Locally dualizable modules abound
Jon F. Carlson, Srikanth B. Iyengar
TL;DR
The paper investigates dualizable objects in stratified tensor triangulated categories arising from algebra and modular representation theory. It shows that, for a countable Noetherian ring $A$ and a nonminimal prime $\\mathfrak p$ with $\\mathrm{height}(\\mathfrak p)\\ge 2$, the $\\mathfrak p$-local $\\mathfrak p$-torsion subcategory $\\varGamma_{\\mathfrak p}D(\\operatorname{Mod}A)$ contains uncountably many indecomposable dualizable objects not retracts of images of perfect complexes, highlighting the failure of density of $\\varGamma_{\\mathfrak p}\\operatorname{Perf}A$. An analogous phenomenon is established for the stable module category $\\operatorname{StMod}(kG)$ of a finite group $G$ over a countable field $k$ of characteristic $p$ when $G$ contains a rank $\\ge 3$ elementary abelian subgroup, with dualizable objects in $\\varGamma_{\\mathfrak p}\\operatorname{StMod}(kG)$ not arising from $\\operatorname{stmod}(kG)$. The method relies on local cohomology functors, Greenlees\\-May duality, Koszul complexes, and support-variety techniques to produce uncountably many indecomposables, while a Krull\\-Schmidt count in the ambient categories bounds retracts from perfect objects. The results reveal substantial richness in the local strata and inform the structure of stratified tensor-triangular categories in both commutative and modular settings.
Abstract
It is proved that given any prime ideal $\mathfrak{p}$ of height at least 2 in a countable commutative noetherian ring $A$, there are uncountably many more dualizable objects in the $\mathfrak{p}$-local $\mathfrak{p}$-torsion stratum of the derived category of $A$ than those that are obtained as retracts of images of perfect $A$-complexes. An analogous result is established dealing with the stable module category of the group algebra, over a countable field of positive characteristic $p$, of an elementary abelian $p$-group of rank at least 3.