# K-theory and Homological Algebra: A Seminar held at the by Tamar Datuashvili (auth.), Hvedri Inassaridze (eds.)

By Tamar Datuashvili (auth.), Hvedri Inassaridze (eds.)

**Contents:****T. Datuashvili**: Homological measurement of extensions of abelian different types and rings.- **J. Gubeladze**: Classical algebraic *K*-theory of monoid algebras.- **H. ****Inassaridze**: *K*-theory of specific normed rings.- **G. ****Janelidze**: Cohomology and extensions of inner modules.- **M. Jibladze**: Coefficients for cohomology of "large" categories.- **T. Kandelaki**: K-theory of Z2-graded Banach categories.- **D. Pataraia**: On Quillen's plus development of excellent groups.- **T. Pirashvili**: Cohomology of small different types in homotopical algebra.- **M. Uridia**: *U*-theory of tangible different types.

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**Additional resources for K-theory and Homological Algebra: A Seminar held at the Razmadze Mathematical Institute in Tbilisi, Georgia, USSR 1987–88**

**Example text**

Thus we have %he exe~% se- 54 •--,AK=~. y. Groups (for the suita- ble class of rings) is also trae. 2. ,~--R'such that either ~ . Of course any field belongs to ~-~. ~ ~ occur in alternating order. It turns out that any local PID belongs to n z for all ~ 3 to show that . Indeed (since ~ (oc) is also PID) we have only SK~(~Coc])=o local regular ring have ~i(A1r)-~O /k ( 2 being a local PID). B~t for any and its arbitrary regular parameter ~" we (this follows from the localization sequence). 1. Let ~ be any ([16]).

It turns out that there exists a normal monoid ~ of rank 2 ( n ~ g ~ C ~ , where n ~ i in the group of the quotients) for which ~ i ( ~ ~ O and ~ is ~[ ~3~) Such counterexamples exist for other regular rings of coeffitients (including some fields) of the normal monoid as well. L The simplest explicit example for which ~K~L given in [I7~ ; it is the submonoid in elements (2,0) , (I,I) , (0,2) ( ~ ~S~#=O ~ L ~ is generated by the me~ns the complex numbers). On the other hand if we consider the sub-monoids in ~-spaces ( ~ is the field of the rational numbers), which are "densely distributed", then we can establish the exact results concerning the functor ~i-analog of the aforementioned ~0 .

We can so that the points / ... ~ _ ~i~)will be sufficiently close to at the same time remaining the internal points of d ation o u r ~-correspondence A~ • In this situ- a~ain can be assumed t o be "flat" (be- cause we don't e x c e e d %he boundary of A ). l~rthermore, by suitab- le choice of the natural numbers mentioned above we can assume % ~ % / is "almost" parallel %0 A~ (with the sufficient exactness). Jr/#. ~ and • Now repeat our procedure of the "blowing up" of free monoids relatively %o the f r e e ments ~-submonoid of j~ &/i~ &~ ~ " .