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HOMOLOGICAL EPIMORPHISMS OF DIFFERENTIAL GRADED ALGEBRAS DAVID PAUKSZTELLO Abstract.Let R and S be differential graded algebras.In this paper we give a characterisation of when a differential graded R -S -bimodule M induces a full embedding of derived categories R M S L ⊗S −:D (S )→D (R ).In particular,this characterisation generalis the theory of Geigle and Lenz-ing’s homological epimorphisms of rings,described in [3].Furthermore,there is an application of the main result to Dwyer and Greenlees’s Morita theory.0.Introduction In [3],Geigle and Lenzing characteri,using the classical derived functors Ext and Tor,when a homomorphism of rings φ:R →S induces a full embed-ding of bounded derived categories,D b (S )֒→D b (R ).Geigle and Lenzing refer to such ring homomorphisms as homological epimorphisms of rings.Differential graded algebras (DGAs)may be regarded as a generalisation of rings.It is,therefore,natural to ask whether the characterisation of Geigle and Lenzing also works for DGAs.It turns out that it does.Moreover,the
characterisation for DGAs is a special ca of a more general result regarding differential graded bimodules (DG bimodules).Given two DGAs R and S and a DG R -S -bimodule M we can look at the functor R M S L
⊗S −:D (S )→D (R )
and ask when this is a full embedding of derived categories.
The ca of homological epimorphisms of DGAs then becomes the situation when M =S ,with S acquiring the left R -structure via a morphism of DGAs φ:R →S .In the work of Keller ([6],Remarks 3.2),DG bimodules are re-garded as generalid morphisms of DGAs.Thus,asking when R M S L
⊗S −is a
full embedding of derived categories is analogous to asking when M is a gener-alid homological epimorphism of DGAs.This more general tting makes the structural reasons behind the characterisation of homological epimorphisms of rings in [3]more transparent.
2DAVID PAUKSZTELLO
This paper is organid in the following way.In Section 1we give a brief exposition of derived functors,compact objects and canonical maps,which will be ud in later ctions.Section 2contains the main results of this paper,characterising when a DG R -S -bimodule M induces a full embedding of derived categories
R M S L
⊗S −:D (S )→D (R ).
In Section 3we consider three examples of the main result.The first example relates to Dwyer and Greenlees’s Morita theory of [2].The cond example is the characterisation of homological epimorphisms of rings given by Geigle and Lenzing in [3].The third example is a generalisation of Geigle and Lenzing’s characterisation of homological epimorphisms of rings to DGAs.
It is worth emphasising again that the motivation of the results in Section 2is to give clear structural reasons why the characterisation of homological epimorphisms of rings by Geigle and Lenzing exists.1.Preliminaries
We begin by reviewing the definitions of DGAs and DG modules,which can be found in [1].
Definition 1.1.A differential graded algebra R over the commutative ground ring K is a graded algebra R = i ∈Z R i over K together with a differential,that is,a K -linear map ∂R :R →R of degree -1with ∂2=0,satisfying the Leibnitz rule
∂R (rs )=∂R (r )s +(−1)|r |r∂R (s )
where r,s ∈R and r is a graded element of degree |r |.
Definition 1.2.A differential graded left R -module (DG left R -module )M is a graded left module M = i ∈Z M i over R (viewed as a graded algebra)together with a differential,that is,a K -linear map ∂M :M →M of degree -1with ∂2=0,satisfying the Leibnitz rule
∂M (rm )=∂R (r )m +(−1)|r |r∂M (m )
where m ∈M and r ∈R is a graded element of degree |r |.DG right R -modules are defined similarly.We shall denote the category of DG left R -modules by DG-Mod(R ).
We denote by R op the opposite DGA of R ,that is,R op consists of the same underlying t but the multiplication is given by r ·s =(−1)|r ||s |sr for r,s ∈R .DG right R -modules can be canonically identified with DG left R op -modules.Thus,we will denote the category of DG right R -modules by D
G-Mod(R op ).Henceforth,“M is a DG R -module”will mean that M is a DG left R -module and “M is a DG R op -module”will mean that M is a DG right R -module.M is
钟静远
HOMOLOGICAL EPIMORPHISMS OF DGAS3 said to be a DG R-S-bimodule if it is a DG R-module and a DG S op-module, with the R and S op structures compatible,that is,r(ms)=(rm)s for all r∈R, s∈S and m∈M.Subscripts are ud to emphasi whether M is a DG left or right module.
