Non-stable $K$-theory for $QB$-rings | Zendy

Pere Ara | Zendy; Francesc Perera | Zendy

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Non-stable $K$-theory for $QB$-rings

Author(s) -

Pere Ara,

Francesc Perera

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.

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