Theory and Applications of Categories, Vol. 10, No. 9, 2002, pp. 180–186.

ON SOME PROPERTIES OF PURE MORPHISMS OF COMMUTATIVE RINGS

BACHUKI MESABLISHVILI

ABSTRACT. We prove that pure morphisms of commutative rings are effective 𝔸descent morphisms where 𝔸 is a (COMMUTATIVE RINGS)^{op}-indexed category given by (i) finitely generated modules, or (ii) flat modules, or (iii) finitely generated flat modules, or (iv) finitely generated projective modules.

1. Introduction

Let E be a category with pullbacks, and let 𝔸: E^{op} [figure] ℂ𝔸𝕋 be an indexed category [PS]. If C is an internal category of E, then one defines an ordinary category 𝔸^{C} of C-diagrams in 𝔸, and the assignment C ↦ 𝔸^{C} induces a pseudo-functor

𝔸^{(−)} : cat(E)^{op} [figure] ℂ𝔸𝕋

of 2-categories (see [JT2]) where cat(E) denotes the 2-category of internal categories of E.

Let p : E [figure] B be a morphism in E. Then, p gives rise to an internal category Eq(p) of E, namely the equivalence relation induced by p, and to a fully faithful (internal) functor

\overline{p} : Eq(p) [figure] B,

where B is the discrete internal category on B. The category 𝔸^{Eq(p)} is called the category of 𝔸-descent data relative to p, and denoted by Des_{𝔸}(p).

The pseudo-functor

𝔸^{(−)} : cat(E)^{op} [figure] ℂ𝔸𝕋

carries \overline{p} into an ordinary functor

ϕ^{p} : 𝔸^{B} [figure] Des_{𝔸}(p),

and one says that p is an effective 𝔸-descent morphism if ϕ^{p} is an equivalence of categories. More details on Descent Theory can be found in [JT1, JT2]. We consider the case where E is the opposite of the category of commutative rings

with unit, and 𝔸 is given by (i) finitely generated modules, or (ii) flat modules, or (iii) finitely generated flat modules, or (iv) finitely generated projective modules.

The aim of the paper is to prove, 1.1. Theorem. Let f : R [figure]S be a pure morphism of commutative rings with unit. Then f is an effective 𝔸-descent morphism.

Received by the editors 2001 December 6 and, in revised form, 2002 March 7.

Transmitted by W. Tholen. Published on 2002 April 12.

2000 Mathematics Subject Classification: 13B02, 13B99, 18A20, 18A22, 18D30.

Key words and phrases: Indexed categories, effective descent morphisms, pure morphisms.

\begin{matrix}c \\ ◯ \end{matrix} Bachuki Mesablishvili, 2002. Permission to copy for private use granted.

2. Preliminaries

Let E be a category with pullbacks and let

𝔸, 𝔹 : E^{op} [figure] ℂ𝔸𝕋

be E-indexed categories. We say that 𝔸 is an E-indexed full subcategory of 𝔹 if there exists an E-indexed functor F : 𝔸 [figure] 𝔹 such that the functor F (e) : 𝔸^{e} [figure] 𝔹^{e}, e ∈ Ob(E), is a fully faithful embedding.

The key role in proving the main result will be Corollary 2.7 of [JT1] that we now state in its indexed version, and since the same sort of proof as given in [JT1] works, we do not give a proof here. 2.1. Theorem. [Janelidze-Tholen] Let 𝔸, 𝔹 : E^{op} [figure]ℂ𝔸𝕋 be E-indexed categories. Suppose 𝔸 is a E-indexed full subcategory of 𝔹. A morphism p : E [figure]B of E which is an effective 𝔹-descent morphism is also an effective 𝔸-descent morphism if and only if the following condition is satisfied: for each x ∈ 𝔹^{B}, p^{∗}(x) ∈ 𝔸^{E}, where p^{∗} : 𝔸^{B} [figure] 𝔸^{E} is the change-of-base functor, implies x ∈ 𝔸^{B}.

Let R be a commutative ring with unit. A morphism f : M [figure] M′ of R-modules is called pure if

1_{L} ⊗_{R} f : L ⊗_{R} M [figure] L ⊗_{R} M′

is a monomorphism for every R-module L.

