# Categorical Bockstein Sequences

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## Abstract

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 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 Positselski (Mosc Math J 11(2):317–402, 2011. arXiv:1006.4343 [math.KT], Section 4).

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## Notes

### Acknowledgements

The author’s conception of the question of constructing the quotient category of an exact category with a twist functor by a natural transformation goes back to my years as a graduate student at Harvard University in the second half of ’90s. My thinking was influenced by conversations with V. Voevodsky and A. Beilinson at the time. The details below were worked out in Moscow in February–March 2010 (as presented in [13, Section 4]) and subsequently in September 2013 (in full generality). The paper was written while I was vacationing in Prague in March–April 2014, visiting Ben Gurion University of the Negev in Be’er Sheva in June–September 2014, visiting the Technion in Haifa in October 2014–March 2015, and working as a postdoc at the University of Haifa in 2016–2018. I am grateful to the anonymous referee for careful reading of the manuscript and many insightful suggestions, which helped to improve the exposition. In particular, the arguments in the paragraphs preceding Lemma 2.4 are largely due to the referee. The author was supported in part by RFBR grants in Moscow, a fellowship from the Lady Davis Foundation at the Technion, and the ISF Grant # 446/15 at the University of Haifa.

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