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)



Submersive and unipotent group quotients among schemes of a countable type over a field $ k$

Author: Paul Cherenack
Journal: Trans. Amer. Math. Soc. 211 (1975), 101-112
MSC: Primary 14M15; Secondary 20G15
MathSciNet review: 0376700
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: An algebraic group G is called submersive if every quotient in affine schemes $ {c^G}:{\text{Spec}}\;A \to {\text{Spec}}\;{A^G}$ which is surjective is also submersive. We prove that every unipotent group is submersive. Suppose G is submersive. We show that if $ {c^G}({\text{Spec}}\;A)$ is open in $ {\text{Spec}}\;{A^G}$ or if some restrictions on the action of G on A are made, $ {c^G}$ is a topological quotient. A criterion for semisimplicity of points is extended to the case where G is unipotent. Finally, applications of the theory are provided.

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

  • [1] Armand Borel, Linear algebraic groups, Notes taken by Hyman Bass, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0251042
  • [2] Paul Cherenack, Basic objects for an algebraic homotopy theory, Canad. J. Math. 24 (1972), 155–166. MR 0291251,
  • [3] Paul Cherenack, The topological nature of algebraic contractions, Comment. Math. Univ. Carolinae 15 (1974), 481–499. MR 0354659
  • [4] Jean A. Dieudonné and James B. Carrell, Invariant theory, old and new, Academic Press, New York-London, 1971. MR 0279102
  • [5] Schémas en groupes. I: Propriétés générales des schémas en groupes, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck. Lecture Notes in Mathematics, Vol. 151, Springer-Verlag, Berlin-New York, 1970 (French). MR 0274458
  • [6] David Mumford, Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Band 34, Springer-Verlag, Berlin-New York, 1965. MR 0214602
  • [7] -, Introduction to algebraic geometry, Harvard Univ. Press, Cambridge, Mass.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 14M15, 20G15

Retrieve articles in all journals with MSC: 14M15, 20G15

Additional Information

Keywords: Submersive group, unipotent group, scheme of a countable type over k, affine scheme, algebraic group quotient, reductive group, relative coequalizer, semistable point, Hilbert's fourteenth problem, Nagata's counterexample, linear systems, Chevalley's theorem, constructible set
Article copyright: © Copyright 1975 American Mathematical Society