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FINITE GENERATION OF SYMMETRIC IDEALS MATTHIAS ASCHENBRENNER AND CHRISTOPHER J. HILLAR
 

Summary: FINITE GENERATION OF SYMMETRIC IDEALS
MATTHIAS ASCHENBRENNER AND CHRISTOPHER J. HILLAR
In memoriam Karin Gatermann (1965­2005).
Abstract. Let A be a commutative Noetherian ring, and let R = A[X] be
the polynomial ring in an infinite collection X of indeterminates over A. Let
SX be the group of permutations of X. The group SX acts on R in a natural
way, and this in turn gives R the structure of a left module over the left group
ring R[SX ]. We prove that all ideals of R invariant under the action of SX are
finitely generated as R[SX ]-modules. The proof involves introducing a certain
well-quasi-ordering on monomials and developing a theory of Gršobner bases
and reduction in this setting. We also consider the concept of an invariant
chain of ideals for finite-dimensional polynomial rings and relate it to the
finite generation result mentioned above. Finally, a motivating question from
chemistry is presented, with the above framework providing a suitable context
in which to study it.
1. Introduction
A pervasive theme in invariant theory is that of finite generation. A fundamen-
tal example is a theorem of Hilbert stating that the invariant subrings of finite-
dimensional polynomial algebras over finite groups are finitely generated [6, Corol-
lary 1.5]. In this article, we study invariant ideals of infinite-dimensional polynomial

  

Source: Aschenbrenner, Matthias - Department of Mathematics, University of California at Los Angeles

 

Collections: Mathematics