Introduction.

Let (Q, m, k, d) be a local ring, M be a finitely generated Q-module, and let X1, X2, … be a countable set of indeterminates over Q. Then we define the general extensions Qg, Mg of Q, M as follows.

DEFINITION. The general extension Qgof Q is the localisation of Q[X1, X2, …] at the prime idealm[X1, X2, …]. The general extension Mgof M is M⊗QQg.

The kernel of the map Q[X1, X2, …] → Qg is zero, since it consists of all polynomials in X1, X2, … annihilated by some polynomial g which has at least one coefficient a unit of Q. But then g is not a zero divisor of Q[X1, X2, …], since if it were there would exist a non-zero element a of Q such that ag = 0, and this is clearly not the case. Hence the map is injective, and the same is true of the induced map QN → Qg where QN is as defined in chapter 7, section 4. We will identify QN with its isomorphic image in Qg. Then the rings QN form an ascending sequence of sub-rings of Qg whose union is Qg. We denote the residue field of Qg by kg and its maximal ideal by mg.

We will also apply the above to the special case when Q is a field F, i.e. Fg and FN will denote the fields F(X1, X2, …) and F(X1, X2, …, XN).