I'm trying to build a little dictionary between old Homological algebra for local rings and the slightly more modern approach via derived functors.
Let $X = SpecA$ be a spectrum of a local ring $(A,m,k)$ and $j_x:Speck=x \to X$ the inclusion of the closed point. Let $\mathcal{M}$ be a coherent sheaf on $X$ (corresponding to a module $M$). Are the following statements correct?
The complex $j_x^!\mathcal{M}$ in $D^+(k)$ has nonzero cohomology precisely in the interval $[depth(M),id(M)]$ (where $id$ stands for injective dimension).
The complex $j^*_x\mathcal{M}$ in $D^-(k)$ has nonzero cohomology precisely in the interval $[-pd(M),-width(M)]$ (projective dimension and width).
Is this correct so far?
