By Seok-Jin Kang, Kyu-Hwan Lee
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Extra resources for Combinatorial and Geometric Representation Theory
Zn is regular in R = R/(z1 , . . , z −1 ). But z +2 , . . , zn is regular in R = R/(z1 , z2 , . . , z , z +1 ), so the result follows from the previous paragraph. 8 Cohen-Macaulay Rings The depth(R) of a ring local ring (R, m) is deﬁned as the maximal length of a regular sequence in m. It can be shown that in a graded Noetherian ring R any two homogeneous maximal regular sequences have the same length. If R is a graded Noetherian ring we put depth(R) = depth(R+ ). The depth of a ring is always less than or equal to its Krull dimension.
F¯r } form a vector space basis of Q(R+ ). Then f1 , f2 , . . , fr minimally generate R as a K algebra. To see r r this consider that R+ = i=1 Rfi . Thus if f ∈ Rd then f = i=1 gi fi for some homogeneous elements gi ∈ R+ . By induction on degree, we see that gi ∈ K[f1 , f2 , . . , fr ] and thus f ∈ K[f1 , f2 , . . , fr ]. Conversely, it is clear that 2 an element of R not lying in R+ cannot be written as a polynomial in lower degree elements of R and thus any homogenous algebra generating set must surject onto to a spanning set for Q(R+ ).
Z , z +1 ), so the result follows from the previous paragraph. 8 Cohen-Macaulay Rings The depth(R) of a ring local ring (R, m) is deﬁned as the maximal length of a regular sequence in m. It can be shown that in a graded Noetherian ring R any two homogeneous maximal regular sequences have the same length. If R is a graded Noetherian ring we put depth(R) = depth(R+ ). The depth of a ring is always less than or equal to its Krull dimension. A local Noetherian ring (R, m) is said to be Cohen-Macaulay if depth(m) = height(m) and a general Noetherian ring R is said to be Cohen-Macaulay if the localization of R at each of its prime ideals is Cohen-Macaulay.
Combinatorial and Geometric Representation Theory by Seok-Jin Kang, Kyu-Hwan Lee