For any R-module M, the abelian group Hom_{𝔸b}(M, ℚ/ℤ) (where ℚ/ℤ denotes the abelian group of rationals mod 1) is an R-module via the action (r.h)(m) = h(r.m) (r ∈ R, m ∈ M, h ∈ Hom_{𝔸b}(M, ℚ/ℤ)). So one can define a functor

C_{R} : (R-mod) [figure] (R-mod)^{op}

given by C_{R}(M) = Hom_{𝔸b}(M, ℚ/ℤ). C_{R} is clearly an exact and faithful functor since ℚ/ℤ is an injective cogenerator for the category of abelian groups (see, for example, [F]). 2.2. Proposition. [see [Me], [L]] A morphism f : M [figure] M′ of R-modules is pure if and only if C_{R}(f) is a split epimorphism of R-modules.

A morphism f : R [figure] S of commutative rings with unit gives rise to two functors (i) f^{∗} : S-mod [figure] R-mod, where for any S-module L, f^{∗}(L) is the underlying abelian

group L with the R-action given by r.l = f(r).l (r ∈ R, l ∈ L); (ii) f_{!} : R-mod [figure] S-mod, where for any R-module M, f_{!}(M) = S ⊗_{R} M. It is well known that the functor f_{!} is left adjoint to the functor f^{∗}. Each component

η_{M} : M [figure] S ⊗_{R} M of the unit η : 1 [figure] f^{∗}f_{!} of the adjunction is given by η_{M}(m) = 1 ⊗_{R} m.

We denote by C_{R}^{S} the composite

C_{R} ∘ f^{∗} ∘ f_{!} : (R-mod) [figure] (R-mod)^{op}

2.3. Proposition. The following statements are equivalent for a morphism f : R [figure] S of commutative rings with unit:

(i) C_{R}(f) is a split epimorphism of R-modules;

(ii) f is pure (i.e., f is pure as an R-module morphism);

(iii) The natural transformation

C_{R}.η : C_{R}^{S} [figure] C_{R}

splits; i.e., there exists a natural transformation

τ : C_{R} [figure] C_{R}^{S}

such that (C_{R}.η).τ ≈ 1.

Proof. (i) ⇔ (ii): It follows from the definition of pure morphisms and Proposition 2.2.

(i) ⇔ (iii): To say C_{R}.η splits is to say Hom_{R}(−, C_{R}(f)) (and hence, by the Yoneda lemma, C_{R}(f)) splits because we have the following commutative (up to an isomorphism) diagram:

[figure: Hom_{R}(−, C_{R}(S)); Hom_{R}(−, C_{R}(R)); Hom_{R}(−,C_{R}(R)); C_{R}^{S}; Hom_{R}(−, C_{R}(S)); ≈; C_{R}^{S}; C_{R}; C_{R}.η; C_{R}; Hom_{R}(−, C_{R}(R)); ≈]

where the vertical morphisms are the canonical isomorphisms.

Recall that an exact sequence in R-mod

0  [figure] M  [figure] M′  [figure] M′′  [figure] 0

is called pure-exact iff the induced sequence

0  [figure] L ⊗_{R} M  [figure] L ⊗_{R} M′  [figure] L ⊗_{R} M′′  [figure] 0

is an exact sequence for every R-module L.

Using Proposition 2.2, we get

2.4. Proposition. [see [L]] The following properties are equivalent for an exact sequence

0  [figure] M  [figure] M′  [figure] M′′  [figure] 0

(i) It is pure-exact;

(ii) The exact sequence in R-mod

0  [figure] C_{R}(M′′)  [figure] C_{R}(M′)  [figure] C_{R}(M)  [figure] 0

splits.

We close this section by recalling a result on flat modules (see, e.g., [S]).

2.5. Proposition. The following properties of an R-module M are equivalent: (i) M is a flat R-module; (ii) Every exact sequence

0  [figure] M′  [figure] M′′  [figure] M  [figure] 0

is pure-exact.

3. Proof of the main theorem

In the sequel we always assume that E is the opposite of the category of commutative rings with unit.

We start by recalling from [Me] the following result:

3.1. Theorem. Let 𝕄 : E^{op} [figure] ℂ𝔸𝕋 be the E-indexed category given by all modules. A morphism in E is an effective 𝕄-descent morphism if and only if it is pure.

