Categorical Bockstein sequences
  We construct the reduction of an exact category with a twist functor with respect to an element of its graded center in presence of an exact-conservative forgetful functor annihilating this central element. The procedure allows, e.g., to recover the abelian/exact category of modular representations of a finite group from the exact category of its l-adic representations. The construction uses matrix factorizations in a nontraditional way. We obtain the Bockstein long exact sequences for the Ext groups in the exact categories produced by reduction. Our motivation comes from the theory of Artin-Tate motives and motivic sheaves with finite coefficients, and our key techniques generalize those of Section 4 in arXiv:1006.4343 
Leonid Positselski
math.KT math.CT math.RT