1, and set β ( f ) = ⊗ ǫ ◦ f . Directcomputation shows that these maps are well defined, natural, and mutual inverses.Thus ϕ ! is the left adjoint of ϕ ∗ .Now we define maps Hom GA ( ϕ ∗ ( N ) , M ) γ . . Hom HA ( N , ϕ ∗ ( M )) δ n n as follows. Given f : ϕ ∗ ( N ) −→ M , for each h ∈ H and each n ∈ N h we set γ ( f )( n ) =( f ( n ⊗ g )) g ∈ ϕ − ( h ) . Conversely, given f : N −→ ϕ ∗ ( M ) , for each g ∈ G and n ∈ N ϕ ( g ) we have f ( n ) ∈ ∏ g ′ ∈ ϕ − ( h ) M g ′ , so we can set δ ( f )( n ⊗ g ) as the g -th component of f ( n ) . Once again direct computation shows that these maps are well defined, natural,and mutual inverses. Injective dimension and change of grading
Recall that G , H are groups and ϕ : G −→ H is a group morphism. We set L = ker ϕ .Throughout this section A denotes a G -graded k -algebra. . . As stated in the Introduction, a G -graded A -module is projective if and only if itis projective as A -module, i.e. the functor ϕ ! preserves the projective dimension of anobject. Our aim is to describe how ϕ ! affects the injective dimension of an object. Webegin by recalling a previous result related to this problem. Proposition ([RZ , Corollaries . . , . . ]) . Let M be an object of Gr G A. Then the fol-lowing hold.(a) pdim GA M = pdim HA ϕ ! ( M ) and injdim GA M ≤ injdim HA ϕ ! ( M ) .(b) pdim GA M ≤ pdim HA ϕ ∗ ( M ) and injdim GA M = injdim HA ϕ ∗ ( M ) . . . The natural inclusion of the direct sum of a family into its product gives rise toa natural transformation η : ϕ ! ⇒ ϕ ∗ . Notice that η ( M ) : ϕ ! ( M ) −→ ϕ ∗ ( M ) is anisomorphism if and only if for each h ∈ H the set supp M ∩ ϕ − ( h ) is finite. If thishappens we say that M is ϕ -finite . The following theorem follows immediately fromProposition . . Theorem.
If an object M of Gr G A is ϕ -finite then injdim GA M = injdim HA ϕ ! ( M ) . Remark. If | L | < ∞ then every G-graded A-module is ϕ -finite. Also, if A is ϕ -finite thenevery finitely generated G-graded A-module is ϕ -finite, so this result applies in many usualsituations. For example, assume A is N r -graded for some r > , i.e. A is Z r -graded and A ξ = if ξ / ∈ N r . Let ψ : Z r −→ Z be the morphism ψ ( z , . . . , z r ) = z + · · · + z r . Then ψ ! ( A ) is Z -graded, and furthermore A z = if z / ∈ N . Since for each z ∈ N the set ψ − ( z ) ∩ N r is finite,the algebra A is ψ -finite. Applying the theorem we see that injdim Z r A A = injdim Z A ψ ! ( A ) . IfA is also noetherian then by [Lev , . Lemma] we see that injdim Z r A A = injdim A A. . . The algebra k [ G ] is a G -graded k -algebra, and hence through ϕ it is also an H -graded algebra, so we may consider the category of H -graded k [ G ] -modules Gr H k [ G ] .The algebra k [ H ] is an object in this category with its usual H -grading and the actionof k [ G ] induced by ϕ . By [Mon , Theorem . . ], the functor − ⊗ k [ H ] : Mod k [ L ] −→ Gr H k [ G ] is an equivalence of categories. In particular the projective dimension of k [ H ] in Gr H k [ G ] equals pdim k [ L ] k . . . Given an object N of Gr H A we denote by S ( N ) the smallest subclass of objects of Gr H A containing the set { ϕ ! ( ϕ ∗ ( N [ h ])) | h ∈ H } and closed under direct sums anddirect summands. Proposition.
