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REFLEXIVITY REVISITED MOHSEN ASGHARZADEH ABSTRACT. We study some aspects of reflexive modules. For example, we search conditions for which reflexive modules are free or close to free modules. 1. INTRODUCTION In this note (R, m, k) is a commutative noetherian local ring and M is a finitely generated R- module, otherwise specializes. The notation stands for a general module. For simplicity, M the notation ∗ stands for HomR( , R). Then is called reflexive if the natural map ϕ : M M M M is bijection. Finitely generated projective modules are reflexive. In his seminal paper M→M∗∗ Kaplansky proved that projective modules (over local rings) are free. The local assumption is really important: there are a lot of interesting research papers (even books) on the freeness of projective modules over polynomial rings with coefficients from a field. In general, the class of reflexive modules is extremely big compared to the projective modules. As a generalization of Seshadri’s result, Serre observed over 2-dimensional regular local rings that finitely generated reflexive modules are free in 1958. This result has some applications: For instance, in the arithmetical property of Iwasawa algebras (see [45]). It seems freeness of reflexive modules is subtle even over very special rings. For example, in = k[X,Y] [29, Page 518] Lam says that the only obvious examples of reflexive modules over R : (X,Y)2 are the free modules Rn. Ramras posed the following: Problem 1.1. (See [19, Page 380]) When are finitely generated reflexive modules free? Over quasi-reduced rings, problem 1.1 was completely answered (see Proposition 4.22). Ram- ras proved any finitely generated reflexive module over BNSI (betti numbers strictly increasing) rings is free. We present some applications of this result. Also, we introduce the class of eventually BNSI rings and we study freeness of reflexive modules over them. We know any nonzero free module decomposes into a direct sum of rank one submodules. Treger conjectured (see [50, Page arXiv:1812.00830v4 [math.AC] 23 Dec 2019 462]): Conjecture 1.2. Let (R, m, k) be a complete local (singular and containing a field) normal domain of dimension 2 where k = k and char(k) = 2. Then R is a cone such as k[[x,y,z]] if and only if 6 (x2+y2+z2) every nonzero reflexive module M decomposes into a direct sum of rank one submodules. 2 connects reflexivity to the finiteness conditions. As an application, we study the following: § When are quasi-reflexive modules flat? 3 collects different notions of reflexivity. Also, Propo- § sition 3.8 supports a question of Iyengar. 4 deals with freeness of reflexive modules over some § 2010 Mathematics Subject Classification. Primary 13D02; 13C10 . Key words and phrases. Betti numbers; free modules; reflexive modules; splitting. 1 2 classes of rings. In 5 we investigate the reflexivity of dual modules. In 6 we settle Conjecture § § 1.2. In 7 we deal with a question of Braun: Let I ✁ R be a reflexive ideal of a normal domain with § id (I) < ∞. Is I ω ? R ≃ R In 8 we descent freeness (resp. reflexivity) from the endomorphism ring to the module. This § is inspired by the paper of Auslander-Goldman. Similarly, we descent some data from the higher tensor products to the module. In particular, we slightly extend some results of Vasconcelos, Huneke-Wiegand and the recent work of Cesnaviˇcius.ˇ Grothendieck solved a conjecture of Samuel, see Theorem 9.1. In 9 we try to understand § this miracles by looking at the mentioned result of Auslander-Goldman. In particular, there is a connection between Problem 1.1 and UFD property of regular (complete-intersection) rings, see Corollary 9.3 as a sample. Samuel remarked that there is no symmetric analogue of Auslander’s theorem on torsion part of tensor powers. We present a tiny remark (also see Corollary 9.9): Observation. Let (R, m) be a regular local ring and M be of rank one. If Symn(M) is reflexive for some n max 2, dim R , then M is free. ≥ { } 2. REFLEXIVITY AND FINITENESS All rings are noetherian. Following Bass, is called torsion-less if ϕ is injective (this some M M times is called semi-reflexive). A torsion-less module is noetherian if and only if is noe- M M∗ therian. Submodules of a torsion-less module are torsion-less. We say is weakly reflexive if ϕ M M is surjective. In general, neither submodule nor quotient of a weakly reflexive module is weakly reflexive. Observation 2.1. Let (R, m) be a zero-dimensional Gorenstein local