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Support theories for non-Noetherian tensor triangulated categories (2312.08596v1)

Published 14 Dec 2023 in math.AT, math.AG, math.CT, and math.RT

Abstract: We extend the support theory of Benson--Iyengar--Krause to the non-Noetherian setting by introducing a new notion of small support for modules. This enables us to prove that the stable module category of a finite group is canonically stratified by the action of the Tate cohomology ring, despite the fact that this ring is rarely Noetherian. In the tensor triangular context, we compare the support theory proposed by W. Sanders (which extends the Balmer--Favi support theory beyond the weakly Noetherian setting) with our generalized BIK support theory. When the Balmer spectrum is homeomorphic to the Zariski spectrum of the endomorphism ring of the unit, the two support theories coincide as do their associated theories of stratification. We also prove a negative result which states that the Balmer--Favi--Sanders support theory can only stratify categories whose spectra are weakly Noetherian. This provides additional justification for the weakly Noetherian hypothesis in the work of Barthel, Heard and B. Sanders. On the other hand, the detection property and the local-to-global principle remain interesting in the general setting.

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References (49)
  1. Paul Balmer. The spectrum of prime ideals in tensor triangulated categories. J. Reine Angew. Math., 588:149–168, 2005.
  2. Paul Balmer. Supports and filtrations in algebraic geometry and modular representation theory. Amer. J. Math., 129(5):1227–1250, 2007.
  3. Paul Balmer. Spectra, spectra, spectra – tensor triangular spectra versus Zariski spectra of endomorphism rings. Algebr. Geom. Topol., 10(3):1521–1563, 2010.
  4. Paul Balmer. Homological support of big objects in tensor-triangulated categories. J. Éc. polytech. Math., 7:1069–1088, 2020.
  5. Paul Balmer. Nilpotence theorems via homological residue fields. Tunis. J. Math., 2(2):359–378, 2020.
  6. Computing homological residue fields in algebra and topology. Proc. Amer. Math. Soc., 149(8):3177–3185, 2021.
  7. Quillen stratification in equivariant homotopy theory. Preprint, 56 pages, available online at arXiv:2301.02212, 2023.
  8. Cosupport in tensor triangular geometry. Preprint, 87 pages, available online at arXiv:2303.13480, 2023.
  9. Thick subcategories of the stable module category. Fund. Math., 153(1):59–80, 1997.
  10. Grothendieck–Neeman duality and the Wirthmüller isomorphism. Compos. Math., 152(8):1740–1776, 2016.
  11. David J. Benson. Representations and cohomology I & II, volume 30 & 31 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, 1998.
  12. Generalized tensor idempotents and the telescope conjecture. Proc. Lond. Math. Soc. (3), 102(6):1161–1185, 2011.
  13. Cohen-Macaulay Rings. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2 edition, 1998.
  14. Stratification and the comparison between homological and tensor triangular support. Q. J. Math., 74(2):747–766, 2023.
  15. Stratification in tensor triangular geometry with applications to spectral Mackey functors. Camb. J. Math., 11(4):829–915, 2023.
  16. Local cohomology and support for triangulated categories. Ann. Sci. Éc. Norm. Supér. (4), 41(4):573–619, 2008.
  17. Stratifying modular representations of finite groups. Ann. of Math. (2), 174(3):1643–1684, 2011.
  18. Stratifying triangulated categories. J. Topol., 4(3):641–666, 2011.
  19. Colocalizing subcategories and cosupport. J. Reine Angew. Math., 673:161–207, 2012.
