Homological dimensions of module - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T05:48:31Z http://mathoverflow.net/feeds/question/33736 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/33736/homological-dimensions-of-module Homological dimensions of module ashpool 2010-07-29T00:07:35Z 2010-07-29T12:50:59Z <p>$(A,\mathfrak{m})$ a Noetherian local ring, $M\neq 0$ a finitely generated $A$-module. As I understand, $\mbox{Ext }^{j}(A/\mathfrak{m}, M) = 0$ for $j&lt;\mbox{depth }(M)$ 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 $\mbox{inj. dim }(M)$ ?</p> http://mathoverflow.net/questions/33736/homological-dimensions-of-module/33740#33740 Answer by Mariano Suárez-Alvarez for Homological dimensions of module Mariano Suárez-Alvarez 2010-07-29T00:18:08Z 2010-07-29T00:18:08Z <p>This is not exactly the same but anyways it is related and very cute...</p> <p>In your context, given a sequence <code>$n_0&lt;n_1&lt;\cdots&lt;n_k$</code> 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.]</p> http://mathoverflow.net/questions/33736/homological-dimensions-of-module/33743#33743 Answer by Graham Leuschke for Homological dimensions of module Graham Leuschke 2010-07-29T01:16:19Z 2010-07-29T12:50:59Z <p>Yes. See Fossum, Foxby, Griffith, and Reiten, <a href="http://archive.numdam.org/article/PMIHES_1975__45__193_0.pdf" rel="nofollow">"Minimal injective resolutions with applications to dualizing modules and Gorenstein modules"</a> (Theorem 1.1) and also Roberts, "<a href="http://archive.numdam.org/article/ASENS_1976_4_9_1_103_0.pdf" rel="nofollow">Two applications of dualizing complexes over local rings</a>". An earlier paper by Foxby, "<a href="http://www.mscand.dk/article.php?id=2032" rel="nofollow">On the mu_i in a minimal injective resolution</a>" settles several special cases, including when $A$ or $M$ is CM, $\mathrm{depth} M \geq \mathrm{depth} A$, or $M$ has finite injective dimension.</p>