On seminormality

AbstractWe give an elementary and essentially self-contained proof that a reduced ring R is seminormal if and only if the canonical map PicR→PicR[X] is an isomorphism, a theorem due to Swan [R. Swan, On seminormality, J. Algebra 67 (1980) 210–229], generalizing some previous results of Traverso [C. Traverso, Seminormality and the Picard group, Ann. Scuola Norm. Sup. Pisa 24 (1970) 585–595]

Constructive Theory of Banach algebras

We present a way to organize a constructive development of the theory of Banach algebras, inspired by works of Cohen, de Bruijn and Bishop. We illustrate this by giving elementary proofs of Wiener's result on the inverse of Fourier series and Wiener's Tauberian Theorem, in a sequel to this paper we show how this can be used in a localic, or point-free, description of the spectrum of a Banach algebra

Seminormal rings (following Thierry Coquand)

International audienceThe Traverso–Swan theorem says that a reduced ring A is seminormal if and only if the natural homomorphism Pic A to Pic A[X] is an isomorphism [C. Traverso, Seminormality and the Picard group, Ann. Sc. Norm. Sup. Pisa 24 (1970) 585-595; R.G. Swan, On seminormality, J. Algebra 67 (1980) 210–229]. We give here all the details needed to understand the elementary constructive proof for this result given by Coquand in [T. Coquand, On seminormality, J. Algebra 305 (2006) 577-584]. This example is typical of a new constructive method. The final proof is simpler than the initial classical one. More important: the classical argument by absurdum using “an abstract ideal object” is deciphered with a general technique based on the following idea: purely ideal objects constructed using TEM and Choice may be replaced by concrete objects that are “finite approximations” of these ideal objects