  We investigate the local properties of Berkovich spaces over Z. Using
Weierstrass theorems, we prove that the local rings of those spaces are
noetherian, regular in the case of affine spaces and excellent. We also show
that the structure sheaf is coherent. Our methods work over other base rings
(valued fields, discrete valuation rings, rings of integers of number fields,
etc.) and provide a unified treatment of complex and p-adic spaces.
