  In this article we extend the Bloch-Wigner exact sequence over local rings,
where their residue fields have more than nine elements. Moreover, we prove Van
der Kallen's theorem on the presentation of the second $K$-group of local rings
such that their residue fields have more than four elements. Note that Van der
Kallen proved this result when the residue fields have more than five elements.
Although we prove our results over local rings, all our proofs also work over
semilocal rings where all their residue fields have similar properties as the
residue field of local rings.
