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Abstract
Given an ideal of forms in an algebra (polynomial ring, tensor algebra, exterior algebra, Lie algebra, bigraded polynomial ring), we consider the Hilbert series of the factor ring. We concentrate on the minimal Hilbert series, which is achieved when the forms are generic. In the polynomial ring we also consider the opposite case of maximal series. This is mainly a survey article, but we give a lot of problems and conjectures. The only novel results concern the maximal series in the polynomial ring.
Highlights
The Hilbert series of a graded commutative algebra is an important invariant in commutative algebra and algebraic geometry
It has since long been known what the Hilbert series of the quotient ring k[x1, . . . , xn]/I can be for a homogeneous ideal I in a polynomial ring over a field k [23]
A much harder question is: If I is generated by forms f1, . . . , fr of degrees d1, . . . , dr, what can the Hilbert series be? Not much is known about this
Summary
The Hilbert series of a graded commutative algebra is an important invariant in commutative algebra and algebraic geometry. It has since long been known what the Hilbert series of the quotient ring k[x1, . Dr, one can construct an algebra with a minimal series in the lexicographical sense, by choosing the forms fi to be generic. (We say that aizi is smaller than bizi lexicographically if for the smallest i for which ai = bi, we have ai < bi.) A form f in a graded k-algebra R is generic if the coefficients are algebraically independent over the prime field of k.
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