A morphism f:M→N of DG R-modules is an R-linear map which is also a morphism of complexes of degree zero.Two morphisms f,g:M→N of DG R-modules are said to be homotopic if there exists a homomorphism of graded modules h:M→N of degree+1such that
f−g=∂N h+h∂M.
Note that homotopy is an equivalence relation.The homotopy category of R, written K(R),is the category who objects are DG R-modules and who morphisms are homotopy class of morphisms of DG R-modules.The derived category of R,D(R),is given by formally inverting the quasi-isomorphisms in K(R)(that is,the morphisms which induce isomorphisms in(co)homology). Recall that K(R)and D(R)are triangulated categories.Since D(R)is trian-gulated,it is equipped with an autoequivalenceΣ:D(R)→D(R),called the suspension functor,which is defined as follows.Let M be a
DG R-module, then:
(ΣM)n=M n+1,and∂ΣM=−∂M.
See[4]for details.
We shall now give brief details of compact objects,derived functors and canonical maps.
1.1.Compact objects.Let R be a DGA.We take the following definition from[5].
Definition1.3.A DG R-module M isfinitely built from R R in D(R)if M can be obtained from R R usingfinitely many distinguished triangles,suspensions, direct summands andfinite coproducts.
Definition1.4.A DG R-module C is called compact in D(R)if the functor Hom D(R)(C,−)commutes with t indexed coproducts in D(R).
Remark1.5.It can be shown that a DG R-module M isfinitely built from R in D(R)if and only if it is compact in D(R)(e[8]).
R
1.2.Derived functors.Let R be a DGA.For DG R-modules M and N, we can construct a complex Hom R(M,N);similarly,for a DG R op-module M and a DG R-module N we can construct a complex M⊗R N.The are analogues of the classical homological functors Hom and tensor.There exist derived functors of the functors analogous to the classical derived functors, denoted RHom R(M,N)and M L⊗R N,e[4].We make a brief note on how the are computed.
4DAVID PAUKSZTELLO
We recall the following definitions from[1].
Definition1.6.A DG R-module P is said to be K-projective if the functor Hom R(P,−)prerves quasi-isomorphisms.A DG R-module I is said to be K-injective if the functor Hom R(−,I)prerves quasi-isomorphisms.A DG R-module F is said to be K-flat if F⊗R−prerves quasi-isomorphisms.Similarly for DG R op-modules.
Definition1.7.Let M be a DG R-module.A K-projective resolution of M is a K-projective DG R-module P together with a quasi-isomorphismπ:P→M.
A K-flat resolution of M is a K-flat DG R-module F together with a quasi-isomorphismθ:F→M.A K-injective resolution of M is a K-injective DG R-module I together with a quasi-isomorphismι:M→I.
Let M and N be DG R-modules.We compute RHom R(M,N)in one of the following two ways.First,we obtain a K-projective resolutionπ:P→M and note that RHom R(M,N)∼=RHom R(P,N)∼=Hom R(P,N),or we obtain a K-injective resolutionι:N→I and note that RHom R(M,N)∼=RHom R(M,I)∼= Hom R(M,I).The derived functor M L⊗R N is computed by substituting by a K-projective or a K-flat resolution of M or N.
1.3.Canonical maps.When considering the classical homological functors of Hom and tensor we have the notion of a“canonical”map.For example,if R and S are rings and M is a left S-module then there is a natural map S M→Hom R(R S S,R S S⊗S S M)given by m→(s→s⊗m).A similar construction can be found for tensor.The natural maps of the classical homological functors induce natural maps of the classical derived functors Ext and Tor.However,in the tting of DGAs where the analogous“hyper-homological”derived functors occur as the right derived Hom functor,RHom,and the left derived tensor functor,L⊗,which are computed by substitution by K-projective or K-injective resolutions,we have to be more careful in constructing“canonical”maps. However,the derived functors RHom and L⊗are adjoints and we may u adjunction tofind a suitable candidate for a“canonical”map.For example, consider the natural map S M→Hom R(R S S,R S S⊗S M)given above.There is a natural isomorphism
Hom R(R S S⊗S S M,R S S⊗S S M)∼=Hom S(S M,Hom R(R S S,R S S⊗S S M)) given by adjunction.It is easy to e that the natural map m→(s→s⊗m) corresponds to the identity map on R S S⊗S S M under the adjunction.
We give an example of the corresponding situation with R and S as DGAs and N as a DG S-module,which we shall u immediately in Section2.Let M be a DG R-S-bimodule.We may ask:what is the canonical map S N→RHom R(R M S,R M S L⊗S S N)for a DG S-module N?
HOMOLOGICAL EPIMORPHISMS OF DGAS5 Here,as above,we have the following isomorphism given by adjunction:
Hom D(R)(R M S L⊗S S N,R M S L⊗S S N)
∼=Hom
(S N,RHom R(R M S,R M S L⊗S S N)).
D(S)
Then we define the canonical map S N→RHom R(R M S,R M S L⊗S S N)to be the image of the identity map on R M S L⊗S S N under the adjunction isomorphism.
2.Main results
Recall that for a functor F:C→D between two categories C and D the terms fully faithful and full embedding are synonymous and will be ud in-terchangeably;the cond usage arising becau of the fact that F being fully faithful means that,up to equivalence,C can be regarded as a full subcategory of D.
Proposition2.1.Let R and S be DGAs and suppo that M is a DG R-S-bimodule.Then the following conditions are equivalent:
(1)For all DG S-modules N the canonical map
N→RHom R(R M S,R M S L⊗S S N)
S
is an isomorphism.
(2)For all DG S-modules N and N′the canonical map
RHom S(S N,S N′)→RHom R(R M S L⊗S S N,R M S L⊗S S N′)
is an isomorphism.
(3)The functor
M S L⊗S−:D(S)→D(R)
R
is a full embedding of derived categories.
Proof:Condition(2)is just a reformulation of what it means for the functor
M S L⊗S−to be fully faithful.
R
The functor R M S L⊗S−:D(S)→D(R)is left adjoint to
RHom R(R M S,−):D(R)→D(S)
.The unit of this adjunction is
N→RHom R(R M S,R M S L⊗S S N).
S
By[7,Theorem IV.3.1],the unit of an adjunction is an isomorphism if and only if the left adjoint is a full embedding.This gives the equivalence(1)⇔(3).2
6DAVID PAUKSZTELLO
Proposition2.2.Let R and S be DGAs and suppo that M is a DG R-S-bimodule.Let Z=RHom S op(R M S,S S S)so that Z obtains the structure S Z R. Then the following conditions are equivalent:
(1)The canonical map
Z R L⊗R R M S→S S S
S
is an isomorphism.
(2)For all DG S-modules N the canonical map
Z R L⊗R(R M S L⊗S S N)→S N
S
is an isomorphism.
(3)For all DG S op-modules N and DG S-modules N′the canonical map
(N S L⊗S S Z R)L⊗R(R M S L⊗S S N′)→N S L⊗S S N′
自讨苦吃is an isomorphism.
Proof:(1)⇒(2).Suppo the canonical map S Z R L⊗R R M S→S S S is an isomorphism.Then,for any DG S-module N we obtain:
新闻200字Z R L⊗R(R M S L⊗S S N)∼−→(S Z R L⊗R R M S)L⊗S S N
S
∼
−→S S S L⊗S S N
∼
−→S N,
debuff是什么意思>铁力士山where thefirst isomorphism is by associativity of left tensor and the cond is by hypothesis.
(2)⇒(3).Suppo that the canonical map S Z R L⊗R(R M S L⊗S S N′)→S N′is an isomorphism for all DG S-modules N′.Then we get the following quence of isomorphisms for DG S op-modules N and DG S-modules N′: (N S L⊗S S Z R)L⊗R(R M S L⊗S S N′)∼−→N S L⊗S(S Z R L⊗R(R M S L⊗S S N′))
∼
−→N S L⊗S S N′.
(3)⇒(1).Suppo that the canonical map
(N S L⊗S S Z R)L⊗R(R M S L⊗S S N′)→N S L⊗S S N′
会计职业道德论文is an isomorphism for all DG S op-modules N and DG S-modules N′.Setting N=N′=S we obtain:
Z R L⊗R R M S∼−→(S S S L⊗S S Z R)L⊗R(R M S L⊗S S S S)
S
∼
−→S S S L⊗S S S S
∼
−→S S S.
