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On Flatness and Coherence with Respect to Modules of Flat Dimension

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On Flatness and Coherence with Respect to Modules of Flat Dimension On flatness and coherence with respect to modules of flat dimension at most one Samir Bouchiba∗ and Mouhssine El-Arabi† Department of Mathematics, Faculty of Sciences, University Moulay Ismail, Meknes, Morocco‡§ November 9, 2020 Abstract This paper introduces and studies homological properties of new classes of modules, fp namely, the F1-flat modules and the F1 -flat modules, where F1 stands for the class of fp right modules of flat dimension at most one and F1 its subclass consisting of finitely fp presented elements. This leads us to introduce a new class of rings that we term F1 - fp coherent rings as they behave nicely with respect to F1 -flat modules as do coherent fp rings with respect to flat modules. The new class of F1 -coherent rings turns out to be a large one and it includes coherent rings, perfect rings, semi-hereditary rings and all rings R such that lim P = F . As a particular case of rings satisfying lim P = F −→ 1 1 −→ 1 1 figures the important class of integral domains. 1 Introduction Throughout this paper, R denotes an associative ring with unit element and the R-modules are supposed to be unital. Given an R-module M, M + denotes the character R-module Q arXiv:2011.03377v1 [math.AC] 6 Nov 2020 of M, that is, M + := Hom M, , pd (M) denotes the projective dimension of M, Z Z R idR(M) the injective dimension of M and fdR(M) the flat dimension of M. As for the global dimensions, l-gl-dim(R) designates the left global dimension of R and wgl-dim(R) the weak global dimension of R. Mod(R) stands for the class of all right R-modules, P(R) stands for the class of all projective right R-modules, I(R) the class of all injective right R-modules and F(R) the class of flat right modules. Also, we denote by F1 (resp., P1) the class of fp fp right R-modules M such that fdR(M) ≤ 1 (resp., pdR(M) ≤ 1) and by F1 (resp., P1 ) the ∗Corresponding author: [email protected] †[email protected] ‡Mathematics Subject Classification 2010 : 13D02; 13D05; 13D07; 16E05; 16E10. § fp fp Key words and phrases: F1-flat module; F1 -flat module; F1 -coherent ring; Tor-torsion theory; global dimension. 1 On flatness and coherence with respect to modules of flat dimension at most one 2 subclass of F1 (resp., P1) consisting of right R-modules which are finitely presented. Any unreferenced material is standard as in [11, 12, 13, 14]. Our main purpose in this paper is to study the homological properties of the Tor- ⊤ fp ⊤ fp orthogonal classes F1 and F1 of F1 and F1 that we term the class of F1-flat modules fp and the class of F1 -flat modules, respectively, over an arbitrary ring R. Let us denote these ⊤ fp fp ⊤ two classes by F1F(R) := F1 and F1 F(R) := F1 . Note that in the context of integral domains, by [9, Lemma 2.3], F1F(R) coincides with the class of torsion-free R-modules. In other words, the F1-flat module notion extends that of the torsion-free module from integral domains to arbitrary rings, and thus our study will permit, in particular, to shed light on homological properties of torsion-free modules in the case when R is a domain. In section fp 2, we introduce and study the F1-flat modules and F1 -flat modules. We mainly seek condi- fp tions on rings R for which the two notions of F1-flat module and F1 -flat module collapse. fp fp In particular, we prove that if lim P1 = F1, then lim F = F1 and thus F1F(R)= F F(R). −→ −→ 1 1 fp This leads us to seek suffisant conditions on a ring R for which lim F = F1 or more gen- −→ 1 erally lim P1 = F1. It is worthwhile pointing out that if R is an integral, then lim P1 = F1 −→ −→ [8, Theorem 3.5] and that this theorem is subsequently generalized by Bazzoni and Herbera in [2, Theorem 6.7 and Corollary 6.8]. Moreover, we exhibit an example of a ring R such fp fp that lim F =6 F1 showing that, in general, the concerned classes F1F(R) and F F(R) are −→ 1 1 different. fp Section 3 