Projectivity and Flatness over the Endomorphism Ring of a Finitely Generated Module

Let A be a ring and Λ a finitely generated A-module. We give necessary and sufficient conditions for projectivity and flatness of a module over the endomorphism ring of Λ.


Introduction.
Suppose that a group G acts as a group of automorphisms on a ring R. Jøndrup [11] has studied when R is projective as an R G -module.García and Del Río [8] have continued Jøndrup's work, and gave necessary and sufficient conditions for the flatness and projectivity of R as an R G -module.In [10], the second author generalized the results from [8] in the following way.Let k be a commutative ring, H a Hopf algebra over k, and Λ a left H-module algebra.Then we can consider the smash product Λ#H and the subring of invariants Λ H . Then we can give necessary and sufficient conditions for the projectivity and flatness over Λ H of a left Λ#H-module P .The results from [8] are recovered if we take k = Z, H = kG, and P = R.
The authors discovered recently that the same methods can be applied to the following situation.Let j : S → Λ be a ring morphism, and consider the centralizer Λ S = {b ∈ Λ | bj(s) = j(s)b, for all s ∈ S}.Then we can give necessary and sufficient conditions for projectivity and flatness over Λ S of 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, Λ is a cyclic module.Such modules appeared recently in [6] and are obtained by considering a ring morphism with a grouplike functional.This will be explained in Section 3.

Main result.
Let A be an associative ring with unit, and Λ ∈ ᏹ A , where ᏹ A is the category of right A-modules and right A-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.Let B = End A (Λ). Then Λ is a (B, A)-bimodule; the left action is just evaluation: b • λ = b(λ), for all b ∈ B and λ ∈ Λ.Then we have an adjoint pair of functors (F , G) connecting the categories of right B-modules and right A-modules: The unit and counit of the adjunction are defined as follows, for M ∈ ᏹ A , N ∈ ᏹ B : 2) The adjointness property then means that we have In particular, η G(M) is monic for every M ∈ ᏹ A .
Recall from [13] that Λ ∈ ᏹ A is called semi-Σ-projective if the functor Hom A (Λ, •) : ᏹ A → Ab sends an exact sequence in ᏹ A of the form to an exact sequence in Ab.Obviously, a projective module is semi-Σ-projective.
(2) It is straightforward to check that the canonical isomorphism B (I) ⊗ B Λ Λ (I) is the composition of the maps and it follows from (1) that ε Λ (I) is an isomorphism.
(3) Putting M = Λ (I) in (2.3), we find From (2), it follows that Hom A (Λ,ε Λ (I) ) is an isomorphism and from (1), it follows that Hom A (Λ, Λ (I) ) B (I) , hence η B (I) is an isomorphism.(4) follows from [13, Theorem 2.1].We include a short proof for the sake of completeness.Take a free resolution B (J) → B (I) → N → 0 of a right B-module N. η is natural, so we have a commutative diagram (J)  GF B (I)  GF (N) 0. (2.7) The top row is exact, the bottom row is exact, since GF (B (I) ) = Hom A (Λ, Λ (I) ) and Λ is semi-Σ-projective.By (3), η B (I) and η B (J) are isomorphisms, and it follows from the five lemma that η N is an isomorphism.
We can now give equivalent conditions for the projectivity and flatness of P ∈ ᏹ B .
Theorem 2.2.Assume that Λ ∈ ᏹ A is finitely generated and let B = End A (Λ).For P ∈ ᏹ B , the following statements are considered: (1) P ⊗ B Λ is projective in ᏹ A and η P is injective; (2) P is projective as a right B-module; (3) P ⊗ B Λ is a direct summand in ᏹ A of some Λ (I) , and η P is bijective; I) , and P

Proof.
(2)⇒(3).If P is projective as a right B-module, then we can find a set I and P ∈ ᏹ B such that B (I) P ⊕ P .Then, obviously, Λ (I) B (I) η is a natural transformation, so we have a commutative diagram From the fact that η B (I) is an isomorphism, it follows that η P (and η P ) are isomorphisms.
(1)⇒ (2).Take an epimorphism f : (2.10) The bottom row is split exact since any functor, in particular, Hom A (Λ, •), preserves split exact sequences.η B (I) is an isomorphism, by Lemma 2.1, and diagram chasing tells us that η P is surjective.By assumption, η P is injective, so η P is bijective, and the top row is isomorphic to the bottom row and therefore split.Thus P ∈ ᏹ B is projective.
(2)⇒( 1).If Λ is projective in ᏹ A , then Λ is a direct summand of A (J) in ᏹ A for some set J. Now P ⊗ B Λ is a direct summand of Λ (I) and also of A (J×I) , so P ⊗ B Λ ∈ ᏹ A is projective.Since (2) implies (3), it follows that η P is injective.
The Govorov-Lazard theorem [9,12] states that M ∈ ᏹ A is flat if and only if it is the inductive limit of a directed system of free modules of finite type.Theorem 2.3 can be viewed as a generalization of this classical result.
Theorem 2.3.Assume that Λ ∈ ᏹ A is finitely presented and let B = End A (Λ).For P ∈ ᏹ B , the following assertions are equivalent: (1) P is flat as a right B-module; (2) for some positive integer n i , and η P is bijective; , and η P is bijective; (4) for some positive integer n i and Hom A (Λ,Q) P in ᏹ B ; (5)  ) as a right A-module and Hom A (Λ,Q) P in ᏹ B .If Λ is semi-Σ-projective as a right A-module, then these conditions are also equivalent to conditions (2) and ( 3) without the assumption that η P is bijective.
Consider the following commutative diagram: (2.12) By Lemma 2.1, the η N i are isomorphisms; the natural homomorphism f is an isomorphism because Λ is finitely presented.Hence η P is an isomorphism.
(5)⇒(1).We have a split exact sequence Consider the following commutative diagram (2.14) We know, from Lemma 2.1, that ε P i is an isomorphism.Both rows in the sequence are split exact, so it follows that ε N i and ε Q i are also isomorphisms.Next, consider the commutative diagram where h and f are the natural homomorphisms.h is an isomorphism because • ⊗ B Λ preserves inductive limits, and f is an isomorphism because Λ ∈ ᏹ A is finitely presented.lim ε Q i is an isomorphism because every ε Q i is an isomorphism, so it follows that ε Q is an isomorphism.From (2.3), we deduce

