Differential geometry over commutative algebras

Much of differential geometry can be transplanted into algebra, starting from the algebra \(C^\infty(M)\) of smooth functions on a manifold \(M\). This approach, developed by a school of Russian mathematicians, is called differential geometry or differential calculus over commutative algebras . A motivating result is a Swan’s theorem .

Literature

Books and surveys

  • Nestruev, 2020: Smooth manifolds and observables (doi)
    • Second edition roughly doubles the size of the 2003 first edition (doi)
    • Jointly authored, Bourbaki-style, by a team lead by A.M. Vinogradov under the pseudonym Jet Nestruev
    • Unlike Bourbaki, the style is friendly and unpretentious and the work stresses connections to physics
  • Krasil’shchik, Lychagin, Vinogradov, 1986: Geometry of jet spaces and nonlinear differential equations (toc ), Ch. 1: Elements of differential calculus in commutative rings
    • Krasil’shchik & Verbovetsky, 1998: Homological methods in equations of mathematical physics (arxiv), Sec. 1: Differential calculus over commutative algebras
    • Krasil’shchik & Prinari, 1998: Lectures on linear differential operators over commutative algebras (pdf)
  • Sardanashvily, 2009: Lectures on differential geometry of modules and rings (arxiv)
  • Deepmala & Mishra, 2015: Differential operators over modules and rings as a path to the generalized differential geometry (pdf)
    • Exposition is rough but bibliography is useful

Papers

  • Swan, 1962: Vector bundles and projective modules (doi)
    • Original paper proving what is now called Swan’s theorem
  • Manoharan, 1992: A non-linear version of Swan’s theorem (doi)
    • Manoharan, 1995: Generalized Swan’s theorem and its application (doi)