𝐾₁ of a curve of genus zero

Roberts, Leslie G.

doi:10.1090/S0002-9947-1974-0332787-X

$K_{1}$ of a curve of genus zero
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Trans. Amer. Math. Soc. 188 (1974), 319-326 Request permission

Abstract:

We determine the structure of the vector bundles on a curve of genus zero and calculate the “universal determinant” ${K_1}$ of such a curve.

References

Cahit Arf, Untersuchungen über quadratische Formen in Körpern der Charakteristik 2. I, J. Reine Angew. Math. 183 (1941), 148–167 (German). MR 8069, DOI 10.1515/crll.1941.183.148
M. Atiyah, On the Krull-Schmidt theorem with application to sheaves, Bull. Soc. Math. France 84 (1956), 307–317. MR 86358, DOI 10.24033/bsmf.1475
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, DOI 10.1007/BF02684690
N. Bourbaki, Éléments de mathématique. 23. Première partie: Les structures fondamentales de l’analyse. Livre II: Algèbre. Chapitre 8: Modules et anneaux semi-simples, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1261, Hermann, Paris, 1958 (French). MR 0098114
S. M. Gersten, Higher $K$-theory of rings, Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf. Seattle Res. Center, Battelle Memorial Inst., 1972) Lecture Notes in Math., Vol. 341, Springer, Berlin, 1973, pp. 3–42. MR 0382398
Daniel Quillen, Higher algebraic $K$-theory. I, Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 341, Springer, Berlin, 1973, pp. 85–147. MR 0338129
Leslie G. Roberts, $K_{1}$ of some abelian categories, Trans. Amer. Math. Soc. 138 (1969), 377–382. MR 251109, DOI 10.1090/S0002-9947-1969-0251109-4
Leslie G. Roberts, $K_{1}$ of projective $r$-spaces, Proc. Amer. Math. Soc. 26 (1970), 587–592. MR 280501, DOI 10.1090/S0002-9939-1970-0280501-0
Leslie G. Roberts, Indecomposable vector bundles on the projective line, Canadian J. Math. 24 (1972), 149–154. MR 297776, DOI 10.4153/CJM-1972-013-9

Real quadrics and

of a curve of genus zero

Leslie Roberts, Base change for $K_{0}$ of algebraic varieties, 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. 122–134. MR 0396577
Winfried Scharlau, Quadratic forms, Queen’s Papers in Pure and Applied Mathematics, No. 22, Queen’s University, Kingston, Ont., 1969. MR 0269679
Shianghaw Wang, On the commutator group of a simple algebra, Amer. J. Math. 72 (1950), 323–334. MR 34380, DOI 10.2307/2372036