Definition. The canonical topology on local points [fcap-0044]
AGENTDRAFTED
Here a local field \(K\) of characteristic different from \(2\) means
\(\mathbb R\) or \(\mathbb C\) with its usual topology, or a complete
discretely valued field with finite residue field and \(\operatorname {char}K\ne 2\),
with its valuation topology. The last case includes finite extensions of
\(\mathbb Q_2\): the characteristic restriction is on \(K\), not its residue
field. This uses the nonarchimedean definition in
[lam2005quadratic, VI.2, Definition 2.1 and the discussion, pp. 151--152],
with the two archimedean fields included explicitly.
For a finite-dimensional \(K\)-vector space \(W\), its canonical topology
is the topology transported from \(K^d\) by any basis, where
\(d=\dim _K W\). It is independent of the basis: a change of coordinates
and its inverse have entries that are finite linear combinations, hence
are continuous. In dimension zero this gives the one-point topology.
Every linear map between such spaces is continuous by the same argument.
More generally, the canonical topology on the local points of an
affine algebraic set is the subspace topology in any finite coordinate
presentation. Explicitly, let \(X(K)\) consist of the points \(x\in K^m\)
satisfying the finite polynomial equations
\[\begin {gathered}
f_i(x)=0\\
(i=1,\ldots ,r).
\end {gathered}\]
Give \(X(K)\) the topology inherited from \(K^m\). Changing the finite
algebraic presentation expresses each new coordinate as a polynomial in
the old ones and conversely. Both coordinate maps are continuous, so this
topology is independent of the presentation. This is the concrete affine
construction in [conrad2011adelic, Proposition 2.1 and its proof, p. 2].
It also shows directly that polynomial maps on local points are continuous.
Let \(A\) be a finite-dimensional unital \(K\)-algebra, with its canonical
vector-space topology. Its multiplication is continuous because its
coordinates are bilinear polynomials. On the group of units \(A^\times \),
the affine-point construction first gives the inverse-graph topology:
\[\begin {gathered}
A^\times \longrightarrow A\times A,\\
x\longmapsto (x,x^{-1}).
\end {gathered}\]
This identifies the units with the solution set of \(xy=yx=1\).
The inverse coordinate matters in the general topological-ring
construction; see [conrad2011adelic, opening of Section 3, pp. 3--4].
Over the local fields specified above, this is also the subspace
topology from \(A\). Indeed, write \(L_x\) for the matrix of left
multiplication by \(x\). Its entries depend linearly on \(x\), and for a
unit \(x\),
\[\begin {gathered}
x^{-1}=L_x^{-1}(1),\\
L_x^{-1}=\frac {\operatorname {adj}(L_x)}{\det L_x}.
\end {gathered}\]
Polynomial continuity and continuity of reciprocal on \(K^\times \) show
that inversion is continuous on the subspace \(A^\times \subseteq A\).
Thus the inverse-graph map is continuous for that subspace topology;
projection to the first coordinate is its continuous inverse.
For the zero algebra the unit group is a singleton and the same topology
agreement is immediate.
For a quadratic form \(Q\) on a finite-dimensional \(V\), apply this
construction to the Clifford algebra
\(A=\mathcal {C}\kern -2pt\ell (Q)\) and to \(\operatorname {End}_K(V)\).
The orthogonal and special orthogonal groups inherit the matrix topology
from \(\operatorname {End}_K(V)\). The actual vector-generated
Lipschitz, Pin, and
Spin carriers inherit the subspace topology from
\(\mathcal {C}\kern -2pt\ell (Q)^\times \), equivalently from \(\mathcal {C}\kern -2pt\ell (Q)\). These topologies depend
on the field topology and the algebraic carriers, without a choice of
positive-definite norm or a replacement of the actual carriers by
normalizer-defined groups. The star-unitary Pin convention and its even
Spin restriction are used here; Remark [fcap-0028] compares the star and
reverse norms.
The coordinate topology on \(W\) is also its module topology: it is
the finest topology making addition and scalar multiplication continuous.
The coordinate topology has these properties; for any other such topology,
the basis map \(K^d\to W\), a finite sum of scalar multiples of fixed basis
vectors, is continuous. This proves the claimed maximality.
Mathlib's moduleTopology defines that intrinsic topology, and
TauCeti's
CliffordAlgebra.instTopologicalSpaceCliffordAlgebra
uses it for \(\mathcal {C}\kern -2pt\ell (Q)\). The agreement with local coordinates follows from
the finite-dimensional argument above; the declaration itself is defined
over a general topological commutative ring.