ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

2. Algebra

1991 MS Classification: 12, 13, 18 Börger, Reinhard and Kemper, Ralf, FernUniversität Hagen, Germany Infinitary Linear Combinations in Cotorsion Modules Let R be an integral domain such that the R-topology (with the collection of all non-zero ideals as a basis of zero-neighbourhoods) is complete and first-countable. If M is a cotorsion R-module in the sense of Matlis (i. e. Hom(K, M) = 0 and Ext^{1}(K, M) = 0 for K := Quot(R)), we can define infinitary linear combinations ∑_{n∈IN} α_{n}x_{n} in M for all α_{n} ∈ R, x_{n} ∈ M with α_{n} → 0 in the R-topology extending the module structure such that ∑_{n∈IN} α_{n}(∑_{m∈IN} β_{n,m}x_{m}) = ∑_{m∈IN}(∑_{n∈IN} α_{n}β_{nm})x_{m} holds whenever the left-hand-side is defined. Moreover, this extension is unique, and each homomorphism between cotorsion modules preserves these infinitary linear combinations. Conversely, each module admitting infinitary linear combinations as above is cotorsion.