Al Shumrani, Mohammed Ahmed Musa
Homotopy theory in algebraic derived categories.
PhD thesis, University of Glasgow.
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In this thesis, we introduce some new notions in the derived category D+(fg) (R) of bounded below chain complexes of finite type over local commutative noetherian ring R with maximal ideal m and residue field K in chapter three and study their relations to each other. Also, we set up the Adams spectral sequence for chain complexes in D+(f,g) (R) in chapter four and study its convergence.
To accomplish this task, we give two background chapters. We give some good account of chain complexes in chapter one. We review some basic homological algebra and give definition and basic properties of chain complexes. Then we study the homotopy category of chain complexes and we end chapter one with section about spectral sequences.
Chapter two is about the derived category of a commutative ring. Section one is about localization of categories and left and right fractions. Then in section two, we give definition of triangulated categories and some of its basic properties and we end section two with definitions of homotopy limits and colimits. In section three, we show that the derived category is a triangulated category. In section four, we give definitions of the derived functors, the derived tensor product and the derived Hom.
In chapter three, we start section one by giving some facts about local rings and we end this section by showing that every bounded below chain complex of finite type has a minimal free resolution. In section two, we show a derived analog of the Whitehead Theorem. In section three, we construct Postnikov towers for chain complexes. In section four, we define the Steenrod algebra. In section five, six and seven, we define irreducible, atomic, minimal atomic, no mod m detectable homology, H*-monogenic, nuclear chain complexes and the core of a chain complex. We show some various results relating these notions to each other and give some examples.
In chapter four, we set up the Adams spectral sequence in section one and study its properties. In section two, we study homology localization and local homology. In section three, we define K-nilpotent completion and we show that the Adams spectral sequence for a chain complex Y converges strongly to the homology of the K-nilpotent completion of Y. In section four, we study the Adams spectral sequence’s convergence where we show that the K-nilpotent completion for a bounded chain complex Y consisting of finitely generated free R-modules in each degree is isomorphic to the localization of Y with respect to the H*(—, K)-theory. In section five, we present some examples.
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