Abstract

Personal recollections about Alexandre Grothendieck and early days of his theory of motives

Highlights

  • Still, during the decades that passed since the vast realm of motives kept rewarding humility of many researchers prepared to be happy with what they could do using the tools they could elaborate

  • Having returned to Moscow in June 1967, after five or six weeks of intense training with Grothendieck, I spent several months writing down his main definitions related to motives and studying necessary background material in the literature

  • To me Grothendieck has written in Russian the letter about this paper (05/02/1969)

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Summary

Early History of Motives

Having returned to Moscow in June 1967, after five or six weeks of intense training with Grothendieck, I spent several months writing down his main definitions related to motives and studying necessary background material in the literature. The second step in the definition of the relevant category of pure motives consists in a formal construction of new objects (and relevant morphisms) that are “pieces” of varieties: kernels and images of projectors, i.e. correspondences p : X → X with p2 = p. This produces a pseudo–abelian, or Karoubian completion of the category. Its basic and universal step consists in the introduction of a linear span of everything that in classical physics was only a set: points of a phase space, field configurations over a domain of space–time etc Such quantum superpositions form linear spaces on which Hilbert–like scalar products are defined, that in turn allow one to speak about probability amplitudes, quantum observations etc. The references at the end of this essay constitute the bare minimum of the relevant research, but the reader will be able to find much additional material bibliography in the survey collection [Mo91] and in [A04], [VoSuFr00], [Ta11]

Motives and Homological Algebra
Motives and Physics
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