TY - JOUR
T1 - Finite separating sets and quasi-affine quotients
AU - Dufresne, Emilie
PY - 2013/2/1
Y1 - 2013/2/1
N2 - Nagata's famous counterexample to Hilbert's fourteenth problem shows that the ring of invariants of an algebraic group action on an affine algebraic variety is not always finitely generated. In some sense, however, invariant rings are not far from affine. Indeed, invariant rings are always quasi-affine, and there always exist finite separating sets. In this paper, we give a new method for finding a quasi-affine variety on which the ring of regular functions is equal to a given invariant ring, and we give a criterion to recognize separating algebras. The method and criterion are used on some known examples and in a new construction.
AB - Nagata's famous counterexample to Hilbert's fourteenth problem shows that the ring of invariants of an algebraic group action on an affine algebraic variety is not always finitely generated. In some sense, however, invariant rings are not far from affine. Indeed, invariant rings are always quasi-affine, and there always exist finite separating sets. In this paper, we give a new method for finding a quasi-affine variety on which the ring of regular functions is equal to a given invariant ring, and we give a criterion to recognize separating algebras. The method and criterion are used on some known examples and in a new construction.
UR - http://www.scopus.com/inward/record.url?scp=84866319630&partnerID=8YFLogxK
U2 - 10.1016/j.jpaa.2012.06.007
DO - 10.1016/j.jpaa.2012.06.007
M3 - Article
AN - SCOPUS:84866319630
SN - 0022-4049
VL - 217
SP - 247
EP - 253
JO - Journal of Pure and Applied Algebra
JF - Journal of Pure and Applied Algebra
IS - 2
ER -