an:06684815
Zbl 1359.13023
Gorchinskiy, Sergey O.; Osipov, Denis V.
Continuous homomorphisms between algebras of iterated Laurent series over a ring
EN
Proc. Steklov Inst. Math. 294, 47-66 (2016); translation from Tr. Mat. Inst. Steklova 294, 54-75 (2016).
00361787
2016
j
13F25
Laurent series; continuous homomorphism
Let \(A\) be a commutative ring, \(A((t))\) be the ring of Laurent series over \(A\) and \({\mathcal L}^n(A)=A((t_1))\ldots((t_n))\) the \(A-\)algebra of iterated Laurent series over \(A\) with the natural topology. The elements of \({\mathcal L}^n(A)\) have the form \(\displaystyle\sum_{l\in\mathbb{Z}^n}a_lt_1^{l_1}\ldots t_n^{l_n}\) where \(l=(l_1,\ldots,l_n)\in\mathbb{Z}^n\) and \(a_l\in A\), with certain restrictions on the set of indices of nonzero coefficients. The goal of this paper is to study the continuous homomorphisms between these types of algebras. First, a description of such homomorphisms is given. Indeed, let \(\phi_1,\ldots,\phi_n\in {\mathcal L}^n(A)^*\) be a collection of \(n\) invertible iterated Laurent series in \(m\) variables, with some restrictive conditions. Then we have a well defined continuous homomorphism of \(A-\)algebras \(\phi:{\mathcal L}^n(A)\longrightarrow {\mathcal L}^m(A)\), which assigned to each \(\displaystyle f=\sum_{l\in\mathbb{Z}^n}a_lt_1^{l_1}\ldots t_n^{l_n}\), \(\displaystyle \phi(f)=\sum_{l\in\mathbb{Z}^n}a_l\phi_1^{l_1}\ldots \phi_n^{l_n}\). Moreover, all the continuous homomorphisms of \(A-\)algebras \(\phi:{\mathcal L}^n(A)\longrightarrow {\mathcal L}^m(A)\) have this form. Then, a criterion of invertibility for endomorphism is given and an explicit formula for the inverse is provided. Other applications are also stated.
Ali Benhissi (Monastir)