It is clear that 𝔸 : E^{op} [figure]ℂ𝔸𝕋 as in the Main Theorem, is an E-indexed full subcategory of 𝕄. Then, in order to prove the Main Theorem, by Theorem 2.1 and Theorem 3.1, it is sufficient to prove the following

3.2. Theorem. Let f : R [figure]S be a pure morphism of commutative rings with unit, and let M ∈ Ob(R-mod). Then

(i) If S⊗_{R}M is a finitely generated S-module, then M is a finitely generated R-module. (ii) If S ⊗_{R} M is a flat S-module, then M is a flat R-module. (iii) If S ⊗_{R} M is a finitely generated flat S-module, then M is a finitely generated

flat R-module.

(iv) If S⊗_{R}M is a finitely generated projective S-module, then M is a finitely generated R-module.

3.3. Remark. The above theorem tells us that the properties of modules of being finitely generated, flat, finitely generated flat, finitely generated projective descend along pure morphisms of commutative rings.

Proof. (i) Suppose S ⊗_{R} M is finitely generated over S. Then we can choose a finite family of elements {s_{i,k}, m_{i,k}}, where s_{i,k} ∈ S, m_{i,k} ∈ M, such that the finite family {∑ s_{i,k} ⊗_{R} m_{i,k}}_{i,k} generates the S-module S ⊗_{R} M. Let M′ denote the R-submodule of M generated by the finite family {m_{i,k}}_{i,k}. It is clear that the induced morphism 1_{S} ⊗_{R} i : S ⊗_{R} M′ [figure] S ⊗_{R} M, where i : M′ [figure] M is the canonical embedding, is an epimorphism. But since f is pure by assumption, the functor

f_{!} : R-mod [figure] S-mod

is faithful (see, for example, [B]); and since any faithful functor between arbitrary categories reflects epimorphisms (see [Mt]), i : M′ [figure] M is an epimorphism, and hence an isomorphism since i is a monomorphism. It means that M is finitely generated R-module.

(ii) Suppose S ⊗_{R} M is a flat S-module. According to Proposition 2.5, it suffices to show that each exact sequence

0  [figure] M′ [figure: u] M′′ [figure: v] M  [figure] 0  (1)

is pure-exact. By Proposition 2.4 this sequence is pure-exact iff the exact sequence in R-mod

0  [figure] C_{R}(M) [figure: C_{R}(v)] C_{R}(M′′) [figure: C_{R}(u)] C_{R}(M′)  [figure] 0  (2)

splits. Since v : M′′ [figure] M is an epimorphism, the sequence in R-mod

S ⊗_{R} M′′ [figure: 1_{S}⊗_{R}v] S ⊗_{R} M  [figure] 0  (3)

is exact. This sequence is also exact in S-mod since the functor

f^{∗} : S-mod [figure] R-mod

is faithful, and hence reflects epimorphisms (see [Mt]).

Therefore, (3) is an exact sequence in S-mod, and so is the sequence

0 [figure] C_{S}(S ⊗_{R} M) [figure: C_{S}(1⊗_{R}v)] C_{S}(S ⊗_{R} M′′)

since the functor

C_{S} : (S-mod) [figure] (S-mod)^{op}

is exact.

C_{S}(S ⊗_{R} M) is an injective S-module since, by assumption, S ⊗_{R} M is a flat Smodule (see, e.g., [F]). Hence the last sequence of S-modules splits. But it is clear that f^{∗} ∘ C_{S} ∘ f_{!} = C_{R}. Hence the exact sequence in R-mod

0 [figure] C_{R}(S ⊗_{R} M) [figure: C_{R}(1⊗_{R}v)] C_{R}(S ⊗_{R} M′′)

also splits, i.e., there is an R-module morphism

h : C_{R}(S ⊗_{R} M′′) [figure] C_{R}(S ⊗_{R} M)

such that h.C_{R}(1 ⊗_{R} v) = 1.

By naturality of τ (see Proposition 2.3.), one has the following commutative diagram

[figure: C_{R}(S ⊗_{R} M); C_{R}(1 ⊗_{R} v); C_{R}(1⊗_{R}v); C_{R}(M); C_{R}(S ⊗_{R} M); τ_{M}; C_{R}(M); C_{R}(M′′); C_{R}(v); C_{R}(M′′); C_{R}(1 ⊗_{R} v); τ_{M′′}]

Let us show that C_{R}(v) splits by the composition

C_{R}(f ⊗_{R} 1).h.τ_{M′′} : C_{R}(M′′) [figure] C_{R}(M).