Set d = pdim H k [ G ] k [ H ] = pdim k [ L ] k . Every H-graded A-module N has aresolution of length at most d by objects of S ( N ) .Proof. We begin by defining a functor D N : Gr H k [ G ] −→ Gr H A . Given an object V of Gr H k [ G ] , the tensor product N ⊗ V is an A -module with action induced by the map ρ : A −→ A ⊗ k [ G ] , and we set D N ( V ) to be the A -submodule L h ∈ H N h ⊗ V h , with theobvious H -grading. Given a morphism f : V −→ W in Gr H k [ G ] , we set D N ( f ) as therestriction and correstriction of Id N ⊗ f .Fix h ∈ H . By definition D N ( k [ G ][ h ]) and ϕ ! ( ϕ ∗ ( N [ h − ]))[ h ] are A -submodules of N ⊗ k [ G ] , and it is immediate to check that in both cases the homogeneous componentof degree h ′ ∈ H is N h ⊗ k [ G ] hh ′ , so in fact these two H -graded A -modules are equal.Furthermore, if P is any projective object in Gr H k [ G ] then there exists an object Q suchthat P ⊕ Q is a free H -graded k [ G ] -module, which is isomorphic to L i ∈ I ( k [ G ])[ h i ] for some index set I , not necessarily finite, with h i ∈ H . Now D N commutes withdirect summs, D N ( P ) is a direct summand of D N ( P ⊕ Q ) ∼ = L i ∈ I D N ( k [ G ][ h i ]) = L i ∈ I ϕ ! ( ϕ ∗ ( N [ h − i ]))[ h i ] , which obviously lies in S ( N ) .For each h ∈ H we define a map n ∈ N h n ⊗ h ∈ D N ( k [ H ]) h ; the direct sum ofthese maps gives us an isomorphism N ∼ = D N ( k [ H ]) . Taking a projective resolution P • of k [ H ] of length d and applying D N , we obtain a complex D N ( P • ) −→ D N ( k [ H ]) ∼ = N ; since k [ G ] is a free k -module, projective k [ G ] -modules are projective over k so this s an exact complex, and from the previous paragraph we see that it is a resolution of N by objects in S ( N ) . . . Let M be a G -graded A -module. Recall that ϕ ∗ ( ϕ ! ( M )) ⊂ M ⊗ k [ G ] consistsof all m ⊗ g ′ with m ∈ M g and ϕ ( g ) = ϕ ( g ′ ) . For each l ∈ L we have a map M [ l ] −→ ϕ ∗ ϕ ! ( M ) whose homogeneous component of degree g ∈ G is given by m ∈ M [ l ] g m ⊗ gl ∈ ϕ ∗ ϕ ! ( M ) . This induces a natural map L l ∈ L M [ l ] −→ ϕ ∗ ϕ ! ( M ) .This map has an inverse, given by m ⊗ g ′ ∈ ϕ ∗ ( ϕ ! ( M )) m ∈ M [ g − g ′ ] , so weget a natural isomorphism ϕ ∗ ( ϕ ! ( M )) ∼ = L l ∈ L M [ l ] . This observation is used in thefollowing lemma. Lemma.
Assume A is left G-graded noetherian. Let I , M be objects of Gr G A with I injective,and let N be a direct summand of ϕ ! ( M ) . Then R i Hom HA ( N , ϕ ! ( I )) = for all i > .Proof. It is enough to show that the result holds for N = ϕ ! ( M ) . In that case we haveisomorphisms Hom HA ( ϕ ! ( M ) , ϕ ! ( I )) ∼ = Hom GA ( M , ϕ ∗ ( ϕ ! ( I ))) ∼ = Hom GA M , M l ∈ L I [ l ] ! .Since this isomorphism is natural in the first variable, we obtain for each i ≥ R i Hom HA ( ϕ ! ( M ) , ϕ ! ( I )) ∼ = R i Hom GA M , M l ∈ L I [ l ] ! .Now by the graded version of the Bass-Papp Theorem (see [GW , Theorem . ] fora proof in the ungraded case, which adapts easily to the graded context), the fact that A is left G -graded noetherian implies that L l ∈ L I [ l ] is injective, and hence the lastisomorphism implies R i Hom HA ( ϕ ! ( M ) , ϕ ! ( I )) = Remark.
