Let $M$ be a finitely generated module over a polynomial ring $R$ over a field $k$. Let $F_{\bullet}$ be a minimal free resolution of $M$ : $0\longrightarrow F_p \longrightarrow ....\longrightarrow F_1 \longrightarrow F_0\longrightarrow M$
In one paper of M.Chardin, he claimed that the maps of $F_{\bullet}\otimes_{R}k$ being zero maps, $\text{Tor}_{i}(M,k)=H_{i}(F_{\bullet}\otimes k)=F_{i}\otimes k$. This claim also appears in the book "The Geometry of Syzygy" of D.Eisenbud, in the proof of proposition 1.7 on page 7.
My question is :
- Why the maps of $F_{\bullet}\otimes_{R}k$ being zero maps if $F_{\bullet}$ is a minimal free resolution
- Why $\text{Tor}_{i}(M,k)=F_{i}\otimes k$