Theory and Applications of Categories, Vol. 7, No. 3, 2000, pp. 38–42.

PURE MORPHISMS OF COMMUTATIVE RINGS ARE EFFECTIVE DESCENT MORPHISMS FOR MODULES – A NEW PROOF

BACHUKI MESABLISHVILI

Transmitted by W. Tholen

ABSTRACT. The purpose of this paper is to give a new proof of the Joyal-Tierney theorem (unpublished), which asserts that a morphism f : R → S of commutative rings is an effective descent morphism for modules if and only if f is pure as a morphism of R-modules.

Let R be a commutative ring with unit and R−mod the category of R-modules. Since, for any R-module M, the group C(M) = Hom_{Ab}(M, ℚ/ℤ) (where Ab is the category of abelian groups and ℚ/ℤ is the rational circle abelian group) becomes an R-module with the action of R on C(M) by (rf)(m) = f(rm) , we can define a functor C : (R−mod)^{op} → R −mod, given by C(M) = Hom_{Ab}(M, ℚ/ℤ). Since the abelian group ℚ/ℤ is an injective cogenerator in the category of abelian groups (see, for example, [1]), the functor C is exact and reflects isomorphisms. We say that a morphism f : M → M′ of R-modules is pure if for any R-module N,

1_{N} ⊗_{R} f : N ⊗_{R} M → N ⊗_{R} M′

is monic. Let f : M → M′ be a morphism of R-modules. The next lemma follows from the commutativity of the diagram

\xymatrix{Hom_{R}(C(M), C(M′)) \ar[d]^{≈} \ar[r]^{Hom_{R}(C(M),C(f))} & Hom_{R}(C(M, C(M))) \ar[d]^{≈} \\ C(C(M) ⊗_{R} M′) \ar[r]_{C(1_{C(M)}⊗_{R}f)} & C(C(M) ⊗_{R} M),}

where the vertical morphisms are the canonical isomorphisms.

1. Lemma. Let f : M → M′ be a morphism of R-modules. The following conditions are equivalent:

(a) f is a pure morphism of R-modules.

(b) C(f) is a split epimorphism of R-modules.

Let f : R → S be a morphism of commutative rings. Recall that a descent datum on an object M ∈ Ob(S −mod) can be described as an S-module morphism θ : M → S ⊗_{R}M such that θ makes

Received by the editors 1999 August 9 and, in revised form, 2000 February 2.

Published on 2000 March 9.

2000 Mathematics Subject Classification: 13C99,18A20,18A30,18A40.

Key words and phrases: Pure morphisms,(effective) Descent morphisms, Split coequalizers.

\begin{matrix}c \\ ◯ \end{matrix} Bachuki Mesablishvili, 2000. Permission to copy for private use granted.

[figure: M; θ; S ⊗_{R} M; μ; M]

and

\xymatrix{M \ar[d]_{θ} \ar[r]^{θ} & S ⊗_{R} M \ar[d]_{1⊗_{R}θ} \\ S ⊗_{R} M \ar[r]_{1⊗_{R}i_{M}} & S ⊗_{R} S ⊗_{R} M}

commutative, where μ denotes the S-module structure on M, and i_{M} : M → S ⊗_{R} M is an R-morphism given by i_{M}(m) = 1 ⊗_{R} m.

Let Des(f) denote the category of pairs (M, θ), θ descent datum on M ∈ Ob(S−mod), in which morphisms (M, θ) → (M′, θ′) are just morphisms g : M → M′ in S − mod which commute with descent data in the obvious sense (see, for example, [2]).

Any object f^{∗}(M) = (S ⊗_{R} M, i_{M}), M ∈ Ob(R − mod) can be equipped with descent data in a canonical way, and this gives rise to a commutative diagram

\xymatrix{R − mod \ar[dr]_{S⊗_{R}−} \ar[r]^{f^{∗}} & Des(f) \ar[d]^{U} \\  & S − mod,}

where U is the forgetful functor. f is said to be a descent morphism if f^{∗} is full and faithful, and an effective descent morphism if f^{∗} is an equivalence.

The functor f^{∗} has a right adjoint f_{∗} which is defined by requiring that

\xymatrix{f_{∗}(M, θ) \ar[r]^{e} & M \ar[r]^{θ} \ar[r]_{i_{M}} & S ⊗_{R} M}

is an equalizer of S-modules for each (M, θ) ∈ Ob(Des(f)). The counit of this adjunction is defined by δ_{M} = μ(1 ⊗_{R} e). The unit ϵ_{M} : M → f_{∗}f^{∗}(M) is obtained from the diagram

\xymatrix{M \ar[dr]^{i_{M}} \ar@{-->}[dd]^{ε_{M}} \\  & S ⊗_{R} M \ar[r]^{1_{S}⊗_{R}i_{M}} \ar[r]_{i_{S⊗_{R}M}} & S ⊗_{R} S ⊗_{R} M. \\ f_{∗}f^{∗}(M) \ar@{-->}[ur]}

It does exist because i_{M} equalizes the two morphisms on the right hand side. From the description of ϵ and δ we obtain immediately the two following propositions.

2. Proposition. f : R → S is a descent morphism if and only if f is pure as a morphism of R-modules.

3. Proposition. A descent morphism f is effective if and only if S ⊗_{R} − preserves the equalizer

\xymatrix{f_{∗}(M, θ) \ar[r]^{e} & M \ar[r]^{θ} \ar[r]_{i_{M}} & S ⊗_{R} M}

for each (M, θ) ∈ Ob(Des(f)).

