By Stephan Körner
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This shows that the length p of the sequence depends only on M. In other words: AI1 the maximal M-sequences have the same number of elements, say p Every M-sequence can be extended to a maximal M-sequence. ) Proposition and Definition 6. Corollary With the above notation, one has depthA M; = depth, M - i. M) is isomorphic O+M%M-Ml-0 Let us again assume that M # 0 Proposition 7. (i) Every M-sequence can be extended to a system ofparameters of M (ii) One has depth, M < dim A/p for every p E Ass(M) and ‘.
Z~) and n = m/p and hence the equivalences: b ) Q [p/pnm’:k] = p e [n/n*:k] = dimA-p. But x1,. ,zp form a subset of a system of parameters of A, so A/(x1, , zP) has dimension dim A - p ; whence the result. c) + b) : Indeed, c) is equivalent to the two conditions: [n/n* : k ] = dimA/p and dim A/p = dim A - p. the following two Corollary properties are equivalent: a) A/p is a regular local ring. b) p is generated by a subset of a regular system of parameters of A Only the implication a) + b) remains to be proved.
Zr of a system of parameters of A such that p E Ass(E), where E = A/(x1,. , z,)A Moreover, according to theorem 4, the module E is a Cohen-1Macaulay~module of dimension dim A/p. The same is thus true for it! 5, (which is applicable since A is A-flat), we have Ass(A/pA) 5 Ass(E). But, according to proposition 13 applied to E , every p’ E Ass(E) is such that dim a/p’ = dim 2, whence the result. Corollary Let E be a finitely generated module over a CohenMacaulay local ring, and let n be an jnteger 2 0.