ON THE PURITY OF THE BRANCH LOCUS OF ALGEBRAIC FUNCTIONS.
ON THE PURITY OF THE BRANCH LOCUS OF ALGEBRAIC FUNCTIONS.
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论代数函数分支轨迹的纯粹性。
DOI：
10.1073/pnas.44.8.791
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发表时间：
1958
影响因子：
11.1
通讯作者：
O. Zariski
中科院分区：
文献类型：
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作者：
O. Zariski
1. 设V/k为绝对不可约的r维正规代数变量，设k = k(V)为V/k的函数域；这里k表示一个任意的场。设K*是K的有限可分代数扩展，设K*是K在K*中的代数闭包，设V*/ K*是V在K*中的归一化。设P*是V*的任意点（不一定是k上的代数点），设P是V的对应点。我们用0表示P在V/k上的局部环，用m表示o的最大理想。设o*和m*对于P*和V*/k*具有类似的意义。众所周知，(1)0 *m是初等理想，m*是伴生的素数理想；(2)剩余域k*(P*) （= o*/m*）是域k(P) （= o/m）的-有限代数推广。定义：当满足以下条件时，点P*（相对于V）是无分支的：
1. Let V/k be an absolutely irreducible, r-dimensional normal algebraic variety and let K = k(V) be the function field of V/k; here k denoted an arbitrary ground field. Let K* be a finite separable algebraic extension of K, let k* be the algebraic closure of k in K*, and let V*/k* be a normalization of V in K*. Let P* be an arbitrary point of V* (not necessarily algebraic over k), and let P be the corresponding point of V. We denote by o the local ring of P on V/k and by m the maximal ideal of o. Let o* and m* have a similar meaning for P* and V*/k*. It is well known that: (1) o*m is a primary ideal, with m* as associated prime ideal; (2) the residue field k*(P*) (= o*/m*) is a -finite algebraic extension of the field k(P) (= o/m). Definition: The point P* is said to be unramified (with respect to V) if thefolltwing conditions are satisfied: