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34 is expounded to the truth that the failure of Nakayama’s Lemma for normal multigradings reasons the idea of minimum loose solution to damage. instance eight. 35 Take S = k[x, y] and set A = Z, with deg(x) = 1 and deg(y) = −1. If I = xy − 1 , that is homogeneous for this multigrading, then i will even be represented because the excellent x2 y 2 − xy, x2 y 2 − 1 . Neither generator should be passed over, even supposing we all know that I is a significant perfect. as a result it simply is senseless to assert that the first multigraded Betti variety of S/I is 1 or 2. however, the 2 unfastened resolutions xy+1 xy−1 zero ← S ←−−−− S ← zero and x2 y 2 −xy x2 y 2 −1 xy zero ← S ←−−−−−−−−−−−−− S 2 ←−−−− S ← zero yield an identical K-polynomial K(S/I; t) = zero. notice that multiplication by means of the measure 0 point xy − 1 induces a level zero multigraded isomorphism S → xy − 1 , so okay( xy − 1 ; t) = K(S; t) = 1 will be anticipated. one in all our purposes for constructing the homogenization options in part eight. three is they paintings even for nonpositive multigradings. specifically, we get that K-polynomials are nonetheless invariant lower than Gr¨ obner degeneration. Theorem eight. 36 repair a time period order on a multigraded loose module F. If okay is a graded submodule of F then K(F/K; t) = K(F/in(K); t). facts. quick from Proposition eight. 28 and Theorem eight. 34. eight. four. K-POLYNOMIALS IN NONPOSITIVE GRADINGS 163 K-polynomials in nonpositive gradings nonetheless continue a few of the enumerative information that they exhibit through Hilbert sequence in Theorem eight. 20 for confident gradings. the way in which out of the most obvious challenge that arbitrary modules— even finitely generated ones—need no longer have Hilbert sequence is to define ourselves out, via limiting our awareness to modules that do have Hilbert sequence. Such modules come up in a few purposes, particularly to homogeneous coordinate jewelry of toric kinds. Definition eight. 37 permit S = k[x] be a polynomial ring multigraded by way of A. A multigraded module M is discreet if dimk (Ma ) is finite for all a ∈ A. even if the grading workforce A could have torsion, the notation Z[[A]] nonetheless is smart for the additive crew of features A → Z, which we nonetheless name Laurent sequence. Modest modules are accurately these graded modules M that experience well-defined Hilbert sequence, which we back denote by means of H(M ; t) ∈ Z[[A]]. instance eight. 38 permit I = xy, yz = y ∩ x, z , and think about the module M = k[x, y, z]/I, with the multigrading as in instance eight. 2. This module is inconspicuous, and the Hilbert functionality Z ⊕ Z/2Z → N of M should be represented as a 2 × ∞ array of integers. In levels starting from (−4, zero) within the decrease left to (4, 1) within the top correct, the Hilbert functionality is zero zero 1 zero 1 zero zero zero zero 1 1 2 2 1 1 2 2 three ↔ y y2 1 x xz z x2 z2 xz 2 xz three x3 x3 z x2 z ... z3 . The Hilbert sequence of M lies within the staff Z[[Z⊕Z/2Z]] = Z[[s, s−1 , t]]/ t2 −1 of Laurent sequence supported on A. The subgroups Z[[s−1 ]][t]/ t2 − 1 and Z[[s]][t]/ t2 −1 are earrings during which all components with consistent coefficients ±1 are devices. The Hilbert sequence of M can for this reason be expressed because the sum H(M ; s, t) = 1 s−2 t + , −2 1 − s t (1 − st)(1 − s) the place the 2 ratios—which are considered as mendacity within the subgroups above—sum the optimistic powers of y and the monomials in k[x, z].