UCLA

We study the derived category of permutation filtered modules. We focus on the case Db(pfmod(E)) of an elementary abelian group E. Our key insight is to consider for each subgroup N ⩽ E a relative subcategory pfmod(E, N) ⊂ pfmod(E) of permutation filtered modules and a relative category pgmod(E, N) of permutation graded modules. The case N = 1 recovers the original categories. We show that the spectrum Spc Db(pfmod(E, N)) as a set consists of a copy of the spectrum Spc Db(pgmod(E, N)) and a colimit colimL>N UL of open subsets UL ⊂ Spc Db(pfmod(E, L)) coming from smaller relative subcategories. This allows an inductive analysis of Spc Db(pfmod(E)).