.16)
Hom A (Λ,ε Q ) is an isomorphism, so η Hom A (Λ,Q) is also an isomorphism, and, since Hom A (Λ,Q) P , η P is an isomorphism.Consider the isomorphisms (2.17) It follows from Lemma 2.1 that Hom A (Λ,P i ) B (I i ) is projective as a right B-module, hence Hom A (Λ,Q i ) is also projective as a right B-module, and we conclude that P ∈ ᏹ B is flat.The final statement is an immediate consequence of Lemma 2.1(4).
Remark 2.4.To Λ ∈ ᏹ A , we can associate a Morita context (cf.[2]), namely, (A, B, Λ, Λ * = Hom A (Λ,A),τ,µ) with connecting maps (2.18) τ is surjective (and a fortiori bijective) if and only if Λ is finitely generated and projective as a right A-module.In this situation, the natural transformation For any N ∈ ᏹ B , we easily compute that We can conclude that η is a natural isomorphism if Λ is finitely generated and projective as a right A-module.This is an alternative proof of a special case of Lemma 2.1.

Main example.
Let Λ and A be rings, i : Λ → A a ring morphism, and χ : A → Λ a right grouplike functional.This means that χ satisfies the following properties, for all a, b ∈ A (see [6]): (1) χ is right Λ-linear; (2) Λ is finitely generated, in fact it is a cyclic module, since for all λ ∈ Λ.It is easy to verify that In particular, We can apply Theorems 2.2 and 2.3, and we find necessary and sufficient conditions for P ∈ ᏹ B to be projective or flat.Observe that Λ is semi-Σ-projective if the functor (•) A sends an exact sequence Λ is finitely presented as a right A-module if Ker χ is a finitely generated right ideal of A. We now give some particular examples of ring morphisms with a grouplike functional.More examples can be found in [6].
Example 3.1.Let H be a Hopf algebra over a commutative ring k, and Λ a left Hmodule algebra.The smash product A = Λ#H is equal to Λ ⊗ H as a k-module, with multiplication defined by Let η H and ε H be the unit and counit of H. Then i = I Λ ⊗ η H : Λ → Λ#H is a ring morphism, and χ = I Λ ⊗ε H : Λ#H → Λ is a left grouplike functional.It is easy to compute that B = Λ A = Λ H is the subring of invariants of Λ.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 where H = kG is a group ring and Λ is a G-module algebra, A is a skew group ring, and we recover [8, Theorems 3 and 5], in which necessary and sufficient conditions for projectivity and flatness of Λ as a Λ G -module are given.
This example can be extended to factorization structures; we refer to [6] for details.
Example 3.2.Let j : S → Λ be a morphism of algebras over a commutative ring k.Then i : Λ → A = S op ⊗Λ, i(λ) = 1 S ⊗λ is a ring morphism, and χ : A → Λ, χ(s ⊗λ) = j(s)λ is a right grouplike functional.We easily compute that is the centralizer of S in Λ.Thus we obtain necessary and sufficient conditions for projectivity and flatness over the centralizer.
Example 3.3.Let Λ be a ring and (Ꮿ, ∆ Ꮿ ,ε Ꮿ ) a Λ-coring (this is a coalgebra in the monoidal category Λ ᏹ Λ ).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 generalizations) 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.Brzeziński'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 ∈ Ꮿ is called grouplike if ∆ Ꮿ (x) = x ⊗ Λ x and ε Ꮿ (x) = 1 Λ (see [4]).The left dual of Ꮿ is then a ring: A = Λ Hom(Ꮿ, Λ) has an associative multiplication (2) . (3.7) The unit is ε Ꮿ .We have a ring morphism and a right grouplike functional We easily compute that and we find necessary and sufficient conditions for projectivity and flatness of a right B-module.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation