 Library Home /
 Search Collections /
 Open Collections /
 Browse Collections /
 UBC Theses and Dissertations /
 On chain maps inducing isomorphisms in homology
Open Collections
UBC Theses and Dissertations
UBC Theses and Dissertations
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.
Item Metadata
Title  On chain maps inducing isomorphisms in homology 
Creator  Nicollerat, Marc Andre 
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  Thesis/Dissertation 
Type  Text 
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  Master of Arts  MA 
Program  Mathematics 
Affiliation  Science, Faculty of; Mathematics, Department of 
Degree Grantor  University of British Columbia 
Campus  UBCV 
Scholarly Level  Graduate 
Aggregated Source Repository  DSpace 
Item Media
Item Citations and Data
License
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.