On the theory of differentials in commutative rings

Abstract

Hitherto the concept of differentials is introduced usually as the dual no- tion of the derivations. For instance the differentials belonging to a function field $K/k$ are the dual elements of the k-derivations of $K$ into itself. If $V$ is a differentiable manifold the local differentials at a point of $V$ are nothing other than the elements of the dual space of the space of tangent vectors at the point. Recently P. Cartier introduced in $[$ 4 $]$ the notion of differentials without referring to the special module of derivations. His theory was used to settle the problem of separability in the extension of fields and the theory of simple points. We shall present here a more general version of the theory of differentials in a commutative R-algebra. In \S 1 we list the notations and terminologies which will be used throughout the rest of the paper. In \S 2 we shall prove some properties of differentials in a commutative algebra. Among others we shall introduce the notion of the quasi-separability for an R-algebra2) S. When $R$ is a field and $S$ is a finite R-module the quasi-separability coincides with the separability of $S/R$. In \S 3 we shall give a theorem by which we can determine the structure of the module of differentials. \S 4 is devoted to the proof of the following theorem. Let $P$ be a point of an algebraic variety $V$ which is defined over $k$ , and let $R$ be the quotient ring of $P$ in $V/k$ . Then under suitable conditions it will be shown that $P$ is a simple point of $V$ if and only if the module of k-differentials $D_{k}(R)$ is a free module. We shall study in \S 5 the contribution of our theory to the theory of ramifications. We shall introduce there the notion of d-different. Quasi-separability and unramifiedness will be shown to be equivalent under some additional assumptions. On the other hand we have another kind of different which is called the homological different in [2]. We can easily prove that the homological different is con- tained in the d-different and in some occasions they have the same radical. But the precise relation between them is yet unknown. The relation between the d-different and the differential index defined in our previous paper [7] will be discussed in the last paragraph. They give rise to the same results in the case where we are studying the covering of algebraic curves. But in the case 1) The numbers in the bracket refer to the bibliography at the end of the paper.