Let f : R → S be pure as a morphism of R-modules. Then by Lemma 1 there is an R-module morphism g : C(R) → C(S) such that c(f)g = 1_{C(R)}.

If (M, θ) ∈ Ob(Des(f)), then we have a commutative diagram

\xymatrix{f_{∗}(M, θ) \ar[r]^{e} & M \ar[d]_{i_{M}} \ar[r]^{θ} \ar[r]_{i_{M}} & S ⊗_{R} M \ar[d]^{i_{S⊗_{R}M}} \\ M \ar[r]^{θ} & S ⊗_{R} M \ar[r]^{1_{S}⊗_{R}θ} \ar[r]_{1_{S}⊗_{R}i_{M}} & S ⊗_{R} S ⊗_{R} M,}

in which the rows are equalizer diagrams.

Applying the functor C to this diagram, we obtain the commutative diagram

\xymatrix{C(S ⊗_{R} S ⊗_{R} M) \ar[d]_{C(i_{S⊗_{R}M})} \ar[r]^{C(1_{S}⊗_{R}θ)} \ar[r]_{C(1_{S}⊗_{R}i_{M})} & C(S ⊗_{R} M) \ar[d]^{C(i_{M})} \ar[r]^{C(θ)} & C(M) \\ C(S ⊗_{R} M) \ar[r]^{C(θ)} \ar[r]_{C(i_{M})} & C(M) \ar[r]_{C(e)} & C(f_{∗}(M, θ)),}

in which the rows are coequalizer diagrams. Now, since for any R-module P we have the isomorphism of functors C(P ⊗_{R} −) → Hom_{R}(−, C(P)), we obtain the commutative diagram

\xymatrix{Hom_{R}(S ⊗_{R} M, C(S)) \ar[r]^{−∘θ} \ar[r]_{−∘i_{M}} \ar[d]^{−∘C(f)} & Hom_{R}(M, C(S)) \ar[d]^{−∘C(f)} \\ Hom_{R}(S ⊗_{R} M, C(R)) \ar[u]^{−∘g} \ar[r]^{−∘θ} \ar[r]_{−∘i_{M}} & Hom_{R}(M, C(R)) \ar[u]^{−∘g} \ar[r]^{−∘e} & Hom_{R}(f_{∗}(M, θ), C(R)).}

Applying the above isomorphism of functors backwards, we deduce that there are Rmorphisms h and h′, such that the diagram

\xymatrix{C(S ⊗_{R} S ⊗_{R} M) \ar[d]^{C(i_{S⊗_{R}M})} \ar[r]^{C(1_{S}⊗_{R}θ)} \ar[r]_{C(1_{S}⊗_{R}i_{M})} & C(S ⊗_{R} M) \ar[d]^{C(i_{M})} \ar[r]^{C(θ)} & C(M) \\ C(S ⊗_{R} M) \ar[u]^{h′} \ar[r]^{C(θ)} \ar[r]_{C(i_{M})} & C(M) \ar[u]^{h} \ar[r]^{C(e)} & C(f_{∗}(M, θ)) \ar@{-->}[u]^{k}}

commutes. Since both left hand side squares commute, there is an R-morphism k : C(f_{∗}(M, θ) → C(M) such that C(θ)h = kC(e). It means that the bottom row becomes a split coequalizer diagram [3] in the category of R-modules,which is split by the morphisms

\xymatrix{C(f_{∗}(M, θ)) \ar[r]^{k} & C(M) \ar[r]^{h} & C(S ⊗_{R} M).}

Since split coequalizers are preserved by any functor, its image under the functor Hom(S, −) is a coequalizer diagram. So

\xymatrix{Hom_{R}(S, C(S ⊗_{R} M)) \ar[r]^{−∘C(θ)} \ar[r]_{−∘C(i_{M})} & Hom_{R}(S, C(M)) \ar[r]^{−∘C(e)} & Hom_{R}(S, f_{∗}(M, θ))}

is a coequalizer diagram, and hence so is

\xymatrix{C(S ⊗_{R} S ⊗_{R} M) \ar[r]^{C(1_{S}⊗_{R}θ)} \ar[r]_{C(1_{S}⊗_{R}i_{M})} & C(S ⊗_{R} M) \ar[r]^{C(1_{S}⊗_{R}e)} & C(S ⊗_{R} f_{∗}(M, θ)).}

The functor C is exact and reflects isomorphisms. Therefore it also reflects coequalizers. It follows that

\xymatrix{S ⊗_{R} f_{∗}(M, θ) \ar[r]^{1_{S}⊗_{R}e} & S ⊗_{R} M \ar[r]^{1_{S}⊗_{R}θ} \ar[r]_{1_{S}⊗_{R}i_{M}} & S ⊗_{R} S ⊗_{R} M}

is an equalizer. But

\xymatrix{M \ar[r]^{θ} & S ⊗_{R} M \ar[r]^{1_{S}⊗_{R}θ} \ar[r]_{1_{S}⊗_{R}i_{M}} & S ⊗_{R} S ⊗_{R} M}

is also an equalizer diagram. Thus we have an isomorphism S ⊗_{R} f_{∗}(M, θ) → M. We obtain

4. Theorem. f : R → S is an effective descent morphism for modules if and only if f is pure as a morphism of R-modules.

Acknowledgement: The work was partially supported by INTAS-97-31961

References

[1] P. T. Johnstone, Topos Theory, Academic Press, New York 1977.

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

[3] S. MacLane, Categories for the Working Mathematician, Springer, New York 1971.

A. Razmadze Mathematical Institute Tbilisi 380093 Georgia

Email: bachi@rmi.acnet.ge

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