Cyclic purity versus purity in excellent Noetherian rings
Cyclic purity versus purity in excellent Noetherian rings
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DOI：
10.1090/s0002-9947-1977-0463152-5
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发表时间：
1977-02
影响因子：
1.3
通讯作者：
M. Hochster
中科院分区：
文献类型：
--
作者：
M. Hochster
给出了Noether环R的一个特征，使得只要R在扩张代数S中是理想闭的（=循环纯的），则R在S中是纯的.事实上，R具有此性质当且仅当R的每个局部环在极大理想处的完备化（A，m）具有以下两个等价性质：（i）对于每个整数N > 0，存在m-准素不可约理想IN C mN。(ii)要么dim_A = 0并且A是Gorenstein，要么深度A > 1并且不存在P ∈ Ass（A）使得dim_A（A/P）= 1并且（A/P）（A/P）可嵌入A中。然后证明了如果R是局部优良的Noether环，使得R是约化的（或者更一般地，使得R是遗传Gorenstein的），或者使得Ass（<$）不包含同高的素数M ® E是内射的。一个特别有趣的例子是N = R和E = S是R的扩张代数：在这种情况下，5中R的循环纯度断言，对于R的每个理想I，IS D R = I（在循环纯度的定义中设E = R/I）。这个条件有时用短语“i？最好是在5”内关闭。非常一般地，如果M/N是M的n阶表示，则N在M中是纯的当且仅当N是M的直和项。我们请读者参阅[3]、[14]、[15]、[20，p. 64]和[24]，了解有关纯洁性的基本事实。我们最初的目标是给出一个真正有用的诺特条件，编辑于1976年2月23日收到。AMS（A/05）科目分类（1970年）。小学13 E05、13 H10、13 B 99、13 C99。（'）作者得到了国家科学基金会的部分资助。美国数学学会1977年
A characterization is given of those Noetherian rings R such that whenever R is ideally closed (= cyclically pure) in an extension algebra S, then R is pure in S. In fact, R has this property if and only if the completion (A, m) of each local ring of R at a maximal ideal has the following two equivalent properties: (i) For each integer N > 0 there is an m-primary irreducible ideal IN C mN. (ii) Either dim^ = 0 and A is Gorenstein or else depth A > 1 and there is no P e Ass(A) such that diia(A/P) = 1 and (A/P) ® (A/P) is embeddable in A. It is then shown that if R is a locally excellent Noetherian ring such that either R is reduced (or, more generally, such that R is genetically Gorenstein), or such that Ass(Ä) contains no primes of coheight M ® E is injective. A case of particular interest is the one where N = R and E = S is an extension algebra of R: in this case the cyclic purity of R in 5 asserts that for every ideal I of R, IS D R = I (let E = R/I in the definition of cyclic purity). This condition is sometimes expressed by the phrase "i? is ideally closed in 5". Quite generally, if M/N is finitely presented, then N is pure in M if and only if N is a direct summand of M. We refer the reader to [3], [14], [15], [20, p. 64], and [24] for basic facts about purity. Our original objective was to give a really useful condition on a Noetherian Received by the editors February 23, 1976. AMS (A/05) subject classifications (1970). Primary 13E05, 13H10, 13B99, 13C99. (') The author was supported, in part, by a grant from the National Science Foundation. « American Mathematical Society 1977