I could really use some help understanding a statement in the last part of the proof of the Jacobian criterion in "Commutative Algebra with a view toward Algebraic Geometry" by D. Eisenbud, namely:
For $R = k[x_1,...,x_n]/I$ we have the usual conormal sequence: $I/I^2 \rightarrow R \otimes \Omega_{k[x_1,...,x_n]/k} \rightarrow \Omega_{R/k} \rightarrow 0$
From here we get that the cokernel of the Jacobian matrix $J$ is equal to $\Omega_{R/k}$. This is clear. But then from this he concludes "thus $K(p) \otimes \Omega_{R_p/k}$ is the cokernel of the matrix $J$ taken modulo $p$", where $K(p)$ is the residue class field at $p$. It might be obvious but I don't see it and it's crucial in understanding the proof. Any help?