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PROJECTIVITY AND FLATNESS OVER THE ENDOMORPHISM
RING OF A FINITELY GENERATED MODULE
S. CAENEPEEL and T. GUÉDÉNON
Received 7 October 2003
LetAbe a ring andΛa finitely generatedA-module. We give necessary and sufficient con-ditions for projectivity and flatness of a module over the endomorphism ring ofΛ.
2000 Mathematics Subject Classification: 16D40, 16W30.
1. Introduction. Suppose that a groupG acts as a group of automorphisms on a ringR. Jøndrup [11] has studied whenRis projective as anRG-module. García and Del Río [8] have continued Jøndrup’s work, and gave necessary and sufficient conditions for the flatness and projectivity ofRas anRG-module. In [10], the second author gen-eralized the results from [8] in the following way. Letkbe a commutative ring,H a Hopf algebra overk, andΛa leftH-module algebra. Then we can consider the smash productΛ#Hand the subring of invariantsΛH. Then we can give necessary and suffi-cient conditions for the projectivity and flatness overΛHof a leftΛ#H-moduleP. The results from [8] are recovered if we takek=Z,H=kG, andP=R.
The authors discovered recently that the same methods can be applied to the follow-ing situation. Letj:S→Λbe a ring morphism, and consider the centralizerΛS= {b∈
Λ|bj(s)=j(s)b, for alls∈S}. Then we can give necessary and sufficient conditions for projectivity and flatness overΛSof an(S,Λ)-bimodule.
In fact, both results are special cases of a more general result which happens to be very elementary. Let Λ be a finitely generated module over a ring A, and B the endomorphism ring of Λ. Adapting the techniques developed in [8, 10], we give, in
Section 2, some necessary and sufficient conditions for projectivity and flatness of a B-module.
It turns out that in our two motivating examples,Λis a cyclic module. Such mod-ules appeared recently in [6] and are obtained by considering a ring morphism with a grouplike functional. This will be explained inSection 3.
2. Main result. LetA be an associative ring with unit, and Λ∈ᏹA, where ᏹA is the category of rightA-modules and rightA-linear maps. All our results hold for left modules as well, but the formulas are slightly more elegant in the right-handed case, a consequence of the fact that we write from left to right. LetB=EndA(Λ). Then Λ
rightB-modules and rightA-modules:
F= •⊗BΛ:ᏹB →ᏹA, G=HomA(Λ,•):ᏹA →ᏹB. (2.1)
ForM∈ᏹA,G(M)=HomA(Λ,M)is a rightB-module, by(f·b)(λ)=f (bλ). The unit and counit of the adjunction are defined as follows, forM∈ᏹA,N∈ᏹB:
ηN:N →HomAΛ,N⊗BΛ, ηN(n)(λ)=n⊗λ,
εM: HomA(Λ,M)⊗BΛ→M, εM(f⊗λ)=f (λ). (2.2)
The adjointness property then means that we have
GεM◦ηG(M)=IG(M), εF(N)◦FηN=IF(N). (2.3)
In particular,ηG(M)is monic for everyM∈ᏹA.
Recall from [13] thatΛ∈ᏹA is called semi-Σ-projective if the functor HomA(Λ,•): ᏹA→Ab sends an exact sequence inᏹAof the form
Λ(J) →Λ(I) →M →0 (2.4)
to an exact sequence in Ab. Obviously, a projective module is semi-Σ-projective.
Lemma2.1. LetΛbe a finitely generated rightA-module, andB=EndA(Λ). For every
setI,
(1) the natural mapκ:B(I)=Hom
A(Λ,Λ)(I)→HomA(Λ,Λ(I))is an isomorphism; (2) εΛ(I)is an isomorphism;
(3) ηB(I)is an isomorphism;
(4) ifΛis semi-Σ-projective as a rightA-module, thenηis a natural isomorphism; in other words, the induction functorF= •⊗AΛis fully faithful.
Proof. (1) is an easy exercise.
