Предварительная защита диссертации Косенко Петра Романовича
Homological properties of smooth crossed products
My research is related to the following natural question: how to do homological algebra for topological modules and topological algebras? The most intuitive way to do this would be as follows: for topological vector (Fréchet, Banach) modules A, B, C we consider the sequences
0 --> A --> B --> C --> 0
which are exact in the usual sense and with all arrows being continuous, "exact" in the respective topological category. But it becomes quickly evident that the respective exact structure carries too much topological information. For example, in this case we cannot even guarantee that modules over the base field are projective!
Helemskii (1970) and Taylor (1972) were among the first to realize that we need an additional condition: all morphisms in such sequences must split in the respective topological category. Then we get a theory which roughly resembles the purely algebraic one, where we can define projectivity, flatness, injectivity, derived functors, and so on.
My thesis consists of three parts: in the first part we treat Arens-Michael envelopes of Laurent tensor algebras, generalizing A. Pirkovskii's approach to (ordinary) tensor algebras. We obtain explicit descriptions of the Arens-Michael envelopes of some concrete finitely generated algebras using our method, proving admissibility of certain relations in Banach algebras. In the second part we derive estimates for homological dimensions for topological Ore extensions, smooth crossed products with Z, via adapting the Cuntz-Quillen construction of relative 1-differentials to the topological setting. In the third part we adapt the above approach to the non-unital case, by constructing certain exact sequences featuring smooth crossed products with T and R, which allow us to derive estimates for projective dimensions for smooth crossed products with T and R. If time permits, we will discuss the possible further directions.
Страница семинара: http://me.hse.ru/pirkovskii/fa_ncg/