From topological rigidity to topological semisimplicity

Laurent Poinsot (1)

1 LIPN - Laboratoire d'Informatique de Paris-Nord

A topological (commutative) ring is said to be rigid when for every set X, the topological dual of the X-fold topological product of the ring is isomorphic to the free module over X. Examples are fields with a ring topology, discrete rings, and normed algebras. Rigidity translates into a dual equivalence between categories of free modules and of " topologically-free " modules and, with a suitable topological tensor product for the latter, it lifts to an equivalence between monoids in this category (some suitably generalized topological algebras) and coalgebras. Several notions of topological semisimplicity for these (commutative) monoids are introduced, and over an algebraically closed field k, with a ring topology, all these notions, in addition to cosemisimplicity for coalgebras and classical Jacobson semisimplicity, are proven to coincide. Accordingly this leads to the new rigidity result that the function algebras of k-valued maps defined on a set, with the topology of simple convergence, are the only topologically semisimple monoids.

Mots clés: Topological dual space; topological basis; topological Jacobson radical; topological semisimplicity; coalgebras; finite duality

Domaines: Algèbre commutative [math.AC]; Topologie générale [math.GN]; Catégories et ensembles [math.CT]

Laurent Poinsot. From topological rigidity to topological semisimplicity. 2018. ⟨hal-01833662v1⟩