Projective modules over subrings of $k[X, Y]$
HTML articles powered by AMS MathViewer
 by David F. Anderson PDF
 Trans. Amer. Math. Soc. 240 (1978), 317328 Request permission
Abstract:
In this paper we study projective modules over subrings of $k[X,Y]$. Conditions are given for projective modules to decompose into free $\oplus$ rank 1 modules. Our main result is that if k is an algebraically closed field and A a subring of $B = k[X,Y]$ with $A \subset B$ integral and ${\text {sing}}(A)$ finite, then all f.g. projective Amodules have the form free $\oplus$ rank 1. We also give several examples of subrings of $k[X,Y]$ which have indecomposable projective modules of rank 2.References

D. F. Anderson, Projective modules over subrings of $k[X,Y]$, Dissertation, Univ. of Chicago, Chicago, Ill., 1976.
—, Projective modules over subrings of $k[X,Y]$ generated by monomials (submitted).
 Hyman Bass, Algebraic $K$theory, W. A. Benjamin, Inc., New YorkAmsterdam, 1968. MR 0249491
 Mark I. Krusemeyer, Fundamental groups, algebraic $K$theory, and a problem of Abhyankar, Invent. Math. 19 (1973), 15–47. MR 335522, DOI 10.1007/BF01418849
 Hideyuki Matsumura, Commutative algebra, W. A. Benjamin, Inc., New York, 1970. MR 0266911
 John Milnor, Introduction to algebraic $K$theory, Annals of Mathematics Studies, No. 72, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971. MR 0349811
 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 10.2307/1970667
 M. Pavaman Murthy and Claudio Pedrini, $K_{0}$ and $K_{1}$ of polynomial rings, Algebraic $K$theory, II: “Classical” algebraic $K$theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 342, Springer, Berlin, 1973, pp. 109–121. MR 0376654
 M. Pavaman Murthy and Richard G. Swan, Vector bundles over affine surfaces, Invent. Math. 36 (1976), 125–165. MR 439842, DOI 10.1007/BF01390007 M. Nagata, On the closedness of singular loci, Inst. Hautes Études Sci. Publ. No. 2, (1952), 512.
 Claudio Pedrini, On the $K_{o}$ of certain polynomial extensions, Algebraic $K$theory, II: “Classical” algebraic $K$theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 342, Springer, Berlin, 1973, pp. 92–108. MR 0371882
 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 J. P. Serre, Sur les modules projectifs, Séminaire DubrielPisot 14 (196061), No. 2.
 C. S. Seshadri, Triviality of vector bundles over the affine space $K^{2}$, Proc. Nat. Acad. Sci. U.S.A. 44 (1958), 456–458. MR 102527, DOI 10.1073/pnas.44.5.456 R. G. Swan, Serre’s problem, Conference on Commutative Algebra, Queen’s Papers in Pure and Appl. Math., No. 42, Kingston, Ont., 1975.
 Wilberd van der Kallen, Le $K_{2}$ des nombres duaux, C. R. Acad. Sci. Paris Sér. AB 273 (1971), A1204–A1207 (French). MR 291158
Additional Information
 © Copyright 1978 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 240 (1978), 317328
 MSC: Primary 13C10; Secondary 13F20, 14F05
 DOI: https://doi.org/10.1090/S00029947197804858275
 MathSciNet review: 0485827