By E. Ramirez De Arellano
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Additional info for Algebraic Geometry and Complex Analysis
It follows that depth(E/zE) = depth(E) - 1 (COT. to prop. 6), whence the fact that EfxE is Cohen-Macaulay. Let E be of dimension n If E is a Cohen-Macaulay Theorem 3. module, then for every system of parameters x = (~1,. , z,) of E , we have the following properties: i) e,(E, n) = e(E/xE) , length of E/xE. i i ) gr,(E) = (E/xE)[X1,. , Xn] iii) Hl(x, E ) = 0 iv) &(x,E) = 0 forsll qZ1. Conversely, if a system of parameters of E satisfies any one of these prop erties, it satisfies all of them and E is a Cohen-Maca&’ module.
Properties and characterizations of regular local Let A be a regular local ring, n = glob dim A , m the maximal ideal of A , k = A/m and M a n~nzem finitely generated A-module. The following proposition compares proj dim, M and depth, M : 2, = KS&, O 1. It follows that: Tor~(Mn, k ) = Torz(Z,-2, k ) Proposition = = Tor,(Z,~,k) = Tor,+l(M, k) = 0: hence M, is free and a) is t,rue.
But if n = m/p, we have the exact sequence: 0 - p/p n In= - m/m2 + nJn2 + 0, and since [n/n’ : k] = dim A/p i we have [p/p n m2 : k] = hta p Thus if ~1,. , zp are elements of p whose images in m/m* form a k-basis of p/p n m2 i then the ideal (~1,. ,zp) is prime and of height p = htn p ; whence p = (21:. , zP) , qed. If p is a prime ideal of a regular ring A 1then the Proposition 23. local ring A, is regular. Indeed, it follows from the properties proved in part C that glob dim A, 5 glob dim A < 03 80 IV.
Algebraic Geometry and Complex Analysis by E. Ramirez De Arellano