  20. Stratification for module categories of finite group schemes. J. Amer. Math. Soc., 31(1):265–302, 2018.
  21. Pure injectives and the spectrum of the cohomology ring of a finite group. J. Reine Angew. Math., 542:23–51, 2002.
  22. Tensor-triangular fields: ruminations. Selecta Math. (N.S.), 25(1):Paper No. 13, 36, 2019.
  23. Nicolas Bourbaki. Commutative algebra. Chapters 1–7. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 1998. Reprint of the 1989 English translation.
  24. The spectrum of the equivariant stable homotopy category of a finite group. Invent. Math., 208(1):283–326, 2017.
  25. The Bousfield lattice for truncated polynomial algebras. Homology Homotopy Appl., 10(1):413–436, 2008.
  26. On the derived category of a graded commutative Noetherian ring. J. Algebra, 373:356–376, 2013.
  27. Spectral spaces, volume 35 of New Mathematical Monographs. Cambridge University Press, Cambridge, 2019.
  28. Hans-Bjørn Foxby. Bounded complexes of flat modules. J. Pure Appl. Algebra, 15(2):149–172, 1979.
  29. J. P. C. Greenlees. Tate cohomology in axiomatic stable homotopy theory. In Cohomological methods in homotopy theory (Bellaterra, 1998), volume 196 of Progr. Math., pages 149–176. Birkhäuser, Basel, 2001.
  30. Robin Hartshorne. Residues and duality. Lecture Notes in Mathematics, No. 20. Springer-Verlag, Berlin-New York, 1966. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64, With an appendix by P. Deligne.
  31. The structure of the Bousfield lattice. In Homotopy invariant algebraic structures (Baltimore, MD, 1998), volume 239 of Contemp. Math., pages 175–196. Amer. Math. Soc., Providence, RI, 1999.
  32. Axiomatic stable homotopy theory. Mem. Amer. Math. Soc., 128(610), 1997.
  33. Nilpotence and stable homotopy theory. II. Ann. of Math. (2), 148(1):1–49, 1998.
  34. Morava K𝐾Kitalic_K-theories and localisation. Mem. Amer. Math. Soc., 139(666):viii+100, 1999.
  35. Eike Lau. The Balmer spectrum of certain Deligne-Mumford stacks. Preprint, 34 pages, available online at arXiv:2101.01446, 2021.
  36. Amnon Neeman. The chromatic tower for D⁢(R)𝐷𝑅D(R)italic_D ( italic_R ). Topology, 31(3):519–532, 1992.
  37. Amnon Neeman. Oddball Bousfield classes. Topology, 39(5):931–935, 2000.
  38. Amnon Neeman. Triangulated categories, volume 148 of Annals of Mathematics Studies. Princeton University Press, 2001.
  39. Amnon Neeman. Colocalizing subcategories of 𝐃⁢(R)𝐃𝑅\mathbf{D}(R)bold_D ( italic_R ). J. Reine Angew. Math., 653:221–243, 2011.
  40. Spatial sublocales and essential primes. Topology Appl., 26(3):263–269, 1987.
  41. Daniel Quillen. The spectrum of an equivariant cohomology ring. I, II. Ann. of Math. (2), 94:549–572; ibid. (2) 94 (1971), 573–602, 1971.
  42. Jeremy Rickard. Derived categories and stable equivalence. J. Pure Appl. Algebra, 61(3):303–317, 1989.
  43. Fred Rohrer. Torsion functors, small or large. Beitr. Algebra Geom., 60(2):233–256, 2019.
  44. Beren Sanders. Higher comparison maps for the spectrum of a tensor triangulated category. Adv. Math., 247:71–102, 2013.
  45. William T. Sanders. Support and vanishing for non-Noetherian rings and tensor triangulated categories. Preprint, 33 pages, available online at arXiv:1710.10199, 2017.
  46. The Stacks project authors. The Stacks project. https://stacks.math.columbia.edu, 2023.
  47. Greg Stevenson. Derived categories of absolutely flat rings. Homology Homotopy Appl., 16(2):45–64, 2014.
  48. Greg Stevenson. The local-to-global principle for triangulated categories via dimension functions. J. Algebra, 473:406–429, 2017.
  49. R. W. Thomason. The classification of triangulated subcategories. Compositio Math., 105(1):1–27, 1997.
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