exact diagram of infinite general linear groups and elementary linear groups
Proposition
Let 
be a ring, 
an ideal
Then there exists a commutative diagram
for the
- infinite general linear group functor
- infinite elementary linear group functor
- natural transformation from infinite elemenatry linear group to general linear group
- explicite description of a pullback of a ring and ideal
with
- exact rows
- and surjective vertical morphisms
Proof
commutes
follows as we have a diagram
where the morphisms are natural with respect to Ring homomorphism.
Then by colim functor preserves naturality we conclude that
is natural with respect to ring homomorphism.
Thus by applying the cokernel functor
we get
a) exactness
b) commutativity
surjective morphism
a) 
follows from lifting generators 
to 
b) 
follows from lifting 
to 
c) 
follows from