We point out that the proof does not use the full Bass-Papp Theorem, just the factthat the direct sum of an arbitrary family of shifted copies of the same injective module is againinjective, so we may wonder whether this property is weaker than G-graded noetherianity. Inthe ungraded case a module is called Σ -injective if the direct sum of arbitrarily many copies ofit is injective. Say that a G-graded A-module is graded Σ -injective if an arbitrary direct sum ofshifted copies of itself is injective. Then by a reasoning analogous to that of [FW , Theorem,pp. - ] one can prove that an algebra is left G-graded noetherian if and only if everyinjective object of Gr G A is graded Σ -injective. We thank MathOverflow user Fred Rohrer forthe reference. . . We are now ready to prove the main result of this section.
Theorem.
Set d = pdim k [ L ] k . Assume A is left G-graded noetherian. For every object M of Gr G A we have injdim GA M ≤ injdim HA ϕ ! ( M ) ≤ injdim GA M + dProof. The first inequality holds by Proposition . . The case where M is of infiniteinjective dimension is trivially true, so let us consider the case where n = injdim GA M is finite. In this case we work by induction. f n = M is injective in Gr G A . Let N be an object of Gr H A , and let P • −→ N be a resolution of N of length d by objects of S ( N ) as in Proposition . . It followsfrom Lemma . that R i Hom HA ( P , ϕ ! ( I )) = P of S ( N ) , so in fact P • is an acyclic resolution of N and R i Hom HA ( N , ϕ ! ( M )) ∼ = H i ( Hom HA ( P • , ϕ ! ( M ))) for each i ≥
0. Thus R i Hom HA ( N , ϕ ! ( M )) = i > d , and since N was arbitrarythis implies that injdim HA ϕ ! ( M ) ≤ d .Now assume that the result holds for all objects of Gr G A with injective dimen-sion less than n . Let M −→ I be an injective envelope of M in Gr G A , and let M ′ be its cokernel. Then injdim GA M ′ = n −
1, and so by the inductive hypothesisinjdim HA ϕ ! ( M ′ ) ≤ n − + d . Now we have an exact sequence in Gr H A of the form0 −→ ϕ ! ( M ) −→ ϕ ! ( I ) −→ ϕ ! ( M ′ ) −→ ϕ ! ( M ) is bounded aboveby the maximum between injdim HA ϕ ! ( I ) + ≤ d + HA ϕ ! ( M ′ ) + ≤ n + d .This gives us the desired inequality. Change of grading at the derived level and dualizing com-plexes
Dualizing complexes for noncommutative rings were introduced by A. Yekutieli inthe context of connected N -graded algebras in order to study their local cohomology;they have proven to be very useful in the study of ring theoretical properties of noncommutative rings, see for example [Yek , Jør , vdB , YZ , WZ , YZ ], etc. Adualizing complex is essentially an object R • in the derived category of Mod A e suchthat the functor R Hom A ( − , R • ) is a duality between D b ( Mod A ) and D b ( Mod A op ) , fora precise definition see Definition . . A graded dualizing complex in principle onlyguarantees dualities at the graded level, but according to Van den Bergh, a Z -gradeddualizing complex is also an ungraded dualizing complex [vdB ]. In this section weshow that in fact a Z r -graded dualizing complex remains a dualizing complex afterregrading. Once you have Theorem . , the proof in the Z r -graded case is no moredifficult than in the Z -graded case, except for the technical complications due to theextra gradings. Still, we felt it was worthwhile to develop these technicalities in orderto obtain a precise statement of Theorem . .Throughout this section k is a field, G is an abelian group, and A is a G -graded k -algebra. We denote by A e the enveloping algebra A ⊗ A op ; since G is abelian both A op and A e are G -graded algebras. . . Let us fix some notation regarding derived categories. Given an abelian category A , we denote by K ( A ) the category of complexes of objects of A