(2) It is straightforward to check that the canonical isomorphismB(I)⊗BΛΛ(I)is the composition of the maps
B(I)⊗
BΛ κ⊗IΛ→HomAΛ,Λ(I)⊗BΛ εΛ(I)
→Λ(I) (2.5)
and it follows from (1) thatεΛ(I)is an isomorphism. (3) PuttingM=Λ(I)in (2.3), we find
HomAΛ,εΛ(I)◦ηHomA(Λ,Λ(I))=IHomA(Λ,Λ(I)). (2.6)
From (2), it follows that HomA(Λ,εΛ(I))is an isomorphism and from (1), it follows that HomA(Λ,Λ(I))B(I), henceη
B(I) is an isomorphism.
we have a commutative diagram
B(J) ηB(J)
B(I) ηB(J)
N
ηN
0
GFB(J) GFB(I) GF(N) 0.
(2.7)
The top row is exact, the bottom row is exact, sinceGF(B(I))=Hom
A(Λ,Λ(I))andΛis semi-Σ-projective. By (3),ηB(I) andηB(J)are isomorphisms, and it follows from the five lemma thatηN is an isomorphism.
We can now give equivalent conditions for the projectivity and flatness ofP∈ᏹB.
Theorem2.2. Assume thatΛ∈ᏹA is finitely generated and letB=EndA(Λ). For
P∈ᏹB, the following statements are considered: (1) P⊗BΛis projective inᏹAandηP is injective; (2) Pis projective as a rightB-module;
(3) P⊗BΛis a direct summand inᏹAof someΛ(I), andηP is bijective;
(4) there exists Q∈ᏹA such that Q is a direct summand of some Λ(I), and P HomA(Λ,Q)inᏹB;
(5) P⊗BΛis a direct summand inᏹAof someΛ(I).
Then (1)⇒(2)(3)(4)⇒(5). IfΛis semi-Σ-projective (in particular,Λis quasi-projective, see[1, Proposition 1.8]) as a rightA-module, then (5)⇒(3); ifΛis projective as a right A-module, then (2)⇒(1).
Proof. (2)⇒(3). IfP is projective as a rightB-module, then we can find a setI and
P∈ᏹBsuch thatB(I)P⊕P. Then, obviously,Λ(I)B(I)⊗
BΛ(P⊗BΛ)⊕(P⊗BΛ). ηis a natural transformation, so we have a commutative diagram
B(I) ηB(I)
P⊕P
ηP⊕ηP
HomAΛ,Λ(I) Hom
AΛ,P⊗BΛ⊕HomAΛ,P⊗BΛ.
(2.8)
From the fact thatηB(I)is an isomorphism, it follows thatηP(andηP) are isomorphisms. (3)⇒(5). TakeQ=P⊗BΛ.
(4)⇒(2). Iff:Λ(I)→Qis a split epimorphism inᏹA, then
HomA(Λ,f ): HomAΛ,Λ(I)B(I) →HomA(Λ,Q)P (2.9)
is also split surjective, hencePis projective as a rightB-module. (4)⇒(5) is trivial.
(5)⇒(3) under the assumption thatΛis semi-Σ-projective; this follows immediately fromLemma 2.1(4).
(1)⇒(2). Take an epimorphism f :B(I)→P inᏹ
diagram
B(I) ηB(I)
f
P
ηP
0
HomAΛ,Λ(I) GF(f ) Hom
AΛ,P⊗BΛ 0.
(2.10)
The bottom row is split exact since any functor, in particular, HomA(Λ,•), preserves split exact sequences.ηB(I)is an isomorphism, byLemma 2.1, and diagram chasing tells us thatηP is surjective. By assumption,ηP is injective, soηP is bijective, and the top row is isomorphic to the bottom row and therefore split. ThusP∈ᏹBis projective.
(2)⇒(1). IfΛis projective inᏹA, thenΛis a direct summand ofA(J)inᏹAfor some set J. Now P⊗BΛis a direct summand of Λ(I)and also ofA(J×I), soP⊗BΛ∈ᏹA is projective. Since (2) implies (3), it follows thatηP is injective.
The Govorov-Lazard theorem [9,12] states thatM∈ᏹAis flat if and only if it is the inductive limit of a directed system of free modules of finite type.Theorem 2.3can be viewed as a generalization of this classical result.
Theorem 2.3. Assume that Λ∈ᏹA is finitely presented and letB=EndA(Λ). For
P∈ᏹB, the following assertions are equivalent: (1) Pis flat as a rightB-module;
(2) P⊗BΛ=lim→Qi, whereQiΛni inᏹA for some positive integer ni, andηP is bijective;
(3) P⊗BΛ=lim→Qi, where Qi∈ᏹA is a direct summand of someΛ(I), and ηP is bijective;
(4) there existsQ=lim
→Qi∈ᏹA, such thatQiΛni for some positive integerniand HomA(Λ,Q)P inᏹB;
(5) there existsQ=lim→Qi∈ᏹA, such thatQiis a direct summand ofΛ(I)as a right A-module andHomA(Λ,Q)PinᏹB.
