# Contemporary Developments in Algebraic K-Theory by (Eds.) M. Karoubi, A. O. Kuku, C. Pedrini By (Eds.) M. Karoubi, A. O. Kuku, C. Pedrini

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Dm ) if (a) In a minimal free resolution of M as above, the free module Fi generated by elements of degree di ; that is, βi, j = 0 when j = di . (b) M is Cohen–Macaulay of codimension m; that is, Fi = 0 for i > m and the annihilator of M is an ideal of codimension m. It is easy to see that if there is a pure module of type d, then d0 < · · · < dm . Much more is true: if M is a pure module, then a result of Herzog and K¨uhl  shows that the Betti table of M is determined by d up to a rational multiple: that is, there is a constant r = r(M) depending on M such that βi,di (M) = r .

An index on the left of a graded module always denotes the selection of the homogeneous component of that degree. If R is standard graded over a field K with maximal homogeneous ideal m all the invariants we are going to study depend actually only on the image of ϕ and not on the map itself as long as ker ϕ ⊆ mF. So, if J = Im ϕ , we will sometimes denote K(ϕ , R) simply by K(J, R) and so on. Fix a basis of the free module F, say {e1 , . . , en }. Given I = {i1 , . . , is } ⊂ [n] with i1 < i2 < · · · < is we write eI for the corresponding basis element ei1 ∧ · · · ∧ eis of s F.

5 Resolutions of Trigraded Artinian Modules of Codimension Three In the case of trigraded artinian modules over the polynomial ring k[x, y, z] where the resolution has pure total degrees, we do not know much. The following are natural questions. • For Betti diagrams with given total degrees, are there, up to translation, only a finite number of extremal rays in the positive cone of such Betti diagrams? Cone of Betti Diagrams of Bigraded Artinian Modules 15 • Suppose the above property holds. From Sect.