Analytic ring

A ring of germs of analytic functions at a point in an analytic space. The following is a more accurate definition. Let be a field with a non-trivial norm (cf. Norm on a field) (which is usually assumed to be complete), and let be the -algebra of power series in with coefficients in which converge on some polycylinder with centre , each series converging on its own polycylinder. A quotient ring of the ring is called an analytic ring over , or an analytic -algebra; usually, is the field of real numbers or the field of complex numbers . Any analytic ring is a local, Noetherian, Hensel ring; its field of residues is isomorphic to . An analytic ring is a regular (and a factorial) ring, and its completion in the topology defined by the maximal ideal coincides with the ring of formal power series . The normalization lemma is true: An integral analytic ring is a finite extension of an analytic ring . Algebras that are finite over are generally called quasi-analytic -algebras. If is a perfect field, an analytic ring is an excellent ring.

References

[1]	J. Dieudonné, A. Grothendieck, "Critères differentiels de régularité pour les localisés des algèbres analytiques" J. of Algebra , 5 (1967) pp. 305–324
[2]	B. Malgrange, "Ideals of differentiable functions" , Tata Inst. (1966)
[3]	S.S. Abhyankar, "Local analytic geometry" , Acad. Press (1964)

References

[a1]	H. Grauert, R. Remmert, "Coherent analytic sheaves" , Springer (1984) (Translated from German)