$(A,\mathfrak{m})$ a Noetherian local ring, $M\neq 0$ a finitely generated $A$-module. As I understand, $\mbox{Ext }^{j}(A/\mathfrak{m}, M) = 0$ for $j<\mbox{depth }(M)$ and for $j>\mbox{inj. dim }(M)$, while $\mbox{Ext }^{j}(A/\mathfrak{m}, M) \neq 0$ for $j=\mbox{depth }(M)$ and $j=\mbox{inj. dim }(M)$. And I cannot help but wonder if $\mbox{Ext }^{j}(A/\mathfrak{m}, M) \neq 0$ for every $j$ between $\mbox{depth }(M)$ and $\mbox{inj. dim }(M)$ ?



Homological dimensions of module
Yes. See Fossum, Foxby, Griffith, and Reiten, ["Minimal injective resolutions with applications to dualizing modules and Gorenstein modules"](http://archive.numdam.org/article/PMIHES_1975__45__193_0.pdf) (Theorem 1.1) and also Roberts, "[Two applications of dualizing complexes over local rings](http://archive.numdam.org/article/ASENS_1976_4_9_1_103_0.pdf)". An earlier paper by Foxby, "[On the mu\_i in a minimal injective resolution](http://www.mscand.dk/article.php?id=2032)" settles several special cases, including when $A$ or $M$ is CM, $\mathrm{depth} M \geq \mathrm{depth} A$, or $M$ has finite injective dimension.


