Let $x$ denote the sequence $(x_1,\ldots,x_m)$ and $x^{'}$ denote $(x_1,\ldots,x_m,y)$. Then we have the following exact sequence of Koszul homologies: (set I=$(x_1,\ldots,x_m)$)
$0\to H_1(x)/yH_1(x)\to H_1(x^{'})\to R/I\to R/I\to 0.$
Suppose the ring is local and $y\in (I:m)\setminus I$. Then I have come across the following exact sequence while reading a note.
$0\to H_1(x)/yH_1(x)\to H_1(x^{'})\to R/(I:y)\to 0.$
Can someone explain how do we get the last term? Thanks.