Generalised Hilbert Numerators Ii
نویسنده
چکیده
Let K[〈X〉] denote the large polynomial ring (in the sense of Halter-Koch [7]) on the set X = {x1, x2, x3, . . . } of indeterminates. For each integer n, there is a truncation homomorphism ρn : K[〈X〉] → K[x1, . . . , xn]. If I is a homogeneous ideal of K[〈X〉], then the N-graded Hilbert series of K[x1,...,xn] ρn(I) can be written as gn(t) (1−t) ; it was shown in [13] that if in addition I is what we call locally finitely generated, then gn(t) → g(t) ∈ Z[[t]], the so-called Hilbert numerator of I . In this article, we generalise this result to N-graded locally finitely generated ideals. For monomial ideals in K[〈X〉], we define the [X]-graded Hilbert numerator as the Hilbert numerator of the contracted monomial ideal in K[X], for which the standard combinatorial and homological methods for calculating multi-graded Hilbert series of monomial ideals in finitely many variables apply. Finally, we show that all polynomial N-graded Hilbert numerators can be obtained from ideals generated in finitely many variables, and that the the closure in Z[[t]] of this set is the set of all N-graded Hilbert numerators. Our main tools are the approximation theorem of [11], relating the initial ideal with the initial ideal of the truncated ideal, and the identification of K[[X]] with the ring of all number-theoretic functions [5] which allows the passing from the characteristic function of the complement of a monomial ideal to its Hilbert numerator to be seen as an example of Möbius inversion.
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