with homotopyclasses of maps of complexes as morphisms, and by D ( A ) the derived category of . As usual we denote by D + ( A ) , D − ( A ) , D b ( A ) the full subcategories of D ( A ) consisting of left bounded, right bounded and bounded complexes. Recall that aninjective resolution of a left bounded complex R • is a quasi-isomorphism R • −→ I • where I • is a left bounded complex formed by injective objects of A . If A has enoughinjectives then every left bounded complex has an injective resolution. Analogousremarks apply for projective resolutions of right bounded complexes.If F : A −→ B is an exact functor between abelian categories, then by the universalproperty of derived categories there is an induced functor D ( A ) −→ D ( B ) , which byabuse of notation we will also denote by F . . . The maps a ∈ A a ⊗ ∈ A e and a ∈ A op ⊗ a ∈ A e induce restrictionfunctors Res A : Gr G A e −→ Gr G A and Res A op : Gr G A e −→ Gr G A op . These functors areexact and preserve projectives and injectives, which can be proved following the linesof the proof in the case G = Z found in [Yek , Lemma . ]. If H is any group and ϕ : G −→ H is a group morphism then it is clear that the associated change of gradingfunctors commute with the restriction functors in the obvious sense. Since restrictionand change of grading functors are exact, they induce exact functors between thecorresponding derived categories. . . There exists a functor
Hom GA : K ( Gr G A e ) op × K ( Gr G A e ) −→ K ( Gr G A e ) defined as follows. Given complexes M • , N • , for each n ∈ Z we set Hom GA ( N • , M • ) n = ∏ p ∈ Z Hom GA ( N p , M p + n ) ,where the product is taken in the category of G -graded A e -modules; this sequence of G -graded A e -modules is made into a complex with differential d n = ∏ p ∈ Z (( − ) n + Hom GA ( d pN , M p + n ) + Hom GA ( N p , d p + nM )) .The action of Hom GA on maps is defined in the usual way.The functor Hom GA has a right derived functor R Hom GA : D ( Gr G A e ) op × D ( Gr G A e ) −→ D ( Gr G A e ) .When M • is an object of D + ( Gr G A e ) such that M i is injective as left A -module foreach i ∈ Z , then R Hom GA ( N • , M • ) ∼ = Hom GA ( N • , M • ) for every object N • of D ( Gr G A e ) . Analogously, if N • is an object of D − ( Gr G A e ) suchthat N i is projective as left A -module for each i ∈ Z , then R Hom GA ( N • , M • ) ∼ = Hom GA ( N • , M • ) for every object M • of D ( Gr G A e ) . This is proved in the case G = Z in [Yek , The-orem . ], and the general proof follows the same reasoning. There is a completelyanalogous functor Hom GA op whose derived functor R Hom GA op has similar properties. . . Let R • be a complex of A e -modules. Seeing A op as a complex of A e -modulesconcentrated in homological degree 0, there is a map A op −→ Hom GA ( R • , R • ) given bysending a ∈ A op to right multiplication by a acting on R • . Now let P • −→ R • be aprojective resolution of R • , so there is an isomorphism R Hom GA ( R • , R • ) ∼ = Hom GA ( P • , P • ) ,and we get a map nat A : A op −→ R Hom Z r A ( R • , R • ) . This map is independent ofthe projective resolution we choose, so we refer to it as the natural map from A op to R Hom Z r A ( R • , R • ) . In the same way there is a natural map from A to R Hom Z r A op ( R • , R • ) .The proof that these maps are independent of the chosen resolution is quite tediousbut elementary; the reader is referred to [Zad , Appendix A] for details. . . Assume that G = Z r for some r ≥
0. We say that A is N r -graded if supp A ⊂ N r ,and that it is connected if A = k . If A is N r -graded then so are A op and A e , and theyare connected if and only if A is connected.The following definition is adapted from [Yek , Definition . ]. Definition.
