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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



$ G\sb{0}$ of a graded ring

Author: Leslie G. Roberts
Journal: Trans. Amer. Math. Soc. 166 (1972), 187-195
MSC: Primary 13J05
MathSciNet review: 0294326
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Abstract: We consider the Grothendieck group $ {G_0}$ of various graded rings, including $ {G_0}(A_n^r)$ where A is a commutative noetherian ring, and $ A_n^r$ is the A-subalgebra of the polynomial ring $ A[{X_0}, \ldots ,{X_n}]$ generated by monomials of degree r. If A is regular, then $ {G_0}(A_n^r)$ has a ring structure. The ideal class groups of these rings are also considered.

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

  • [1] Hyman Bass, Algebraic 𝐾-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491
  • [2] H. Bass, A. Heller, and R. G. Swan, The Whitehead group of a polynomial extension, Inst. Hautes Études Sci. Publ. Math. 22 (1964), 61–79. MR 0174605
  • [3] Armand Borel and Jean-Pierre Serre, Le théorème de Riemann-Roch, Bull. Soc. Math. France 86 (1958), 97–136 (French). MR 0116022
  • [4] N. Bourbaki, Éléments de mathématique. Fasc. XXXI. Algèbre commutative. Chapitre 7: Diviseurs, Actualités Scientifiques et Industrielles, No. 1314, Hermann, Paris, 1965 (French). MR 0260715
  • [5] A. V. Geramita and L. G. Roberts, Algebraic vector bundles on projective space, Invent. Math. 10 (1970), 298–304. MR 0480519,
  • [6] A. Grothendieck, Éléments de géométrie algébrique. II, Inst. Hautes Études Sci. Publ. Math. No. 8 (1961). MR 36 #177b.
  • [7] -, Éléments de géométrie algébrique. III, Inst. Hautes Études Sci. Publ. Math. No. 11 (1961). MR 36 #177c.
  • [8] M. Pavaman Murthy, Vector bundles over affine surfaces birationally equivalent to a ruled surface, Ann. of Math. (2) 89 (1969), 242–253. MR 0241434,
  • [9] P. Samuel, Lectures on unique factorization domains, Notes by M. Pavman Murthy. Tata Institute of Fundamental Research Lectures on Mathematics, No. 30, Tata Institute of Fundamental Research, Bombay, 1964. MR 0214579
  • [10] SGA 6 (1966-67) Théorie globale des intersections et théorème de Riemann-Roch, Sém. Inst. Hautes Études Sci. dirigé par P. Bertelot, A. Grothendieck et L. Illusie (to appear).
  • [11] L. G. Roberts, $ {G_0}$ of certain subrings of a graded ring, Department of Math., Queen's University, Kingston, 1971 (Preprint).

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Keywords: Grothendieck group, $ {G_0}$, graded ring, ideal class group
Article copyright: © Copyright 1972 American Mathematical Society