Download e-book for kindle: Combinatorial and Geometric Representation Theory by Seok-Jin Kang, Kyu-Hwan Lee

By Seok-Jin Kang, Kyu-Hwan Lee

ISBN-10: 0821832123

ISBN-13: 9780821832127

ISBN-10: 1091987254

ISBN-13: 9781091987258

ISBN-10: 1351341251

ISBN-13: 9781351341257

ISBN-10: 3719737527

ISBN-13: 9783719737528

ISBN-10: 7119938398

ISBN-13: 9787119938394

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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.

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Combinatorial and Geometric Representation Theory by Seok-Jin Kang, Kyu-Hwan Lee


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