Let A be a connected N r -graded noetherian algebra. A Z r -graded dualizingcomplex over A is a bounded complex R • of A e -modules with the following properties.(a) The cohomology modules of Res A ( R • ) and Res A op ( R • ) are finitely generated.(b) Both Res A ( R • ) and Res A op ( R • ) have finite injective dimension.(c) The maps nat A : A op −→ R Hom Z r A ( R • , R • ) and nat A op : A −→ R Hom Z r A op ( R • , R • ) are isomorphisms in D ( Gr Z r A e ) . A dualizing complex in the ungraded sense is an object of D ( Mod A e ) which com-plies with the ungraded analogue of the previous definition. Our objective is to showthat a Z r -graded dualizing complex remains a dualizing complex if we change (orforget) the grading. Since being finitely generated is independent of grading, item (a)of the definition remains true if we change or forget the grading. To see how item(b) behaves with respect to change of grading requires a derived version of Theorem . , while item (c) is also invariant by change of grading by a simple argument. Weprovide the details in the following lemmas, in a slightly more general context. . . Recall that given a group morphism ϕ : G −→ H , a G -graded k -vector space M issaid to be ϕ -finite if supp M ∩ ϕ − ( h ) is a finite set for each h ∈ H . Lemma.
Let ϕ : G −→ H be a group morphism and set L = ker ϕ . Let R • be a boundedcomplex of G-graded A-modules.(a) If the cohomology modules of R • are ϕ -finite then injdim GA R • = injdim HA ϕ ! ( R • ) (b) Let d = pdim k [ L ] k . If A is left G-graded noetherian then the following inequalities hold injdim GA R • ≤ injdim HA ϕ ! ( R • ) ≤ injdim GA R • + d . roof. Let R • −→ I • be an injective resolution of minimal length. It is enough to provethe statement with I • instead of R • .Suppose I • has ϕ -finite cohomology modules. Recall that there is a natural trans-formation η : ϕ ! ⇒ ϕ ∗ , and that η ( M ) is an isomorphism if an only if M is ϕ -finite. The class of ϕ -finite G -graded A -modules is closed by extensions, so applying[Har , Proposition . ] (in the reference “thick” stands for “closed by extensions”)we get that the map ϕ ! ( I • ) −→ ϕ ∗ ( I • ) is a quasi-isomorphism, and since ϕ ∗ preservesinjectives it is an injective resolution, so injdim GA R • ≥ injdim HA ϕ ! ( R • ) . If the inequalitywere strict, then we could truncate ϕ ∗ ( I • ) to obtain a shorter complex of the form · · · −→ ϕ ∗ ( I j − ) −→ ϕ ∗ ( I j ) −→ ϕ ∗ ( coker d j ) −→ −→ · · · with ϕ ∗ ( coker d j ) an injective H -graded A -module. Since ϕ ∗ preserves injective dimen-sion by Proposition . , this would contradict the fact that I • is a minimal resolutionof R • , so in fact injdim GA R • = injdim HA ϕ ! ( R • ) . This proves item (a)For item (b), assume first that I • is bounded. We proceed by induction on s , thelength of I • . The case s = . . Now let t ∈ Z bethe minimal homological degree such that I t =
0, and consider the exact sequence ofcomplexes0 −→ I > t −→ I • −→ I t −→ I t is seen as a complex concentrated in homological degree t and I > t is thesubcomplex of I • formed by all components in homological degree larger than t . Thusthere is a distinguished triangle ϕ ! ( I > t ) −→ ϕ ! ( I • ) −→ ϕ ! ( I t ) −→ in D ( Gr H A ) . Bythe inductive hypothesis the inequality holds for the first and third complexes of thetriangle, so a simple argument with long exact sequences shows that the correspondinginequality holds for ϕ ! ( I • ) .Finally, if I • is not bounded then we only have to prove that ϕ ! ( I • ) does nothave finite injective dimension. Now ϕ ∗ preserves injective dimensions, and since ϕ ∗ ( ϕ ! ( I • )) ∼ = L l ∈ L I [ l ] • has infinite injective dimension, so does ϕ ! ( I • ) . . Lemma.
