Algebra (Translation of Cours d'algèbre) - download pdf or read online

By Roger Godement

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Let (0) c Q' ~ N' that lies over nary). ,Nh} } known that there exist that be a saturated infinitely many height one q' ~ U = (U{N" and N' ~ M]) {q'MR ; q' ~ N' of them have height one, by ; N" E I})U(U{M S in is cate- Assume it is q' E Spec S ; M S such is a maximal Then there are infinitely many pri~ne and q' ~ U} , and all but finitely many [Mc-l, T h e o r e m 7]. depth q'NR = depth q' = depth N' + 1 , since Also, q' q'NR c N is contained and in a 33 maximal ideal Therefore M existence and S only if q' ~_ U[N" and since ; N" E I} Therefore of infinitely many height one S is catenary.

4). 5), level. ideal in Then R' altitude R > i H-local domain.

Depth PI = depth P + i = 2 . is not catenary. (with ideal in and let of this. 5) be a local domain, h > i . height PI -<- h - I . pose that R such that number of repetitions and let and depth one prime holds, clearly, holds, L = RSN L' = (R')(RS_N) to prove that L , where ideal in such that is an N let Rs be is a maximal L , and let p'NL = p , so H-domain, it may clearly 26 be assumed that altitude = altitude L - 1 , as will now be shown. 5). 4), R holds. 4) is catenary, , then if N' L' by height M' E over R , is a maximal L = altitude , so R} .

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Algebra (Translation of Cours d'algèbre) by Roger Godement


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