A counterexample to a 1961 ``theorem'' in homological algebra
For a fixed cardinal \(\alpha\) let \(\mathcal S\) denote an essentially small, additive category closed under the formation of coproducts of \(\leq\alpha\) of its objects. If \(\mathcal C\) denotes the category of all (contravariant) additive functors \(F:\mathcal S\longrightarrow {\mathcal A}b\) into the category of abelian groups, denote by \(\mathcal F\) its full subcategory of additive functors that carry coproducts of \(\leq\alpha\) objects into products of abelian groups. Then it is shown that: (a) \(\mathcal F\) is an abelian subcategory of \(\mathcal C\), (b) \(\mathcal F\) satisfies AB4\(^*\) and AB3, (c) the Yoneda map \(Y:\mathcal S\longrightarrow \mathcal F\) preserves coproducts of \(\leq \alpha\) objects, (d) the inclusion \(\mathcal F\hookrightarrow {\mathcal C}\) has a left adjoint, (e) if any morphism \(q\longrightarrow r\) in \(\mathcal S\) can be completed to an exact sequence \(p\longrightarrow q\longrightarrow r\) and if the coproduct of \(\leq\alpha\) exact sequences in \(\mathcal S\) is exact in \(\mathcal S\), then \(\mathcal F\) satisfies AB4. In a special case: \(\alpha=\aleph_0\) and \(\mathcal S\) the category of normed, non-archimedean, complete abelian groups of cardinality not exceeding the continuum, the author then shows that all the requirements for the general \(\mathcal S\) above are satisfied, thus the corresponding properties hold as well, in particular AB4 and AB\(4^*\). The crucial example is the injective direct sequence (i.e., all the maps are injective) \(\mathbf B:\quad J_p\buildrel{p}\over\longrightarrow J_p\buildrel{p}\over\longrightarrow J_p\buildrel{p}\over\longrightarrow\dots\) of the \(p\)-adic integers with \(p\)-multiplications as the maps. Looking into the Yoneda image of this system, it is proved that \(\varinjlim Y(\mathbf B)=\varinjlim^1Y(\mathbf B)=0\) and that, although \(Y(\mathbf B)\) is an injective direct system indexed by a countable set, the natural map \(Y(\mathbf B)_0\longrightarrow\varinjlim \mathbf B\) is not a monomorphism. If \(\mathbf A:\quad J_p\buildrel{1}\over\longrightarrow J_p\buildrel{1}\over\longrightarrow J_p\buildrel{1}\over\longrightarrow\dots\), then for the exact sequence \(0\longrightarrow \mathbf A\longrightarrow \mathbf B\longrightarrow \mathbf C\longrightarrow 0\) (for an appropriate \(\mathbf C\)) it is shown that \(\varinjlim^1Y(\mathbf C)\neq 0\), although \(Y(\mathbf C)\) is an injective direct system with the countable index set. The author implies (without saying it) that all the statements proved will still be correct after dualizing them. In the Appendix, \textit{P. Deligne} gives a six page illumination of the argument by examining the category of inverse systems of modules for which the derived inverse limit(s) vanish; related material can be found in Chapter 2 of the reviewer's monograph [\textit{R. M. Dimitrić}, ``Slender modules and rings'', Cambridge University Press (2004; to appear)]. As for the title of the paper: It refers to \textit{J.-E. Roos} [C. R. Acad. Sci., Paris, Sér. AB 252, 3702-3704 (1961; Zbl 0102.02501)]. The dualized conterexamples would disprove Roos' Proposition 5 that the notions of flabby and surjective inverse systems are identical for countable index sets (under the general assumptions in Roos' paper), as well as that for surjective inverse systems all the derived inverse limit functors vanish. To be fully convincing though, it would be desirable to actually construct, in a concrete manner, (counter) example(s) of countably indexed surjective inverse systems with non-vanishing \(\varinjlim^1\). The situation is not as tragic as it seems, for Roos' results are certainly still valid in categories with enough projectives (such as module categories; see also, for instance Proposition 13.2.2 in: \textit{A. Grothendieck} [``Éléments de géométrie algébrique'', Chapitre 0, Publ. Math. Inst. Hautes Étud. Sci. 11, 349-423 (5-79) (1961; Zbl 0118.36206)]. \textit{J.-E. Roos} seems to have been aware of a neccessity of additional conditions for his statements, as evidenced from his unpublished papers from that period [e.g. ``Derived functors of infinite products and projective objects in abelian categories'', 16p. (1962)]. His primary interests were Grothendieck abelian categories and he conjectured that such categories with AB4\(^*\) must have enough projectives.
scientific article; zbMATH DE number 1777249