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Algebraic Cohomology and Deformation Theory

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Deformation Theory of Algebras and Structures and Applications

Part of the book series: NATO ASI Series ((ASIC,volume 247))

Abstract

We should state at the outset that the present article, intended primarily as a survey, contains many results which are new and have not appeared elsewhere. Foremost among these is the reduction, in §28, of the formal deformation theory of a smooth compact complex algebraic variety χ to that of a single ring built from χ. Others include the relationship between the classical Hodge decomposition of the cohomology of an analytic manifold and the more recent Hodge decomposition of the cohomology of a commutative algebra, the invariance of the Euler characteristic of an algebra under deformation, the correspondence between the deformation theories for Morita equivalent algebras, much of the work on the deformation of presheaves (diagrams) of algebras, and the explicit description of the (algebraic) Hodge decomposition for regular affine algebras. However, in line with the goals of a survey article, we have tried to maximize the exposition, including details only in so far as they aid in this purpose. Many proofs are sketched; many others, including the most difficult, are omitted altogether.

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Author information

Authors and Affiliations

  1. U. of Pennsylvania, 19104-6395, Phila., Pa, USA

    Murray Gerstenhaber

  2. S.U.N.Y./A.B., 14214-3093, Buffalo, NY, USA

    Samuel D. Schack

Authors
  1. Murray Gerstenhaber
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  2. Samuel D. Schack
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Editor information

Editors and Affiliations

  1. CWI, University of Utrecht, Amsterdam, The Netherlands

    Michiel Hazewinkel

  2. University of Pennsylvania, Philadelphia, PA, USA

    Murray Gerstenhaber

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© 1988 Kluwer Academic Publishers

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Gerstenhaber, M., Schack, S.D. (1988). Algebraic Cohomology and Deformation Theory. In: Hazewinkel, M., Gerstenhaber, M. (eds) Deformation Theory of Algebras and Structures and Applications. NATO ASI Series, vol 247. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-3057-5_2

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  • DOI: https://doi.org/10.1007/978-94-009-3057-5_2

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-7875-7

  • Online ISBN: 978-94-009-3057-5

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