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Continuous homomorphisms between algebras of iterated Laurent series over a ring. (English. Russian original) Zbl 1359.13023
Proc. Steklov Inst. Math. 294, 47-66 (2016); translation from Tr. Mat. Inst. Steklova 294, 54-75 (2016).
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.

##### MSC:
 13F25 Formal power series rings
##### Keywords:
Laurent series; continuous homomorphism
Full Text:
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