On spinor bundles

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Elsevier Science B.V. (North-Holland)
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A. A. Beĭlinson's spectral sequence [Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 68–69;] gives a way of reconstructing any coherent sheaf on P n from the cohomology on P n of this sheaf and its twists. In this article the author shows that a similar construction is possible on the Grassmannian of 2-planes in C 4 , where twisting with the hyperplane section bundle on P n is replaced by tensoring with various "spinor bundles'', namely, associated bundles of the universal bundle. Indeed it is clear that the author's generalization applies to an arbitrary Grassmannian. The author illustrates his theorem with an example (Proposition 5), and in fact most of this paper is concerned with calculations of cohomology for this example. These calculations may be effected alternatively by means of the Bott-Borel-Weil theorem [ R. Bott , Ann. of Math. (2) 66 (1957), 203–248;]. The paper is plagued with misprints: e.g., in Proposition 5 replace E(1) by E(−1) , ⊗ by ⊕ , and (n+2 2) by (n+2 3) .
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A.A. Beilinson Coherent sheaves on n and problems in linear algebra Funct. Analysis, 12 (3) (1978), pp. 68–69 H. Cartan, S. Eilenberg Homological Algebra Princeton Univ. Press, Princeton NJ (1966) M.G. Eastwood, R. Penrose, R.O. Wells Cohomology and masless fields Comm. Math. Physics, 78 (1981), pp. 305–381 R. Hartshorne Algebraic Geometry (Springer, Berlín 1977)