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Complexes and Diffrerential Graded Modules

Apassov, Dmitri LU (1999) In Doctoral theses in mathematical sciences 1999:3.
Abstract
The main topic of the thesis is the generalization of some traditional module-theoretic homological applications to complexes of modules and to differential graded modules over differential graded rings.



We introduce three possible generalizations of the classical notion of annihilator of an <i>R</i>-module. For linear functors D (<i>R</i>) -> D (<i>R</i>), preserving the triangulation, certain inclusion results for these annihilators are obtained.



We study ascent and descent of Gorenstein and Cohen-Macaulay properties along a local homomorphism <i>f</i>:<i>R -> S</i> in the presence of a finite <i>S</i>-module which is of... (More)
The main topic of the thesis is the generalization of some traditional module-theoretic homological applications to complexes of modules and to differential graded modules over differential graded rings.



We introduce three possible generalizations of the classical notion of annihilator of an <i>R</i>-module. For linear functors D (<i>R</i>) -> D (<i>R</i>), preserving the triangulation, certain inclusion results for these annihilators are obtained.



We study ascent and descent of Gorenstein and Cohen-Macaulay properties along a local homomorphism <i>f</i>:<i>R -> S</i> in the presence of a finite <i>S</i>-module which is of finite flat dimension over <i>R</i> thus generalizing the concept of <i>homomorhism of finite flat dimension</i> introduced by Luchezar Avramov and Hans-Bjørn Foxby.



The two approaches of the classical homological algebra to homological dimensions (the resolutional approach and the functorial one) give rise to different invariants in the category of differential graded modules over a differential graded ring. We study this dichotomy and establish the simultaneous finiteness of the resolutional and the functorial flat dimension in one special case.



We also generalize the notion of <i>the canonical module of a Cohen-Macaulay ring</i> to the case of a genuine differential graded ring. (Less)
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author
opponent
  • Fröberg, Ralf, Stockholm University
organization
publishing date
type
Thesis
publication status
published
subject
keywords
algebraic geometry, field theory, Number Theory, fiber of a local homomorphism, DG dualizing module, homological dimensions, differential graded rings, almost finite module, Cohen-Macaulay rings, local homomorphism, Gorenstein rings, annihilator, complex of modules, algebra, group theory, Talteori, fältteori, algebraisk geometri, gruppteori
in
Doctoral theses in mathematical sciences
volume
1999:3
pages
55 pages
publisher
Centre for Mathematical Sciences, Lund University
defense location
Mathematics Building, Sölvegatan 18, Room MH:C
defense date
1999-05-22 10:00
external identifiers
  • other:ISRN: LUNFMA-1011-1999
ISSN
1404-0034
ISBN
91-628-3578-5
language
English
LU publication?
yes
id
c8b1cb4f-4fa4-4a34-aa93-2d0f5a2d2e97 (old id 39582)
date added to LUP
2007-06-20 10:57:41
date last changed
2016-09-19 08:44:55
@phdthesis{c8b1cb4f-4fa4-4a34-aa93-2d0f5a2d2e97,
  abstract     = {The main topic of the thesis is the generalization of some traditional module-theoretic homological applications to complexes of modules and to differential graded modules over differential graded rings.<br/><br>
<br/><br>
We introduce three possible generalizations of the classical notion of annihilator of an &lt;i&gt;R&lt;/i&gt;-module. For linear functors D (&lt;i&gt;R&lt;/i&gt;) -&gt; D (&lt;i&gt;R&lt;/i&gt;), preserving the triangulation, certain inclusion results for these annihilators are obtained.<br/><br>
<br/><br>
We study ascent and descent of Gorenstein and Cohen-Macaulay properties along a local homomorphism &lt;i&gt;f&lt;/i&gt;:&lt;i&gt;R -&gt; S&lt;/i&gt; in the presence of a finite &lt;i&gt;S&lt;/i&gt;-module which is of finite flat dimension over &lt;i&gt;R&lt;/i&gt; thus generalizing the concept of &lt;i&gt;homomorhism of finite flat dimension&lt;/i&gt; introduced by Luchezar Avramov and Hans-Bjørn Foxby.<br/><br>
<br/><br>
The two approaches of the classical homological algebra to homological dimensions (the resolutional approach and the functorial one) give rise to different invariants in the category of differential graded modules over a differential graded ring. We study this dichotomy and establish the simultaneous finiteness of the resolutional and the functorial flat dimension in one special case.<br/><br>
<br/><br>
We also generalize the notion of &lt;i&gt;the canonical module of a Cohen-Macaulay ring&lt;/i&gt; to the case of a genuine differential graded ring.},
  author       = {Apassov, Dmitri},
  isbn         = {91-628-3578-5},
  issn         = {1404-0034},
  keyword      = {algebraic geometry,field theory,Number Theory,fiber of a local homomorphism,DG dualizing module,homological dimensions,differential graded rings,almost finite module,Cohen-Macaulay rings,local homomorphism,Gorenstein rings,annihilator,complex of modules,algebra,group theory,Talteori,fältteori,algebraisk geometri,gruppteori},
  language     = {eng},
  pages        = {55},
  publisher    = {Centre for Mathematical Sciences, Lund University},
  school       = {Lund University},
  series       = {Doctoral theses in mathematical sciences},
  title        = {Complexes and Diffrerential Graded Modules},
  volume       = {1999:3},
  year         = {1999},
}