Homological dimensions of module - MathOverflow

Homological dimensions of module

2010-07-29T00:07:35Z

$(A,\mathfrak{m})$ 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 $\mbox{inj. dim }(M)$ ?

Answer by Mariano Suárez-Alvarez for Homological dimensions of module

2010-07-29T00:18:08Z

This is not exactly the same but anyways it is related and very cute...

In your context, given a sequence $n_0<n_1<\cdots<n_k$ of non-negative integers, there exists a local ring and a module such that $\mathrm{depth}R=n_0$, $\dim R=n_k$ and the local cohomology groups $H^\bullet_{\mathfrak m}(R)$ is non-zero precisely in the degrees $n_i$; see [E. G. Evans Jr. and P. A. Griffith, Local cohomology modules for normal domains, J. London Math. Soc. (2) 19 (1979), 277–284.]

Answer by Graham Leuschke for Homological dimensions of module

2010-07-29T01:16:19Z

Yes. See Fossum, Foxby, Griffith, and Reiten, "Minimal injective resolutions with applications to dualizing modules and Gorenstein modules" (Theorem 1.1) and also Roberts, "Two applications of dualizing complexes over local rings". An earlier paper by Foxby, "On the mu_i in a minimal injective resolution" settles several special cases, including when $A$ or $M$ is CM, $\mathrm{depth} M \geq \mathrm{depth} A$, or $M$ has finite injective dimension.