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$ K\sb{1}$ of projective $ r$-spaces


Author: Leslie G. Roberts
Journal: Proc. Amer. Math. Soc. 26 (1970), 587-592
MSC: Primary 14.55
DOI: https://doi.org/10.1090/S0002-9939-1970-0280501-0
MathSciNet review: 0280501
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Abstract: Let $ A$ be a commutative ring, and let $ X$ = projective $ r$-space over $ A$. Then we prove that $ {K_1}$ of the category of locally free sheaves of finite type on $ X$ is isomorphic to the direct sum of $ r + 1$ copies of $ {K_1}(A)$.


References [Enhancements On Off] (What's this?)

  • [1] H. Bass, Algebraic $ K$-theory, Benjamin, New York, 1968. MR 0249491 (40:2736)
  • [2] A. Grothendieck, Élements de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes, Inst. Hautes Études Sci. Publ. Math. No. 8 (1961). MR 36 #l77b.
  • [3] SGA 6 (1966-67) Théorie globale des intersections et Théorème de Riemann-Roch, Inst. Hautes Études Sci.
  • [4] L. Roberts, $ {K_1}$ of some Abelian categories, Trans. Amer. Math. Soc. 138 (1969), 377-382. MR 0251109 (40:4340)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1970-0280501-0
Keywords: $ {K_1}$, locally free sheaf, projective $ r$-space
Article copyright: © Copyright 1970 American Mathematical Society

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