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By Tor H Gulliksen

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Example text

X xl x. he zero-ideal. he empty t,hen shall be inter- sequel~ce • An R-sequence is called a maximal R-seqnence in not an \{-sequence for any . y E a noetherian any R-sequence in a if a xl'···'x s is xl,···,xs,y J\ote that. since R is can be extended to a maxi- mal one. 2. "'J. - 25 - Then s-tq == n • Pr~of. If s == 0 then cver'y element of ~ llence 'We prove the proposition by induction on n let, valid for k s > :::::::: 1 O:!! and 1 S Cl Now let k-l a. ,x . 1 1 J<. 1n +0 an d q == n • s == k ~.

S to Ule rCiHier. ). f:X - Y Let be a homomorphism of j{-algebras inducing an isomorphism Let- s X • he a cycle in Y' == Y = and has a canonical extension "to f' :X' - Y' an K-homomorphism X' Put f~:_: H(X) ~ H(Y) inducing an isomorpllism , f~:: H (X ,) --< H( Y' ) P f'oof -- • deg S If E X is odd, then for define and if deg S ff( ~ . e-htforward fll: 1 . verifications show that in both cases is an isomorphism. he map ff:X'~Y' fl We restrict. s(i») _. becomes an K-homomorphism.

Proof. a 1S he shall first see that we may assume tllat contained ill the square of tile maximal idf"al - 41 - m of ') i m- ~ Indeeci, if let, Put is {·egular. 16) in RI . 1 , tile image of j'l j'l j{ ~ 2.. If anci £1 is generated hy an is generated by an R- 1! 1 We have a canonical map sequence if and only if sequence. Let Xl f: ~l \ E2i . er a finite numb er of steps. ln{I; j{ = a. 1)) E: complet~e intersection if and only if "'J Dim K - Dim and let rJ E Since R is regular, it is a Nacaulay hence the last equation is equivalent to sayinr- that rv is an K-sequence.

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