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 On chain maps inducing isomorphisms in homology
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On chain maps inducing isomorphisms in homology Nicollerat, Marc Andre
Abstract
Let A be an abelian category, I the full subcategory of A consisting of injective objects of A, and K(A) the category whose objects are cochain complexes of elements of A, and whose morphisms are homotopy classes of cochain maps. In (5), lemma 4.6., p. 42, R. Hartshorne has proved that, under certain conditions, a cochain complex X˙ ε. KA) can be embedded in a complex I˙ ε. K(I) in such a way that I˙ has the same cohomology as X˙. In Chapter I we show that the construction given in the two first parts of Hartshorne's Lemma is natural i.e. there exists a functor J : K(A) → K(I) and a natural transformation [formula omitted] (where E : K(I) → K(A) is the embedding functor) such that [formula omitted] is injective and induces isomorphism in cohomology. The question whether the construction given in the third part of the lemma is functorial is still open. We also prove that J is left adjoint to E, so that K(I) is a reflective subcategory of K(A). In the special case where A is a category [formula omitted] of left Amodules, and [formula omitted] the category of cochain complexes in [formula omitted] and cochain maps (not homotopy classes), we prove the existence of a functor [formula omitted] In Chapter II we study the natural homomorphism [formula omitted] where A, B are rings, and M, L, N modules or chain complexes. In particular we give several sufficient conditions under which v is an isomorphism, or induces isomorphism in homology. In the appendix we give a detailed proof of Hartshorne's Lemma. We think that this is useful, as no complete proof is, to our knowledge, to be found in the literature.
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Title 
On chain maps inducing isomorphisms in homology

Creator  
Publisher 
University of British Columbia

Date Issued 
1973

Description 
Let A be an abelian category, I the full subcategory of A consisting of injective objects of A, and K(A) the category whose objects are cochain complexes of elements of A, and whose morphisms are homotopy classes of cochain maps.
In (5), lemma 4.6., p. 42, R. Hartshorne has proved that, under certain conditions, a cochain complex X˙ ε. KA) can be embedded in a complex I˙ ε. K(I) in such a way that I˙ has the same cohomology as X˙.
In Chapter I we show that the construction given in the two first parts of Hartshorne's Lemma is natural i.e. there exists a functor
J : K(A) → K(I) and a natural transformation [formula omitted]
(where E : K(I) → K(A) is the embedding functor) such that [formula omitted] is
injective and induces isomorphism in cohomology. The question whether the construction given in the third part of the lemma is functorial is still open.
We also prove that J is left adjoint to E, so that K(I) is a reflective subcategory of K(A).
In the special case where A is a category [formula omitted] of left Amodules, and [formula omitted] the category of cochain complexes in [formula omitted] and cochain maps (not homotopy classes), we prove the existence of a functor [formula omitted]
In Chapter II we study the natural homomorphism [formula omitted]
where A, B are rings, and M, L, N modules or chain complexes. In particular we give several sufficient conditions under which v is an isomorphism, or induces isomorphism in homology.
In the appendix we give a detailed proof of Hartshorne's Lemma. We think that this is useful, as no complete proof is, to our knowledge, to be found in the literature.

Genre  
Type  
Language 
eng

Date Available 
20110401

Provider 
Vancouver : University of British Columbia Library

Rights 
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.

DOI 
10.14288/1.0080520

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Campus  
Scholarly Level 
Graduate

Aggregated Source Repository 
DSpace

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Rights
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.