## 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.

## References

- 1.Buchweitz, R.-O.: Maximal Cohen–Macaulay modules and Tate-cohomology over Gorenstein rings (Manuscript, 1986) p. 155. http://hdl.handle.net/1807/16682
- 2.Bühler, T.: Exact categories. Expos. Math.
**28**(1), 1–69 (2010). arXiv:0811.1480 [math.HO]MathSciNetCrossRefGoogle Scholar - 3.Efimov, A.I., Positselski, L.: Coherent analogues of matrix factorizations and relative singularity categories. Algebra Number Theory
**9**(5), 1159–1292 (2015). arXiv:1102.0261 [math.CT]MathSciNetCrossRefGoogle Scholar - 4.Eisenbud, D.: Homological algebra on a complete intersection, with an application to group representations. Trans. Am. Math. Soc.
**260**(1), 35–64 (1980)MathSciNetCrossRefGoogle Scholar - 5.Harrison, D.K.: Infinite Abelian groups and homological methods. Ann. Math.
**69**(2), 366–391 (1959)MathSciNetCrossRefGoogle Scholar - 6.Keller, B.: Exact categories. Appendix A to the paper: chain complexes and stable categories. Manuscr. Math.
**67**(4), 379–417 (1990)CrossRefGoogle Scholar - 7.Keller, B.: On Gabriel–Roiter’s axioms for exact categories. Appendix to the paper: P. Dräxler, I. Reiten, S. O. Smalø Ø. Solberg. Exact categories and vector space categories. Trans. Am. Math. Soc.
**351**(2), 647–692 (1999)CrossRefGoogle Scholar - 8.Neeman, A.: The derived category of an exact category. J. Algebra
**135**(2), 388–394 (1990)MathSciNetCrossRefGoogle Scholar - 9.Orlov, D.: Triangulated categories of singularities and D-branes in Landau–Ginzburg models. Proc. Steklov Math. Inst.
**246**(3), 227–248 (2004). arXiv:math.AG/0302304 MathSciNetzbMATHGoogle Scholar - 10.Orlov, D.: Matrix factorizations for nonaffine LG-models. Mathematische Annalen
**353**(1), 95–108 (2012). arXiv:1101.4051 [math.AG]MathSciNetCrossRefGoogle Scholar - 11.Polishchuk, A., Vaintrob, A.: Matrix factorizations and singularity categories for stacks. Annales de l’Institut Fourier (Grenoble)
**61**(7), 2609–2642 (2011). arXiv:1011.4544 [math.AG]MathSciNetCrossRefGoogle Scholar - 12.Positselski, L., Vishik, A.: Koszul duality and Galois cohomology. Math. Res. Lett.
**2**(6), 771–781 (1995). arXiv:alg-geom/9507010 MathSciNetCrossRefGoogle Scholar - 13.Positselski, L.: Mixed Artin–Tate motives with finite coefficients. Mosc. Math. J.
**11**(2), 317–402 (2011). arXiv:1006.4343 [math.KT]MathSciNetzbMATHGoogle Scholar - 14.Positselski, L.: Artin–Tate motivic sheaves with finite coefficients over an algebraic variety. Proc. Lond. Math. Soc.
**111**(6), 1402–1430 (2015). arXiv:1012.3735 [math.KT]MathSciNetCrossRefGoogle Scholar - 15.Positselski, L.: Weakly curved \(\text{A}_{{\infty }}\)-algebras over a topological local ring. Electronic (preprint). arXiv:1202.2697 [math.CT]
- 16.Positselski, L.: Contramodules. Electronic (preprint). arXiv:1503.00991 [math.CT]
- 17.Positselski, L.: Contraadjusted modules, contramodules, and reduced cotorsion modules. Mosc. Math. J.
**17**(3), 385–455 (2017). arXiv:1605.03934 [math.CT]MathSciNetGoogle Scholar - 18.Voevodsky, V.: On motivic cohomology with \({\bf Z}/l\)-coefficients. Ann. Math.
**174**(1), 401–438 (2011). arXiv:0805.4430 [math.AG]MathSciNetCrossRefGoogle Scholar