ring. Then the properties of weakly reflexive and finitely generated are the same. In particular, (weakly) reflexivity is not closed under taking direct limits. Proof. Here we use the concept of Gorenstein-projective. For its definition, see [14, 4.2.1]. Since R is zero-dimensional and Gorenstein, any module is Gorenstein-projective (see [14, 4.4.8]). It follows by definition that any module is a submodule of a projective module. Over local rings, and by the celebrated theorem of Kaplansky, any projective module is free. Combining these, any module is a submodule of a free module. It follows by definition that any module is torsion-less. Now, let be weakly reflexive. Thus, is reflexive. It is shown in [44, Corollary 2.4.(4)] that M M over any commutative artinian ring, every reflexive module is finitely generated. By this, is M finitely generated. Conversely, assume that M is finitely generated. Since R is zero-dimensional and Gorenstein, M is reflexive. To see the particular case, we remark that any module can be written as a directed limit of finitely generated modules. We use this along with the first part to get the claim. Discussion 2.2. The zero dimensional assumption is important. Indeed, by [29, Ex. 2.8’(2)], LN Z is reflexive. This is one of two extra-credit exercises in the book [29]. There are flat modules that are not reflexive, e.g. the vector space LN Q and the abelian group Q. In order to handle this drawback, let R be a normal domain of dimension bigger than zero with 3 fraction field Q(R). Following Samuel, is called quasi-reflexive if = Tp Spec(R),ht(p)=1 p M M ∈ M where we compute the intersection in := Q(R). Yuan proved that any flat module is M0 M ⊗R quasi-reflexive (see [57, Lemma 2]). Lemma 2.3. Let (R, m) be a normal domain of dimension bigger than zero and be quasi-reflexive. Then M there is a family of finitely generated and reflexive modules Mi i I such that = limMi. { } ∈ M −→ Proof. There is a directed family Ni i I of finitely generated submodules of such that = 1 { } ∈ M M limN . Let Spec (R) := p Spec(R), ht(p) = 1 . For each i, we set M := 1 (N )p i { ∈ } i Tp Spec (R) i −→ 1 ∈ where we compute the intersection in . Let p Spec (R). We note that M (N )p p M0 ∈ i ⊂ i ⊂ M and so Mi Tp Spec1(R) p = . Suppose Ni Nj. It follows that (Ni)p (Nj)p p. ⊂ ∈ M M ⊂ ⊂ ⊂ M Consequently, \ (N )p \ (N )p \ p = . i ⊂ j ⊂ M M p Spec1(R) p Spec1(R) p Spec1(R) ∈ ∈ ∈ This says that M M . Let x . There is a j I such that x N . Since N M , we i ⊂ j ⊂ M ∈ M ∈ ∈ j j ⊆ j have x M limM , i.e., limM . The reverse inclusion holds by definition. In sum, = ∈ j ⊂ j M ⊆ i M −→ −→ limMi. Clearly, Ni∗∗ is finitely generated. In view of Fact 5.1 (see below) Ni∗∗ is reflexive. Again, 1 let−→p Spec (R). Note that Rp is a discrete valuation ring. Over such a ring any torsion-free is ∈ free, and so reflexive. Since Ni is finitely generated, HomR(Ni, R) commutes with the localization. Thus, (Ni)p (Ni)p∗∗ (Ni∗∗)p. By e.g. [57, Proposition 1], Ni∗∗ Tp Spec1(R)(Ni∗∗)p. We put ≃ ≃ ≃ ∈ these together, N∗∗ \ (N∗∗)p \ (N )p. i ≃ i ≃ i p Spec1(R) p Spec1(R) ∈ ∈ From this, M N . In particular, M is finitely generated and reflexive. This completes the i ≃ i∗∗ i proof. Proposition 2.4. Let (R, m) be a regular local ring of dimension at most two. There is no difference between flat modules and quasi-reflexive modules. In particular, any direct limit of quasi-reflexive modules is quasi-reflexive. Proof. Recall that any flat module is quasi-reflexive. Conversely, let be quasi-reflexive. By M the above lemma, there is a family of finitely generated reflexive modules M such that = { i} M limMi. By the mentioned result of Serre, each Mi is free. Clearly, direct limit of free modules is flat.−→ We apply this to observe that is flat. To see the particular case, we mention that direct M limit of flat modules is again flat. Corollary 2.5. Let (R, m, k) be a normal domain of dimension bigger than zero. The following are equiva- lent: i) there is no difference between flat modules and quasi-reflexive modules, ii) R is a regular local ring of dimension at most two. Proof. i) ii): The second syzygy of k is reflexive and so quasi-reflexive. By the assumption it ⇒ is flat. Finitely generated flat modules over local rings are free. Thus, second syzygy of k is free. This in turn is equivalent with p. dim(k) 2. In view of local-global-principle, R is regular and ≤ of dimension at most two. 4 ii) i): This is in the previous Proposition.
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