IfΛis semi-Σ-projective as a rightA-module, then these conditions are also equivalent to conditions (2) and (3) without the assumption thatηP is bijective.
Proof. (1)⇒(2).P=lim
→Ni, withNi=Bni. TakeQi=Λni, then
lim→Qilim→Ni⊗BΛlim→Ni⊗BΛP⊗BΛ. (2.11)
Consider the following commutative diagram:
P=lim →Ni ηP
limηNi
lim→HomAΛ,Ni⊗BΛ
f
HomAΛ,lim→Ni⊗BΛ HomAΛ,lim→Ni⊗BΛ.
(2.12)
(2)⇒(3) and (4)⇒(5) are obvious.
(2)⇒(4) and (3)⇒(5). PutQ=P⊗BΛ. ThenηP :P→HomA(Λ,P⊗BΛ)is the required isomorphism.
(5)⇒(1). We have a split exact sequence
0 →Ni →Pi=Λ(I) →Qi →0. (2.13)
Consider the following commutative diagram
0 FGNi
εNi
FGPi
εPi
FGQi
εQi
0
0 Ni Pi Qi 0.
(2.14)
We know, fromLemma 2.1, thatεPi is an isomorphism. Both rows in the sequence are split exact, so it follows that εNi and εQi are also isomorphisms. Next, consider the commutative diagram
lim→HomAΛ,Qi⊗BΛ f⊗BIΛ HomA(Λ,Q)⊗BΛ
εQ
lim →
HomAΛ,Qi⊗BΛ h
limεQi
Q,
(2.15)
wherehandf are the natural homomorphisms.his an isomorphism because• ⊗BΛ preserves inductive limits, andf is an isomorphism becauseΛ∈ᏹA is finitely pre-sented. limεQi is an isomorphism because everyεQi is an isomorphism, so it follows thatεQis an isomorphism. From (2.3), we deduce
HomAΛ,εQ◦ηHomA(Λ,Q)=IHomA(Λ,Q). (2.16)
HomA(Λ,εQ) is an isomorphism, so ηHomA(Λ,Q) is also an isomorphism, and, since HomA(Λ,Q)P,ηP is an isomorphism. Consider the isomorphisms
P ηP HomAΛ,P⊗BΛ HomAΛ,HomA(Λ,Q)⊗BΛ
Hom(Λ,εQ)
HomA(Λ,Q)
f
lim→HomAΛ,Qi.
(2.17)
It follows fromLemma 2.1that HomA(Λ,Pi)B(Ii) is projective as a rightB-module, hence HomA(Λ,Qi)is also projective as a rightB-module, and we conclude thatP∈ᏹB is flat.
The final statement is an immediate consequence ofLemma 2.1(4).
Remark2.4. ToΛ∈ᏹA, we can associate a Morita context (cf. [2]), namely,(A,B,Λ, Λ∗=HomA(Λ,A),τ,µ)with connecting maps
τ:Λ⊗AΛ∗ →B, τλ⊗λ∗(µ)=λλ∗(µ),
τis surjective (and a fortiori bijective) if and only ifΛis finitely generated and projective as a rightA-module. In this situation, the natural transformation
γ:G1= •⊗AHomA(Λ,A) →G=HomA(Λ,•) (2.19)
withγM(m⊗Aλ∗)(λ)=mλ∗(λ)is a natural isomorphism. For anyN∈ᏹB, we easily compute that
γN⊗BΛ=ηN◦
IN⊗τ. (2.20)
We can conclude thatηis a natural isomorphism ifΛis finitely generated and projective as a rightA-module. This is an alternative proof of a special case ofLemma 2.1.
3. Main example. LetΛandAbe rings,i:Λ→Aa ring morphism, andχ:A→Λa right grouplike functional. This means thatχsatisfies the following properties, for all a,b∈A(see [6]):
(1) χis rightΛ-linear; (2) χ(χ(a)b)=χ(ab); (3) χ(1A)=1Λ.
ThenΛ∈ᏹA, with action
λ a=χ(λa). (3.1)
Λis finitely generated, in fact it is a cyclic module, since
λ=χ1Aλ=χ1Aλ=χ1Aλ1Λ=1Λ1Aλ (3.2)
for allλ∈Λ. It is easy to verify that
HomA(Λ,M)MA=m∈M|ma=mχ(a),∀a∈A. (3.3)
In particular,
B=HomA(Λ,Λ)ΛA=b∈Λ|bχ(a)=χ(ba), ∀a∈A. (3.4)
We can apply Theorems2.2and2.3, and we find necessary and sufficient conditions for P∈ᏹBto be projective or flat. Observe thatΛis semi-Σ-projective if the functor(•)A sends an exact sequenceΛ(J)→Λ(I)→M→0 inᏹA to a free resolutionB(J)→B(I)→ MA→0.Λ∈ᏹ
Ais projective if(•)Ais exact.Λis finitely presented as a rightA-module if Kerχis a finitely generated right ideal ofA. We now give some particular examples of ring morphisms with a grouplike functional. More examples can be found in [6].
Example3.1. LetHbe a Hopf algebra over a commutative ringk, andΛa left
H-module algebra. The smash productA=Λ#H is equal toΛ⊗H as ak-module, with multiplication defined by
Let ηH and εH be the unit and counit of H. Then i=IΛ⊗ηH :Λ→Λ#H is a ring morphism, andχ=IΛ⊗εH:Λ#H→Λis a left grouplike functional. It is easy to compute
thatB=ΛA=ΛHis the subring of invariants ofΛ. Thus we find necessary and sufficient conditions for projectivity and flatness of a module over the subring of invariants, and we recover the results obtained by the second author in [10].
In the particular situation whereH=kGis a group ring andΛis aG-module algebra, Ais a skew group ring, and we recover [8, Theorems 3 and 5], in which necessary and sufficient conditions for projectivity and flatness ofΛas aΛG-module are given.
This example can be extended to factorization structures; we refer to [6] for details.
Example3.2. Letj:S→Λbe a morphism of algebras over a commutative ringk.
Theni:Λ→A=Sop⊗Λ,i(λ)=1S⊗λis a ring morphism, andχ:A→Λ,χ(s⊗λ)=j(s)λ is a right grouplike functional. We easily compute that
B=ΛA=b∈Λ|bj(s)=j(s)b, ∀s∈S=ΛS (3.6)
is the centralizer of S inΛ. Thus we obtain necessary and sufficient conditions for projectivity and flatness over the centralizer.
Example3.3. LetΛbe a ring and(Ꮿ,∆Ꮿ,εᏯ)aΛ-coring (this is a coalgebra in the
monoidal categoryΛᏹΛ). Corings were introduced by Sweedler [14]. The language of
corings allows us to generalize, unify, and formulate more elegantly many results about generalized Hopf modules. This is due to the fact that Hopf modules (and their general-izations) are in fact comodules over a coring. This observation can be found implicitly in the proof of [7, Theorem 2.11]. It is tributed to Takeuchi in a Mathematical Review (MR 2000c 16047) written by Masuoka, but the referee informs us that it was already known by Sweedler. Only recently the importance of the coring viewpoint became clear. Brzezi´nski’s [3] is the first of a series of papers on this subject; we refer to [5] for more references and the actual state of the art.
x∈Ꮿis called grouplike if∆Ꮿ(x)=x⊗ΛxandεᏯ(x)=1Λ(see [4]). The left dual of
Ꮿis then a ring:A=ΛHom(Ꮿ,Λ)has an associative multiplication
(f#g)(c)=gc(1)f
c(2). (3.7)
The unit isεᏯ. We have a ring morphism
i:Λ →A, i(λ)(c)=εᏯ(c)λ, (3.8)
and a right grouplike functional
χ:A →Λ, χ(f )=f (x). (3.9)
We easily compute that
B=ΛA=b∈Λ|bf (x)=f (xb),∀f∈A, (3.10)
Acknowledgment. This research is supported by the Project G.0278.01
“Construc-tion and applica“Construc-tions of noncommutative geometry: from algebra to physics” from FWO Vlaanderen.
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S. Caenepeel: Faculty of Applied Sciences, Vrije Universiteit Brussel (VUB), 1050 Brussels, Bel-gium
E-mail address:[email protected]
T. Guédénon: Faculty of Applied Sciences, Vrije Universiteit Brussel (VUB), 1050 Brussels, Bel-gium
