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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.

References

  1. 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. 2.
    Bühler, T.: Exact categories. Expos. Math. 28(1), 1–69 (2010). arXiv:0811.1480 [math.HO]MathSciNetCrossRefMATHGoogle Scholar
  3. 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]MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Eisenbud, D.: Homological algebra on a complete intersection, with an application to group representations. Trans. Am. Math. Soc. 260(1), 35–64 (1980)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Harrison, D.K.: Infinite Abelian groups and homological methods. Ann. Math. 69(2), 366–391 (1959)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Keller, B.: Exact categories. Appendix A to the paper: chain complexes and stable categories. Manuscr. Math. 67(4), 379–417 (1990)CrossRefMATHGoogle Scholar
  7. 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. 8.
    Neeman, A.: The derived category of an exact category. J. Algebra 135(2), 388–394 (1990)MathSciNetCrossRefMATHGoogle Scholar
  9. 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 MathSciNetMATHGoogle Scholar
  10. 10.
    Orlov, D.: Matrix factorizations for nonaffine LG-models. Mathematische Annalen 353(1), 95–108 (2012). arXiv:1101.4051 [math.AG]MathSciNetCrossRefMATHGoogle Scholar
  11. 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]MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Positselski, L., Vishik, A.: Koszul duality and Galois cohomology. Math. Res. Lett. 2(6), 771–781 (1995). arXiv:alg-geom/9507010 MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Positselski, L.: Mixed Artin–Tate motives with finite coefficients. Mosc. Math. J. 11(2), 317–402 (2011). arXiv:1006.4343 [math.KT]MathSciNetMATHGoogle Scholar
  14. 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]MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Positselski, L.: Weakly curved \(\text{A}_{{\infty }}\)-algebras over a topological local ring. Electronic (preprint). arXiv:1202.2697 [math.CT]
  16. 16.
    Positselski, L.: Contramodules. Electronic (preprint). arXiv:1503.00991 [math.CT]
  17. 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. 18.
    Voevodsky, V.: On motivic cohomology with \({\bf Z}/l\)-coefficients. Ann. Math. 174(1), 401–438 (2011). arXiv:0805.4430 [math.AG]MathSciNetCrossRefMATHGoogle Scholar

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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Natural SciencesUniversity of HaifaHaifaIsrael
  2. 2.Laboratory of Algebraic GeometryNational Research University Higher School of EconomicsMoscowRussia
  3. 3.Sector of Algebra and Number TheoryInstitute for Information Transmission ProblemsMoscowRussia
  4. 4.Mathematical InstituteCzech Academy of SciencesPrague 1Czech Republic

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