A well-studied and important numerical invariant of a graded ideal over a graded polynomial ring S is the Hilbert function. It gives the sizes of the graded components of the ideal.
The Hilbert function encodes important information (for example, dimension and multiplicity). Hilbert’s insight was that it is determined by finitely many of its values.
In many recent papers and books, Hilbert functions are studied using clever computations with binomials; we mention the binomial-approach briefly and avoid such computations whenever possible. Instead our arguments are founded upon Macaulay’s key idea in 1927: There exist highly structured monomial ideals - lex ideals - which attain all Hilbert functions. Lex ideals play an important role in many results on Hilbert functions. The pivotal property is that a lex ideal grows as slowly as possible.
Another exciting direction of research is to parametrize all graded ideals in S with a fixed Hilbert function, and then study their (common) properties and the structure of the parameter space. Lex ideals play crucial role in Hartshorne’s Theorem that Grothendieck’s Hilbert scheme is connected.
KeywordsSimplicial Complex Betti Number Hilbert Series Hilbert Function Monomial Ideal
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