introduces the F1 -coherent rings. This new class of rings behaves nicely with fp respect to F1 -flat modules as do coherent rings with respect to flat modules. In particular, fp fp it is proven that a ring R is F1 -coherent if and only if any product of F1 -flat modules is F fp-flat if and only if lim Pfp = lim F fp. It turns out that the class of F fp-coherent rings is a 1 −→ 1 −→ 1 1 large one and it includes integral domains, coherent rings, semi-hereditary rings and perfect fp rings which permits to unify all these classes of rings into one. Also, the class of F1 -coherent rings includes all rings such that lim P1 = F1. Finally, we characterize rings R for which all −→ fp R-modules are F1 -flat and discuss the homological dimensions of the R-modules as well as the global dimensions of R in terms of the homological dimensions of the F1-flat modules. fp 2 F1-flat and F1 -flat modules fp This section introduces and studies the notions of F1-flat and F1 -flat modules as being the ⊤ fp ⊤ fp fp Tor-orthogonal classes F1 and F1 of F1 and F1 . Observe that F1F(R) ⊆F1 F(R). We fp mainly seek conditions on rings R for which the two notions of F1-flat module and F1 -flat fp fp module collapse. In particular, we prove that if lim F = F1, then F1F(R) = F F(R). −→ 1 1 fp Also, we exhibit an example R such that lim F =6 F1 showing that, in general, the con- −→ 1 fp cerned classes F1F(R) and F1 F(R) are different. Let C be a class of right R-modules and D be a class of left R-modules. We put ⊤ R R C = ker Tor1 (C, -)= {left R-modules M : Tor1 (C, M)=0 for all C ∈ C} On flatness and coherence with respect to modules of flat dimension at most one 3 and ⊤ R R D = ker Tor1 (-, D)= {right R-modules N : Tor1 (N, D)=0 for all D ∈ D}. A pair (A, B) of classes of R-modules is called a Tor-torsion theory if A = ⊤B and B = A⊤. Let C be a class of right R-modules. Then it is easy to check that (⊤(C⊤), C⊤) is a Tor-torsion theory. Also, we put C := ⊤(C⊤). Note that lim C ⊆ C as C is stable under direct limits. −→ A Tor-torsion theory (Ab , B) is said to be generated by Cb if Ab= C (and thus B = C⊤). Let (A1, B1) and (A2, B2) two Tor-torsion theories generated by C1 andb C2, respectively. Then the two pairs (A1, B1) and (A2, B2) coincide if and only if C1 = C2. b b We begin by proving the following lemma of general interest. Lemma 2.1. Let R be a ring and C and D be classes of right R-modules. 1) (lim C)⊤ = C⊤ = C⊤. −→ b 2) If C⊆D⊆ C, then C = D. 3) If lim C = limb D, thenb C⊤b= D⊤ and C = D. −→ −→ b b Proof. 1) Note that C⊤ = C⊤ and that C⊤ ⊆ (lim C)⊤ ⊆ C⊤ as C ⊆ lim C ⊆ C. Then the −→ −→ result easily follows. b b b 2) Assume that C⊆D⊆ C. Then C ⊆ D ⊆ C. Now, as C = C, we get C = D, as desired. 3) It follows easily from (1).b b b bb bb b b b 1 Definition 2.2. 1) A left R-module M is said to be F1-flat if TorR(H, M)=0 for each ⊤ right module H ∈ F1, that is, M ∈ F1 . The class of all left F1-flat modules is denoted by F1F(R). fp 1 fp 2) A left R-module M is said to be F1 -flat if TorR(H, M)=0 for each right module H ∈F1 , fp ⊤ fp fp that is, M ∈F1 . The class of all left F1 -flat modules is denoted by F1 F(R). fp Next, we list some properties of F1-flat and F1 -flat modules. Proposition 2.3. Let R be a ring. Then fp 1) F1F(R) ⊆F1 F(R). fp 2) F1F(R) and F1 F(R) are stable under direct sums and direct limits. fp 3) F1F(R) and F1 F(R) are stable under submodules. fp 4) Any left ideal of R is F1-flat and F1 -flat. R Proof. 1) and 2) are clear as the functor Torn (H, −) commutes with direct sums and direct limits for any right R-module H and each positive integer n. 3) Let N be a submodule of a left F1-flat module M. Let H ∈ F1be a right module and M consider the short exact sequence 0 −→ N −→ M −→ −→ 0 of left modules. Then N applying the functor H ⊗R −, we get the exact sequence M TorR H, −→ TorR(H, N) −→ TorR(H, M). 2 N 1 1 On flatness and coherence with respect to modules of flat dimension at most one 4 M Now, as TorR(H, M)=0 since M is F -flat and TorR H, =0 as fd (H) ≤ 1, we deduce 1 1 2 N R R that Tor1 (H, N)=0.
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