Let $I \subseteq S=\mathbb{k}\left[x_{1}, \ldots, x_{d}\right]$ be an ideal, $<$ be a monomial order on $S$ and let $T=S[t]=\mathbb{k}\left[x_{1}, \ldots, x_{d}, t\right]$. There exists $\omega \in \mathbb{Z}_{>0}^{d}$ such that $in_{\omega}(I)=in_{<}(I)$. We take $J=$ hom $_{\omega}(I) \subseteq T$ and $R=T / J$.

In the book of David Eisenbud (Commutative algebra: with a view toward algebraic geometry) Theorem 15.17, shows that $R \otimes_{\mathrm{k}[t]} \mathbb{k}(t) \cong S / I \otimes_{\mathbb{k}} \mathbb{k}(t)$.

Questions

1. Is it an isomorphism of $\mathbb{K}$-algebras?
2. Is it an isomorphism of $\mathbb{K}[t]$-modules?
3. Do you recommend another book to see the proof?