Study of blowing-aps.

设A是Noether环，I是A的理想，如果I是正高度的，且I的Rees代数R(I)是Cohen-Macaulay，则R(I)称为A的算术Cohen-Macaulay化.给出了A有算术Macaulay化的充要条件.也就是说，A有算术Macaulay化当且仅当A满足(C_1)A是泛悬链线；(C2)A的所有局部化的形式纤维都是Cohen-Macaulay；(C3)任何有限生成的A-代数B的Cohen-Macaulay轨迹在Spec B中是开的；(Qu)对于任何一对素理想p⊂q，ht q=ht q/p+ht p；证明了A是Cohen-Macaulay环的同态像当且仅当A满足(C1)-(C3)且(Cd)A有余维函数.

Let A be a Noetherian ring and I an ideal of A. If I is of positive height and the Rees algebra R(I) of I is Cohen-Macaulay, then R(I) is called an Arithmetic Cohen-Macaulayfication of A. We give a necessary and sufficient condition for A to have an arithmetic Macaulayfication. That is, A has an arithmetic Macaulayfication if and only if A satisfies(C1)A is universally catenary ;(C2)all the formal fiber of any localization of A are Cohen-Macaulay ;(C3)the Cohen-Macaulay locus of any finitely generated A-algebra B is open in Spec B ;(QU)for any pair of prime ideals p ⊂ q, ht q = ht q/p + ht p ;(UM)A has no embedded primes.In consequence of this result, we show that A is a homomorphic image of a Cohen-Macaulay ring if and only if A satisfies (C1)-(C3) and(CD)A has a codimension function.