Open AccessNon-stable $K$-theory for $QB$-rings
Author(s) -
Pere Ara,
Francesc Perera Domènech
Publication year - 2007
Publication title -
mathematica scandinavica
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.553
H-Index - 30
eISSN - 1903-1807
pISSN - 0025-5521
DOI - 10.7146/math.scand.a-15024
Subject(s) - mathematics , invertible matrix , lift (data mining) , surjective function , quotient , pure mathematics , class (philosophy) , prime (order theory) , ring (chemistry) , combinatorics , discrete mathematics , chemistry , organic chemistry , artificial intelligence , computer science , data mining
We study the class of $QB$-rings that satisfy the weak cancellation condition of separativity for finitely generated projective modules. This property turns out to be crucial for proving that all (quasi-)invertible matrices over a $QB$-ring can be diagonalised using row and column operations. The main two consequences of this fact are: (i) The natural map $(\mathrm{GL}_1(R)\to K_1(R)$ is surjective, and (ii) the only obstruction to lift invertible elements from a quotient is of $K$-theoretical nature. We also show that for a reasonably large class of $QB$-rings that includes the prime ones, separativity always holds.