By Rafael Villarreal

ISBN-10: 0824705246

ISBN-13: 9780824705244

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Hr is a regular sequence, then ht (p) = r for any minimal prime p of I . 13 one has the inequality ht (p) = dim(/? p ) < r. (b) Set J = (hi,. . ,hi-i) and K = (hi, . . ,ht). Assume ht (P) — i - I for any minimal prime P over J. Since hi is regular on R/J and K/ J is a principal ideal one has ht (K/ J) = 1, thus there is a prime ideal po minimal over J such that J C po C p and consequently ht (p) > i. Using (a) one gets ht (p) = i. 15 If (R,m,k) is a local ring, then dim(R) < dinifc(m/m 2 ) and R has finite Krull dimension.

Proof. See [217, Chapter 4]. 5 Let M be an N-graded R-module. If N is a graded submodule of M , then N has an irredundant primary decomposition N = jVi n • • • n Nr such that: (a) Ni is a graded submodule and AsSft(M/Ni) = {pi} for all i, (b) N ^ NI n • • • n JVj_i n Ni+i n • • • n Nr for all i, and Proof. 23. Q Finding primary decompositions of graded ideals in polynomial rings over fields is a difficult task, the punch line is that there is not yet a best general strategy for computing primary decompositions [77]; for some of the main algorithms that can been used see [95, 127, 255, 313].

We proceed by induction on n > 2. kn > Q-in for i > k. One may assume k < m; for otherwise fa > • • • > fan and one can use induction. Since fa > • • • > fa, by induction there is w' = (l,u>2, . . , w n -i) such that w' • fa > w' • fa for 2 < i < k. On the other hand for every i > A; one can choose Si € N+ so that w1 • fa + ain8i > w' • fa + ainSi. To finish the proof set •u;n = max{(5j| k * w • ai for i > 2. 4 Lei jR = A;[x 1; . . ,xn] be a polynomial ring over a field k and let f be a polynomial in R \ k. *

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