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Algebraic geometry --- 512.73 --- Cohomology theory of algebraic varieties and schemes --- 512.73 Cohomology theory of algebraic varieties and schemes

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Algebraic geometry --- Number theory --- 512.73 --- #KVIV:BB --- Cohomology theory of algebraic varieties and schemes --- 512.73 Cohomology theory of algebraic varieties and schemes

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Category theory. Homological algebra --- K-theory --- Congresses. --- 512.73 --- -Algebraic topology --- Homology theory --- Cohomology theory of algebraic varieties and schemes --- Congresses --- -Cohomology theory of algebraic varieties and schemes --- 512.73 Cohomology theory of algebraic varieties and schemes --- -512.73 Cohomology theory of algebraic varieties and schemes --- Algebraic number theory --- K-theory - Congresses

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This dissertation consists of two parts. In the first part, we study the low-dimensional cohomology of group extensions and Lie algebra extensions. A classical tool to study the cohomology groups of a group or Lie algebra extension is the associated spectral sequence. From the spectral sequence, one can deduce a seven-term exact sequence relating the low-dimensional cohomology groups of the groups or Lie algebras in the extension. However, not all the maps in this sequence are explicit. In this thesis, we use the low-dimensional interpretations of the cohomology groups to construct alternative, explicit maps that yield a seven-term exact sequence of the same form as the classical sequence. We also give an easy description of these maps on cocycle level. Very recently, Huebschmann has showed that the new maps coincide with the original maps. The second part of this dissertation discusses the second cohomology of finitely generated, torsion-free nilpotent groups (T-groups) with coefficients in a trivial module that is torsion-free and finitely generated as an abelian group. In particular, we give a formula for the cohomology group of a two-step T-group in terms of certain data from the associated graded Lie ring. To get this result, we use polynomial methods and abelian models of group extensions.

academic collection --- 512.73 <043> --- Cohomology theory of algebraic varieties and schemes--dissertaties --- Theses

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Following Quillen's approach to complex cobordism, the authors introduce the notion of oriented cohomology theory on the category of smooth varieties over a fixed field. They prove the existence of a universal such theory (in characteristic 0) called Algebraic Cobordism. Surprisingly, this theory satisfies the analogues of Quillen's theorems: the cobordism of the base field is the Lazard ring and the cobordism of a smooth variety is generated over the Lazard ring by the elements of positive degrees. This implies in particular the generalized degree formula conjectured by Rost. The book also contains some examples of computations and applications.

Cobordism theory --- Cobordism theory. --- Homology theory. --- Cohomology theory --- Contrahomology theory --- Homology theory --- Algebraic topology --- Differential topology