symmetric monoidal (∞,1)-category of spectra
For each prime number the ring of -adic integers is the formal completion of the ring at the prime ideal . Geometrically this means that is the ring of functions on a formal neighbourhood of inside Spec(Z) (this is discussed in more detail below). Algebraically it means that the elements in look like formal power series where the formal variable is the prime number .
To the person on the street, it may be described as (the ring of) numbers written in base , but allowing infinite expansions to the left. Thus, numbers of the form
where , added and multiplied with the usual method of carrying familiar from adding and multiplying ordinary integers.
Alternatively, it is the metric completion of the ring of integers with respect to the -adic absolute value. Since addition and multiplication of integers are uniformly continuous with respect to the -adic absolute value, they extend uniquely to a uniformly continuous addition and multiplication on . Thus is a topological ring.
Also , see at analytic completion.
Hence one also speaks of the -adic completion of the integers. See completion of a ring (which generalizes 2&3).
The ring of -adic integers has the following properties:
As a topological space, it is compact, Hausdorff, and totally disconnected (i.e., is a Stone space). Moreover, every point is an accumulation point, and there is a countable basis of clopen sets – a Stone space with these properties must be homeomorphic to Cantor space.
This plays a central role for instance in the function field analogy. It is highlighted for instance in (Hartl 06, 1.1, Buium 13, section 1.1.3). See also at arithmetic jet space and at ring of Witt vectors.
Examples of sequences of local structures
|geometry||point||first order infinitesimal||formal = arbitrary order infinitesimal||local = stalkwise||finite|
|smooth functions||derivative||Taylor series||germ||smooth function|
|curve (path)||tangent vector||jet||germ of curve||curve|
|smooth space||infinitesimal neighbourhood||formal neighbourhood||germ of a space||open neighbourhood|
|function algebra||square-0 ring extension||nilpotent ring extension/formal completion||ring extension|
|arithmetic geometry||finite field||p-adic integers||localization at (p)||integers|
|Lie theory||Lie algebra||formal group||local Lie group||Lie group|
|symplectic geometry||Poisson manifold||formal deformation quantization||local strict deformation quantization||strict deformation quantization|
Introductions and surveys include
Bernard Le Stum, One century of -adic geometry – From Hensel to Berkovich and beyond talk notes, June 2012 (pdf)
The synthetic differential geometry-aspect of the -adic numbers is highlighted for instance in