Zainalov B. R.
First nontrivial group of homologies of simplicial schemes of unimodular frames over the Dedekind ring
Ukr. Mat. Zh. - 2018. - 70, № 9. - pp. 1231-1248
We prove a theorem on generation of the first nontrivial group of homologies of a simplicial scheme of unimodular frames over the Dedekind ring by standard cycles.
Ukr. Mat. Zh. - 2016. - 68, № 2. - pp. 202-216
We consider a simplicial scheme, its geometric realization, and the required properties of the simplicial scheme of unimodal frames and prove both a sufficiently strong theorem on acyclicity for the simplicial scheme of unimodal frames and a theorem stating that the first nontrivial group of homologies is generated by standard cycles over the rings with infinite fields of residues.
Ukr. Mat. Zh. - 2012. - 64, № 11. - pp. 1464-1476
We study the problem of stabilization in the higher $K$ -theory of rings and improved stabilization. The triviality of the group of standard cycles is established in the case of arithmetic-type rings. Some applications of the obtained results to the problem of homological stabilization are presented.