Let G , H be abelian groups and ϕ : G −→ H a group morphism. Assume A isG-graded noetherian. Let S • , R • be bounded complexes of G-graded A e -modules such that thecohomology modules of R • are finitely generated as left A-modules.(a) The map ϕ ! ( R Hom GA ( R • , S • )) −→ R Hom HA ( ϕ ! ( R • ) , ϕ ! ( S • )) is an isomorphism.(b) The composition ϕ ! ( A ) ϕ ! ( nat A ) / / ϕ ! ( R Hom GA ( R • , R • )) / / R Hom HA ( ϕ ! ( R • ) , ϕ ! ( R • )) equals nat ϕ ! ( A ) : ϕ ! ( A ) −→ R Hom HA ( ϕ ! ( R • ) , ϕ ! ( R • )) roof. The map from item (a) is obtained as follows. Let P • −→ R • be a projective reso-lution. Then ϕ ! ( P • ) −→ ϕ ! ( R • ) is also a projective resolution since ϕ ! is exact and pre-serves projectives. Now by definition of Hom GA ( R • , S • ) , we have ϕ ! ( Hom GA ( P • , S • )) ⊂ Hom HA ( ϕ ! ( P • ) , ϕ ! ( S • )) , and the desired map is the inclusion. Once again this map isindependent of the chosen projective resolution. Clearly item (b) follows from this.If R • and S • are concentrated in homological degree 0, item (a) is a well-known re-sult, see for example [RZ , Proposition . . ]. The general result follows by standardarguments using [Har , Proposition I. . (i)]. . . We are now ready to prove the main result of this section.
Theorem.
Let A be a connected N r -graded noetherian k -algebra and let R • be a Z r -gradeddualizing complex over A.(a) Let s > and let ϕ : Z r −→ Z s be a group morphism such that ϕ ! ( A ) is N s -graded con-nected. Then ϕ ! ( R • ) is a Z s -graded dualizing complex over ϕ ! ( A ) of injective dimension injdim Z r A R • .(b) Let O : D ( Gr Z r A e ) −→ D ( Mod A e ) be the forgetful functor. Then O ( R • ) is a dualiz-ing complex over A in the ungraded sense, of injective dimension at most injdim Z r A R • + .Proof. Let us prove item (a). As we have already noticed, ϕ ! commutes with the re-striction functors and does not change the fact that a bimodule is finitely generated asleft or right A -module, so ϕ ! ( R • ) complies with item (a) of Definition . . Since A is Z r -graded noetherian it is also Z s -graded noetherian, and hence ϕ ! ( A ) is locally finite;this implies that A is ϕ -finite, otherwise ϕ ! ( A ) would have a homogeneous componentof infinite dimension. Since the cohomology modules of R • are finitely generated, theyare also ϕ -finite and hence by item (a) of Lemma . injdim Z s A ϕ ! ( R • ) = injdim Z r A R • , soitem (b) of Definition . also holds for R • . Finally item (c) of the definition followsimmediately from item (b) of Lemma . .We now prove item (b). Let ψ : Z r −→ Z be the map ψ ( z , . . . , z r ) = z + · · · + z r .Then A is ψ -finite and ψ ! ( A ) is connected N -graded, so by the first item ψ ! ( R • ) is a Z -graded dualizing complex over A of injective dimension injdim Z r A R • . Now a similarreasoning as the one we used for the first item, but this time using item (b) of Lemma . , shows that O ( ψ ! ( R • )) = O ( R • ) is a dualizing complex and gives the bound for itsinjective dimension. References [CQ ] W. Chin and D. Quinn, Rings graded by polycyclic-by-finite groups , Proc. Amer. Math. Soc. ( ), no. , – .