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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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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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Ordered algebraic structures --- 512.73 --- #WWIS:ALTO --- Cohomology theory of algebraic varieties and schemes --- 512.73 Cohomology theory of algebraic varieties and schemes --- Algèbre homologique. --- Algebra, Homological --- Algèbres d'opérateurs

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K-theory --- K-théorie --- 512.73 --- 515.14 --- Algebraic topology --- Homology theory --- Cohomology theory of algebraic varieties and schemes --- 515.14 Algebraic topology --- 512.73 Cohomology theory of algebraic varieties and schemes --- K-théorie

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Spencer Bloch's 1979 Duke lectures, a milestone in modern mathematics, have been out of print almost since their first publication in 1980, yet they have remained influential and are still the best place to learn the guiding philosophy of algebraic cycles and motives. This edition, now professionally typeset, has a new preface by the author giving his perspective on developments in the field over the past 30 years. The theory of algebraic cycles encompasses such central problems in mathematics as the Hodge conjecture and the Bloch-Kato conjecture on special values of zeta functions. The book begins with Mumford's example showing that the Chow group of zero-cycles on an algebraic variety can be infinite-dimensional, and explains how Hodge theory and algebraic K-theory give new insights into this and other phenomena.

Algebraic cycles. --- 512.73 --- Cycles, Algebraic --- Geometry, Algebraic --- Cohomology theory of algebraic varieties and schemes --- 512.73 Cohomology theory of algebraic varieties and schemes --- Algebraic cycles --- Mathematics. --- Math --- Science