Indeed, we know that h.C_{R}(1 ⊗_{R} v) = 1, and, moreover, commutativity of the diagram implies τ_{M′′}.C_{R}(v) = C_{R}(1 ⊗_{R} v).τ_{M}. Then

\begin{aligned}C_{R}(f ⊗_{R} 1).h.τ_{M′′}.C_{R}(v) & = C_{R}(f ⊗_{R} 1).h.C_{R}(1 ⊗_{R} v).τ_{M} \\ & = C_{R}(f ⊗_{R} 1).τ_{M}. \end{aligned}

Since the composite of f ⊗_{R} 1 with the canonical isomorphism M ≅ R ⊗_{R} M is the M-component μ_{M} of μ : 1 [figure] f^{∗}f_{!}, Proposition 2.3 gives C_{R}(f ⊗_{R} 1).τ_{M} = 1. So

(C_{R}(f ⊗_{R} 1).h.τ_{M′′}).C_{R}(v) = 1.

It means that the sequence (2) splits, and, by Proposition 2.4, the sequence (1) is pureexact. Consequently, M is a flat R-module by Proposition 2.5.

(iii) It follows immediately from (i) and (ii).

(iv) If S_{R} ⊗ M is finitely generated , then, in particular, it is finitely generated over S , and since every projective module is flat, by (i) and (ii), M is finitely generated flat module over R.

Since every finitely generated projective module is finitely presented, there exists an exact sequence in S-mod

0  [figure] X  [figure] S^{n}  [figure] S ⊗_{R} M  [figure] 0  (4)

where X is a finitely generated S-module.

Moreover, since M is finitely generated over R, there exists an exact sequence in R-mod

0  [figure] M′  [figure] R^{m}  [figure] M  [figure] 0,

and since M is a flat R-module, by Proposition 2.5, this sequence is pure-exact. Therefore the sequence in S-mod

0  [figure] S ⊗_{R} M′  [figure] S ⊗_{R} R^{m}  [figure] S ⊗_{R} M  [figure] 0  (5)

is exact.

Apply Schanuel’s lemma to (4) and (5); we get

X ⊕ (S ⊗_{R} R^{m}) ≈ S^{n} ⊕ (S ⊗_{R} M′).

It follows that S ⊗_{R} M′ is finitely generated over S, and by (i), M′ is finitely generated over R. Therefore, M is finitely presented over R. But, as we have seen, M is a flat R-module, and since every finitely presented flat module is projective (see, for example, [F]), M is a finitely generated projective R-module.

This completes the proof.

Acknowledgements

This work was partially supported by INTAS-97-31961.

References

[B] F. Borceux, Handbook of Categorical Algebra, Vol. 1, Cambridge University Press, 1994.

[F] C. Faith, Algebra: Rings, Modules and Categories, Vol. 1, Springer-Verlag, 1973.

[JT1] G. Janelidze and W. Tholen, Facets of Descent, I, Appl. Categorical Structures, 2 (1994), 245–281.

[JT2] G. Janelidze and W. Tholen, Facets of Descent, II, Appl. Categorical Structures, 5 (1997), 229–248.

[L] T. Y. Lam, Lectures on Modules and Rings, Graduate Texts in Math. Vol. 109, Springer-Verlag, 1999.

[Me] B. Mesablishvili, Pure Morphisms of Commutative Rings Are Effective Descent Morphisms for Modules – A New Proof, Theory and Applications of Categories, 7 (2000), 38–42.

[Mt] B. Mitchell, Theory of Categories, Academic Press, 1965.

[S] B. Stenstrőm, Rings of Quotients, Lecture Notes in Mathematics, 237, Springer-Verlag, 1971.

[PS] R. Paré and D. Schumacher, Abstract Families and the Adjoint Functor Theorems, Lecture Notes in Mathematics, 661, Springer-Verlag 1978, 1–125.

A. Razmadze Mathematical Institute of the Georgian Academy of Sciences, M.Alexidze St. 1, Tbilisi 380093, Georgia

Email: bachi@rmi.acnet.ge

This article may be accessed via WWW at http://www.tac.mta.ca/tac/ or by anony-

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