[Eks ] E. K. Ekstr ¨om, The Auslander condition on graded and filtered Noetherian rings , Ann´ee (Paris, / ), Lecture Notes in Math., vol. , Springer, Berlin, , pp. – .[FW ] C. Faith and E. A. Walker, Direct-sum representations of injective modules , J. Algebra ( ), – . FF ] R. Fossum and H.-B. Foxby, The category of graded modules , Math. Scand. ( ), – .[GW ] K. R. Goodearl and R. B. Warfield Jr., An introduction to noncommutative Noetherian rings , nd ed.,London Mathematical Society Student Texts, vol. , Cambridge University Press, Cambridge, .[Har ] R. Hartshorne, Residues and duality , Lecture notes of a seminar on the work of A. Grothendieck,given at Harvard / . With an appendix by P. Deligne. Lecture Notes in Mathematics, No. , Springer-Verlag, Berlin, .[Jør ] P. Jørgensen, Local cohomology for non-commutative graded algebras , Comm. Algebra ( ),no. , – .[Lev ] T. Levasseur, Some properties of noncommutative regular graded rings , Glasgow Math. J. ( ),no. , – .[Mon ] S. Montgomery, Hopf algebras and their actions on rings , CBMS Regional Conference Series inMathematics, vol. , Published for the Conference Board of the Mathematical Sciences, Wash-ington, DC, .[NVO ] C. N˘ast˘asescu and F. Van Oystaeyen, Methods of graded rings , Lecture Notes in Mathematics,vol. , Springer-Verlag, Berlin, .[RZ ] L. Rigal and P. Zadunaisky, Twisted Semigroup Algebras , Alg. Rep. Theory ( ), – .[PP ] A. Polishchuk and L. Positselski, Hochschild (co)homology of the second kind I , Trans. Amer. Math.Soc. ( ), no. , – .[vdB ] M. van den Bergh, Existence theorems for dualizing complexes over non-commutative graded andfiltered rings , J. Algebra ( ), no. , – .[WZ ] Q.-S. Wu and J. J. Zhang, Applications of dualizing complexes , Proceedings of the Third Interna-tional Algebra Conference (Tainan, ), Kluwer Acad. Publ., Dordrecht, , pp. – .[Yek ] A. Yekutieli, Dualizing complexes over noncommutative graded algebras , J. Algebra ( ), no. , – .[Yek ] A. Yekutieli, Another proof of a theorem of Van den Bergh about graded-injective modules ( ).Available at http://arxiv.org/abs/1407.5916 .[YZ ] A. Yekutieli and J. J. Zhang, Rings with Auslander dualizing complexes , J. Algebra ( ), no. , – .[YZ ] , Rigid dualizing complexes over commutative rings , Algebr. Represent. Theory ( ),no. , – .[Zad ] P. Zadunaisky, Homological regularity properties of quantum flag va-rieties and related algebras , . PhD Thesis. Available online at http://cms.dm.uba.ar/academico/carreras/doctorado/desde . A.S.:IMAS-CONICET y Departamento de Matem´aticaFacultad de Ciencias Exactas y Naturales,Universidad de Buenos Aires,Ciudad Universitaria, Pabell ´on , Buenos Aires, Argentina. [email protected] .Z. :Instituto de Matem´atica e Estat´ıstica,Universidade de S˜ao Paulo.Rua do Mat ˜ao, CEP - - S˜ao Paulo - SP [email protected]@ime.usp.br