A counterexample to a 1961 ``theorem'' in homological algebra
type: Item
instance of: scholarly article
title: A counterexample to a 1961 ``theorem'' in homological algebra
zbMATH Open document ID: 1025.18007
author: Amnon Neeman
published in: Inventiones Mathematicae
publication date: 2002-08-08T00:00:00Z
review text: 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…
Mathematics Subject Classification ID: 18E10
Mathematics Subject Classification ID: 18A30
Mathematics Subject Classification ID: 18A15
zbMATH DE Number: 1777249
zbMATH Keywords: derived colimit functors
zbMATH Keywords: AB3
zbMATH Keywords: AB4
zbMATH Keywords: AB3\(^*\)
zbMATH Keywords: AB4\(^*\)
zbMATH Keywords: vanishing of colim
zbMATH Keywords: injective direct systems
zbMATH Keywords: surjective inverse systems
reviewed by: Radoslav M. Dimitrić
MaRDI profile type: Publication
full work available at URL: s002220100197
OpenAlex ID: W2031763812
Wikidata QID: Q55888596
DOI: 10.1007/S002220100197
Recommended article: Q3822679
Recommended article: Q3830033
Recommended article: Counterexamples to a conjecture of Grothendieck
Recommended article: A counterexample to the \(a\)-`theorem'
Recommended article: Q3752948
Recommended article: Counterexample to a conjecture of H. Hopf
Recommended article: A counterexample in the theory of prehomogeneous vector spaces
Recommended article: A counterexample to the Gröbner ring conjecture
Recommended article: A counterexample to the Ch(G)-conjecture
Recommended article: A Counterexample to the Equivariant Simple Loop Conjecture
description: scientific article; zbMATH DE number 1777249