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On the derived category and K-functor of coherent sheaves on intersections of quadrics. (English. Russian original) Zbl 0679.14005
The study and classification of coherent sheaves on diverse classes of algebraic varieties is of central importance in algebraic geometry and, nowadays, also in mathematical physics. In 1978, I. N. Bernstein, I. M. Gelfand and S. I. Gelfand invented an approach that allowed to translate the classification problem for coherent sheaves on \({\mathbb{P}}^ n\) into a classification problem for graded modules over the Grassmann algebra with \(n+1\) generators. The basic link between categories of coherent sheaves on \({\mathbb{P}}^ n\) and the derived category of certain modules over the Grassmann algebra seemed to be so crucial and general that several attempts have been made to generalize it to other base varieties (than \({\mathbb{P}}^ n)\) and graded algebras. For example, the author himself succeeded to describe the category of coherent sheaves on Grassmann varieties by such a correspondence [cf. the author, Funct. Anal. Appl. 17, 145-146 (1983); translation from Funkt. Anal. Prilozh. 17, No.2, 78- 79 (1983; Zbl 0571.14007)].
In the present paper, the author applies this method to the case of coherent sheaves on complete intersections of quadrics. He describes the derived category of coherent sheaves on complete intersections of quadrics in terms of modules over certain graded Clifford algebras, interprets this link in the language of higher K-theory, and compares his results with the classical approach to the study of intersections of quadrics.