Minimal Generators for Invariant Ideals in Infinite Dimensional Polynomial Rings
نویسنده
چکیده
Let K be a field, and let R = K[X] be the polynomial ring in an infinite collection X of indeterminates over K. Let SX be the symmetric group of X. The group SX acts naturally on R, and this in turn gives R the structure of a left module over the group ring R[SX ]. A recent theorem of Aschenbrenner and Hillar states that the module R is Noetherian. We address whether submodules of R can have any number of minimal generators, answering this question positively. As a corollary, we show that there are invariant ideals of R with arbitrarily large minimal Gröbner bases. We also describe minimal Gröbner bases for monomially generated submodules.
منابع مشابه
m at h . A C ] 2 6 A ug 2 00 6 MINIMAL GENERATORS FOR INVARIANT IDEALS
Let K be a field, and let R = K[X] be the polynomial ring in an infinite collection X of indeterminates over K. Let S X be the symmetric group of X. The group S X acts naturally on R, and this in turn gives R the structure of a left module over the (left) group ring R[S X ]. A recent theorem of Aschenbrenner and Hillar states that the module R is Noetherian. We prove that submodules of R can ha...
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