The Splitting Criterion of Kempf and the Babylonian Tower Theorem

Abstract

We show that the idea used by Kempf (1990 Kempf , G. R. ( 1990 ). A criterion for the splitting of a vector bundle . Forum Math. 2 : 477 – 480 . [CSA] [Crossref] , [Google Scholar]) in order to obtain a splitting criterion for vector bundles on projective spaces leads to an elementary proof of the Babylonian tower theorem for this class of bundles, a result due to Barth and Van de Ven (1974 Barth , W. , Van de Ven , A. ( 1974 ). A decomposability criterion for algebraic 2-bundles on projective spaces . Invent. Math. 25 : 91 – 106 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]) in the rank 2 case and to Sato (1977 Sato , E. ( 1977 ). On the decomposability of infinitely extendable vector bundles on projective spaces and Grassmann varieties . J. Math. Kyoto Univ. 17 : 127 – 150 . [CSA] [Google Scholar] 1978 Sato , E. ( 1978 ). The decomposability of an infinitely extendable vector bundle on the projective space, II . In: International Symposium on Algebraic Geometry . Kyoto University, Kyoto, Kinokuniya Book Store , Tokyo , pp. 663 – 672 . [Google Scholar]) and Tyurin (1976 Tyurin , A. N. ( 1976 ). Finite dimensional vector bundles over infinite varieties . Math. USSR Izv. 10 : 1187 – 1204 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) in the case of arbitrary rank. As a byproduct, we obtain a slight improvement of the numerical criterion of Flenner (1985 Flenner , H. ( 1985 ). Babylonian tower theorems on the punctured spectrum . Math. Ann. 271 : 153 – 160 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) in the particular case under consideration.