Hy. Im looking for a proof that SL_n(IZ)->SL_n(IZ/mIZ) is onto. Can you help me?
Have I interpreted the question correctly? If so, then I think that the result is false.
For , denote by its image in the quotient ring I assume that denotes the set of all matrices with entries in with determinant and that the map from to is the natural quotient map given by
Now take m = 7 and let Then but any lifting of A to a matrix in will have determinant of the form 8+28k (mod 49), which can never be equal to 1.
Edit. Stupid mistake: 8 + 28k can be equal to 1 (mod 49). In fact, 8 + 12×28 = 7^3 + 1; and in fact is a lifting of that matrix A to an element of So it looks to me as though the result may be true after all. But it's not at all obvious!
Yes you have interpretet the question correctly! More precisely i think i need this result to prove (pro-finite completion) in the case (this is actually what i want to prove). With the solution of the congruence subgroup problem one can get (projective limes) where denotes the kernel of . Because it shouldn't be difficult to prove it remains to prove . Thus i need to show that this natural projection is onto. Maybe one can solve this using matrices of the type where has 1 at the place and 0 else. But have no idea how to show...