Comments on Lefschetz's Principle
Comments on Lefschetz's Principle
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对莱夫谢茨原理的评论
DOI：
10.1080/00029890.1958.11991979
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发表时间：
1958
期刊：
影响因子：
--
通讯作者：
A. Seidenberg
中科院分区：
文献类型：
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作者：
A. Seidenberg
本说明的目的纯粹是批判性的。在他的书[1]中，S。莱夫谢茨说：“在本附录中，我们打算证明，在某种意义上，特征为零的基场上的代数几何可以简化为复代数几何。这是毫无疑问的深层原因，为什么特征零代数几何提出没有新的结果以上复杂的代数几何。莱夫谢茨接着解释说，定义在基场K（假设代数封闭）上的一个簇V实际上是由有限数量的量决定的，即定义V的方程中的系数。由这些量在有理数域上生成的场同构地嵌入到复数域C中，一个复簇V* 由与决定V的那些方程对应的方程产生。莱夫谢茨说：“重要的事实是，在从V到V* 的过程中，V的所有严格代数性质都被保留了下来。“但问题当然是，假设复形的假设和结论是用代数术语表达的，那么用拓扑方法得到的复形的基场结果是否真的是代数的，也就是说，是否真的与基场无关。因此，莱夫谢茨只是假设了他要证明的东西。莱夫谢茨没有讨论典型的例子，而只是陈述了三个应用。其中前两个涉及皮卡德数p的表面。这些定理不容易理解，人们可能想知道它们对于说明一个简单的元数学原理有多合适。我们会回到这些例子，但首先考虑一个更简单的例子。考虑特征为0的代数闭基场K上的射影平面中的两条曲线f（x0，x11 x2）= 0，g（x0，x11 x2）= 0。在复数情况下，已知两条曲线至少在一点相交，即C中存在数a0，a1，a2，但不都= 0，使得f（a0，a1，a2）= 0，g（a0，a1，a2）= 0。我们大概可以得出这样的结论：K也是如此。让我们沿着这个论点继续下去，看看。在上述准备之后，我们可以假设K是C的一个子域，并且是有理数域的有限扩张的代数闭包。然后我们得出结论，这两条曲线有一个共同点。是的，但是“点”这个词的意思已经改变了。我们要证明的定理是，在K中存在满足f= 0和g= 0的三元组（不是所有的三元组都= 0），我们只是得出结论，在K的扩张域中存在这样的三元组。
The object of the present note is purely critical. In his book [1], S. Lefschetz says:" In the present appendix we propose to show that, in a certain sense, algebraic geometry over a groundfield of characteristic zero may be reduced to complex algebraic geometry. This is without question the deep reason why characteristic zero algebraic geometry presents no new results over and above complex algebraic geometry." Lefschetz goes on to explain that a variety V defined over the groundfield K (assumed algebraically closed) is actually determined by a finite number of quantities, namely, the coefficients in equations defining V. The field generated over the rational number field by these quantities is embedded isomorphically into the complex number field C and a complex variety V* arises from the equations corresponding to those determining V." The important fact," says Lefschetz," is that in the passage from V to V* all the strictly algebraic properties of V are preserved." But of course the question is whether results obtained by topological methods for the groundfield of the complexes, supposing their hypothesis and conclusion to be couched in algebraic terms, really are algebraic, that is, really are independent of the groundfield. Lefschetz thus merely assumes what he set out to prove.Lefschetz discusses no typical example, but merely states three applications. The first two of these concern Picard's number p for surfaces. These theorems are not easy to understand, and one may wonder how appropriate they are for illustrating what is asserted to be a simple metamathematical principle. We'll come back to these examples but first consider a simpler one. Consider two curves f (x0, x11 x2)= 0, g (x0, x11 x2)= 0 in the projective plane over the algebraically closed groundfield K of characteristic 0. In the complex case, it is known that the two curves meet in at least one point, that is, there exist numbers ao, a1, a2 inC, not all= 0, such thatf (ao, a11 a2)= 0, g (ao, a1, a2)= 0. Presumably we are to conclude that the same is true for K. Let us follow the argument up and see. After the preparations indicated above, we may suppose K to be a subfield of C and to be the algebraic closure of a finite extension of the rational number field. We then conclude that the two curves have a point in common. Yes, but the meaning of the word point has shifted. The theorem we set out to prove was that there are triples of numbers (not all= 0) in K which satisfy f= 0 and g= 0 and we have only concluded that there are such triples in an extension field of K.*