Constructing modules with prescribed cohomological support

Abstract

A cohomological support, $\operatorname{Supp}^*_{\mathcal A}(M)$, is defined for finitely generated modules $M$ over a left noetherian ring $R$, with respect to a ring $\mathcal A$ of central cohomology operations on the derived category of $R$-modules. It is proved that if the $\mathcal A$-module $\operatorname{Ext}^*_R(M,M)$ is noetherian and $\operatorname{Ext}^*_R(M,R)=0$ for $i\gg0$, then every closed subset of $\operatorname{Supp}^*_{\mathcal A}(M)$ is the support of some finitely generated $R$-module. This theorem specializes to known realizability results for varieties of modules over group algebras, over local complete intersections, and over finite dimensional algebras over a field. The theorem is also used to produce large families of finitely generated modules of finite projective dimension over commutative local noetherian rings.