Abstract
We give necessary and sufficient conditions for stratification and costratification to descend along a coproduct preserving, tensor-exact R-linear functor between R-linear tensor-triangulated categories which are rigidly-compactly generated by their tensor units. We then apply these results to non-positive commutative DG-rings and connective ring spectra. In particular, this gives a support-theoretic classification of (co)localizing subcategories, and thick subcategories of compact objects of the derived category of a non-positive commutative DG-ring with finite amplitude, and provides a formal justification for the principle that the space associated to an eventually coconnective derived scheme is its underlying classical scheme. For a non-positive commutative DG-ring A, we also investigate whether certain finiteness conditions in D(A) (for example, proxy-smallness) can be reduced to questions in the better understood category D(H0A).
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References
P. Balmer, I. Dell’Ambrogio and B. Sanders, Grothendieck–Neeman duality and the Wirthmüller isomorphism, Compositio Mathematica 152 (2016), 1740–1776.
T. Barthel, N. Castellana, D. Heard and G. Valenzuela, Stratification and duality for homotopical groups, Advances in Mathematics 354 ( 2019), Article no. 106733.
T. Barthel, N. Castellana, D. Heard and G. Valenzuela, On stratification for spaces with Noetherian mod p cohomology, American Journal of Mathematics 144(2022), 895–941.
K. A. Beck and S. Sather-Wagstaff, Krull dimension for differential graded algebras, Archiv der Mathematik 101 (2013), 111–119.
D. Benson and J. P. C. Greenlees, Stratifying the derived category of cochains on BG for G a compact Lie group, Journal of Pure and Applied Algebra 218 (2014), 642–650.
D. Benson, S. B. Iyengar and H. Krause, Local cohomology and support for triangulated categories, Annales Scientifiques de l’École Normale Supérieure 41 (2008), 573–619.
D. Benson, S. B. Iyengar and H. Krause, Stratifying modular representations of finite groups, Annals of Mathematics 174 (2011), 1643–1684.
D. Benson, S. B. Iyengar and H. Krause, Stratifying triangulated categories, Journal of Topology 4 (2011), 641–666.
D. Benson, S. B. Iyengar and H. Krause, Colocalizing subcategories and cosupport, Journal für die Reine und Angewandte Mathematik 673 (2012), 161–207.
B. Briggs, E. Grifo and J. Pollitz, Constructing nonproxy small test modules for the complete intersection property, Nagoya Mathematical Journal 246 (2022), 412–429.
B. Briggs, S. B. Iyengar, J. Letz, and J. Pollitz, Locally complete intersection maps and the proxy small property, International Mathematics Research Notices 2022 (2022), 12625–12652.
I. Dell’Ambrogio and D. Stanley, Affine weakly regular tensor triangulated categories, Pacific Journal of Mathematics 285 (2016), 93–109.
W. G. Dwyer, J. P. C. Greenlees and S. B. Iyengar, Duality in algebra and topology, Advances in Mathematics 200 (2006), 357–402.
W. G. Dwyer, J. P. C. Greenlees and S. B. Iyengar, Finiteness in derived categories of local rings, Commentarii Mathematici Helvetici 81 (2006), 383–432.
M. J. Hopkins, Global methods in homotopy theory, in Homotopy Theory (Durham, 1985), London Mathematical Society Lecture Note Series, Vol. 117, Cambridge University Press, Cambridge, 1987, pp. 73–96.
M. J. Hopkins and J. H. Smith, Nilpotence and stable homotopy theory. II, Annals of Mathematics 148 (1998), 1–49.
M. Hovey, J. H. Palmieri and N. P. Strickland, Axiomatic stable homotopy theory, Memoirs of the American Mathematical Society 128 (1997).
P. Jørgensen, Amplitude inequalities for differential graded modules, Forum Mathematicum 22 (2010), 941–948.
J. C. Letz, Local to global principles for generation time over commutative noetherian rings, Homology, Homotopy and Applications 23 (2021), 165–182.
A. Neeman, The chromatic tower for D(R), Topology 31 (1992), 519–532.
A. Neeman, The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, Journal of the American Mathematical Society 9 (1996), 205–236.
A. Neeman, Triangulated Categories, Annals of Mathematics Studies, Vol. 148, Princeton University Press, Princeton, NJ, 2001.
A. Neeman, Colocalizing subcategories of D(R), Journal für die Reine und Angewandte Mathematik 653 (2011), 221–243.
J. Pollitz, The derived category of a locally complete intersection ring, Advances in Mathematics 354 (2019), Article no. 106752.
M. Porta, L. Shaul and A. Yekutieli, On the homology of completion and torsion, Algebras and Representation Theory 17 (2014), 31–67.
S. Schwede and B. Shipley, Stable model categories are categories of modules, Topology 42 (2003), 103–153.
L. Shaul, Homological dimensions of local (co)homology over commutative DG-rings, Canadian Mathematical Bulletin 61 (2018), 865–877.
L. Shaul, Injective DG-modules over non-positive DG-rings, Journal fo Algebra 515 (2018), 102–156.
L. Shaul, Completion and torsion over commutative DG rings, Israel Journal of Mathematics 232 (2019), 531–588.
L. Shaul, The Cohen–Macaulay property in derived commutative algebra, Transactions of the American Mathematical Society 373 (2020), 6095–6138.
L. Shaul, Smooth flat maps over commutative DG-rings, Mathematische Zeitschrift 299 (2021), 1673–1688.
B. Shipley, Hℤ-algebra spectra are differential graded algebras, American Journal of Mathematics 129 (2007), 351–379.
A. Yekutieli, Duality and tilting for commutative DG-rings, https://arxiv.org/abs/1312.6411.
A. Yekutieli, Derived Categories, Cambridge Studies in Advanced Mathematics, Vol. 183, Cambridge University Press, Cambridge, 2020.
Acknowledgements
Both authors were supported by the grant GA ČR 20-02760Y from the Czech Science Foundation. Shaul was also supported by Charles University Research Centre program No. UNCE/SCI/022. The authors thank the referee for suggestions that helped improving this manuscript.
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Shaul, L., Williamson, J. Lifting (co)stratifications between tensor triangulated categories. Isr. J. Math. 261, 249–280 (2024). https://doi.org/10.1007/s11856-023-2578-5
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DOI: https://doi.org/10.1007/s11856-023-2578-5