Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

$G_{0}$ of a graded ring


Author: Leslie G. Roberts
Journal: Trans. Amer. Math. Soc. 166 (1972), 187-195
MSC: Primary 13J05
DOI: https://doi.org/10.1090/S0002-9947-1972-0294326-X
MathSciNet review: 0294326
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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?)

  • Hyman Bass, Algebraic $K$-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491
  • 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 174605
  • Armand Borel and Jean-Pierre Serre, Le thĂ©orĂšme de Riemann-Roch, Bull. Soc. Math. France 86 (1958), 97–136 (French). MR 116022
  • N. Bourbaki, ÉlĂ©ments de mathĂ©matique. Fasc. XXXI. AlgĂšbre commutative. Chapitre 7: Diviseurs, ActualitĂ©s Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1314, Hermann, Paris, 1965 (French). MR 0260715
  • A. V. Geramita and L. G. Roberts, Algebraic vector bundles on projective space, Invent. Math. 10 (1970), 298–304. MR 480519, DOI https://doi.org/10.1007/BF01418777
  • A. Grothendieck, ÉlĂ©ments de gĂ©omĂ©trie algĂ©brique. II, Inst. Hautes Études Sci. Publ. Math. No. 8 (1961). MR 36 #177b. ---, ÉlĂ©ments de gĂ©omĂ©trie algĂ©brique. III, Inst. Hautes Études Sci. Publ. Math. No. 11 (1961). MR 36 #177c.
  • M. Pavaman Murthy, Vector bundles over affine surfaces birationally equivalent to a ruled surface, Ann. of Math. (2) 89 (1969), 242–253. MR 241434, DOI https://doi.org/10.2307/1970667
  • P. Samuel, Lectures on unique factorization domains, Tata Institute of Fundamental Research Lectures on Mathematics, No. 30, Tata Institute of Fundamental Research, Bombay, 1964. Notes by M. Pavman Murthy. MR 0214579
  • SGA 6 (1966-67) ThĂ©orie globale des intersections et thĂ©orĂšme de Riemann-Roch, SĂ©m. Inst. Hautes Études Sci. dirigĂ© par P. Bertelot, A. Grothendieck et L. Illusie (to appear). L. G. Roberts, ${G_0}$ of certain subrings of a graded ring, Department of Math., Queen’s University, Kingston, 1971 (Preprint).

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 13J05

Retrieve articles in all journals with MSC: 13J05


Additional Information

Keywords: Grothendieck group, <IMG WIDTH="30" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="${G_0}$">, graded ring, ideal class group
Article copyright: © Copyright 1972 American Mathematical Society