FORMAL AND RIGID GEOMETRY: AN INTUITIVE INTRODUCTION AND SOME APPLICATIONS

FORMAL AND RIGID GEOMETRY: AN INTUITIVE INTRODUCTION, AND SOME APPLICATIONS JOHANNES NICAISE Abstract. We give an informal introduction to formal and rigid geometry over complete discrete valuation rings, and we discuss some applications in al- gebraic and arithmetic geometry and singularity theory, with special emphasis on recent applications to the Milnor fibration and the motivic zeta function by J. Sebag and the author. 1. Introduction Let R be a complete discrete valuation ring, with quotient field K, and residue field k. We choose a uniformizing parameter pi, i.e. (pi) is the unique maximal ideal of R. Geometers may take R = C[[t]], the ring of formal power series over the complex numbers, with K = C((t)), k = C, pi = t, while number theorists might prefer to think of R = Zp, the ring of p-adic integers, with K = Qp, k = Fp, pi = p. A formal scheme over R consists of an algebraic variety over k, together with algebraic information on an infinitesimal neighbourhood of this variety. If X is a variety over R, we can associate to X its formal completion X?, a formal scheme over R, in a natural way. An important aspect of the formal scheme X? is the following phenomenon.

• containing additional algebraic


• complete discrete valuation


• open neighbourhoods


• x∞ over


• formal schemes


• formal scheme


• quotient field