Two differential themes in characteristic zero This is also dedicated to Wolmer Vasconcelos on his sixtieth fifth birthday Aron Simis 2 Departamento de Matem´atica, CCEN Universidade Federal de Pernambuco 50740-540 Recife, PE, Brazil Foreword As a commutative algebraist one has time and again found oneself intrigued by the way in which differentials and derivations play their frequently unexpected act in ring and module theory (for instances of my own bewilderment or ignorance see [1], [2], [3], [5], [7], [8],[10], [11], [12],[13], [14],[16], [17]). At the jeopardy of loosing any readers left, I will allow myself yet another perplexity. This paper is divided in two sections, as conceived from the title. Both deal with a finitely generated k-algebra A, where k is a field mainly of characteristic zero. The first section is concerned with k-derivations of A. There are as of today withstanding questions concerning the structure of the module Der k (A) of k-derivations of A, such as the Zariski– Lipman conjecture - not yet settled in dimension two - and its natural extended versions such as the homological Zariski–Lipman conjecture of Herzog–Vasconcelos. It appears to me that some of the difficulty inherent to these questions has to do with the lack of resiliency in taking derivations with extended values. In this section we will mainly assume that A is a finitely generated k-subalgebra of a polynomial ring B := k[t]= k[t 1 ,...,t d ]. Besides the A-module of k-derivations of A, it is natural in this context to consider the A-module Der k (A, B) of A with values in B. True, one can consider a similar derivation module for any ring extension A ⊂ B, but one will lack the possibility of taking partial derivatives with respect to parameters which is, after all, the old tool in classical differential geometry (cf. [1]). Clearly, there is a natural inclusion of A-modules Der k (A) ⊂ Der k (A, B). The latter is naturally a B-module and we will be interested in determining its structure. As mentioned above, the main tool is the jacobian matrix Θ(g) of g with respect to t, where g is a set of generators of A over k. A preliminary basic result about the rank of this matrix is the fact that rank Θ(g) = dim A when k is a field of characteristic zero. 0 AMS 1980 Mathematics Subject Classification (1985 Revision). Primary 13H10, 14C17; Secondary 13C40, 13D10, 13N05, 13C15, 14B07, 14E25, 14M05, 14M25. 2 Partially supported by a grant from CNPq and PRONEX 1