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Approximation by perfect complexes detects Rouquier dimension (2401.10146v2)

Published 18 Jan 2024 in math.AG and math.AC

Abstract: This work explores bounds on the Rouquier dimension in the bounded derived category of coherent sheaves on Noetherian schemes. By utilizing approximations, we exhibit that Rouquier dimension is inherently characterized by the number of cones required to build all perfect complexes. We use this to prove sharper bounds on Rouquier dimension of singular schemes. Firstly, we show Rouquier dimension doesn't go up along \'{e}tale extensions and is invariant under \'{e}tale covers of affine schemes admitting a dualizing complex. Secondly, we demonstrate that the Rouquier dimension of the bounded derived category for a curve, with a delta invariant of at most one at closed points, is no larger than two. Thirdly, we bound the Rouquier dimension for the bounded derived category of a (birational) derived splinter variety by that of a resolution of singularities.

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References (40)
  1. Homology of perfect complexes. Adv. Math., 223(5):1731–1781, 2010.
  2. Ko Aoki. Quasiexcellence implies strong generation. J. Reine Angew. Math., 780:133–138, 2021.
  3. Michael Artin. Algebraic approximation of structures over complete local rings. Inst. Hautes Études Sci. Publ. Math., (36):23–58, 1969.
  4. Paul Balmer. The derived category of an Étale extension and the separable neeman–thomason theorem. Journal of the Institute of Mathematics of Jussieu, 15(3):613–623, 2016.
  5. Faisceaux pervers. Actes du colloque “Analyse et Topologie sur les Espaces Singuliers”. Partie I, volume 100 of Astérisque. Paris: Société Mathématique de France (SMF), 2nd edition edition, 2018.
  6. On the Rouquier dimension of wrapped Fukaya categories and a conjecture of Orlov. Compos. Math., 159(3):437–487, 2023.
  7. Bounded t𝑡titalic_t-structures, finitistic dimensions, and singularity categories of triangulated categories. arXiv preprint 2401.00130, 2023.
  8. Tilting on non-commutative rational projective curves. Math. Ann., 351(3):665–709, 2011.
  9. On the derived categories of gentle and skew-gentle algebras: homological algebra and matrix problems. arXiv preprint arXiv:1706.08358, 2017.
  10. Singular curves and quasi-hereditary algebras. Int. Math. Res. Not., 2017(3):895–920, 2017.
  11. Apostolos Beligiannis. Some ghost lemmas. Lecture Notes for the Conference “The Representation Dimension of Artin Algebras”, 2008.
  12. Hochschild dimensions of tilting objects. Int. Math. Res. Not., 2012(11):2607–2645, 2012.
  13. A category of kernels for equivariant factorizations. II: Further implications. J. Math. Pures Appl. (9), 102(4):702–757, 2014.
  14. Bhargav Bhatt. Derived splinters in positive characteristic. Compos. Math., 148(6):1757–1786, 2012.
  15. High frobenius pushforwards generate the bounded derived category. arXiv preprint arXiv:2303.18085, 2023.
  16. A. Bondal and M. van den Bergh. Generators and representability of functors in commutative and noncommutative geometry. Mosc. Math. J., 3(1):1–36, 2003.
  17. Éléments de géométrie algébrique. Inst. Hautes Études Sci. Publ. Math., 4, 8, 11, 17, 20, 24, 28, 32, 1961–1967.
  18. Robin Hartshorne. Residues and duality. Lecture notes of a seminar on the work of A. Grothendieck, given at Havard 1963/64. Appendix: Cohomology with supports and the construction of the f!superscript𝑓f^{!}italic_f start_POSTSUPERSCRIPT ! end_POSTSUPERSCRIPT functor by P. Deligne, volume 20 of Lect. Notes Math. Springer, Cham, 1966.
  19. Robin Hartshorne. Algebraic geometry. Corr. 3rd printing, volume 52 of Grad. Texts Math. Springer, Cham, 1983.
  20. Resolutions of toric subvarieties by line bundles and applications. arXiv preprint arXiv:2303.03763, 2023.
  21. D. Huybrechts. Fourier-Mukai transforms in algebraic geometry. Oxford Math. Monogr. Oxford: Clarendon Press, 2006.
  22. V. B. Jatoba. Strong generators in Dperf⁢(X)subscript𝐷perf𝑋D_{\mathrm{perf}}(X)italic_D start_POSTSUBSCRIPT roman_perf end_POSTSUBSCRIPT ( italic_X ) for schemes with a separator. Proc. Am. Math. Soc., 149(5):1957–1971, 2021.
  23. Yujiro Kawamata. Semi-orthogonal decomposition of a derived category of a 3-fold with an ordinary double point. Recent Developments in Algebraic Geometry: To Miles Reid for his 70th Birthday, 478:183, 2022.
  24. G. Maxwell Kelly. Chain maps inducing zero homology maps. Proc. Camb. Philos. Soc., 61:847–854, 1965.
  25. Sándor J. Kovács. A characterization of rational singularities. Duke Math. J., 102(2):187–191, 2000.
  26. Henning Krause. Homological theory of representations, volume 195 of Camb. Stud. Adv. Math. Cambridge: Cambridge University Press, 2022.
  27. Henning Krause. The finitistic dimension of a triangulated category, 2023.
  28. Pat Lank. Descent conditions for generation in derived categories. arXiv preprint arXiv:2308.08080, 2023.
  29. Janina C. Letz. Local to global principles for generation time over commutative Noetherian rings. Homology Homotopy Appl., 23(2):165–182, 2021.
  30. Quasi-perfect scheme-maps and boundedness of the twisted inverse image functor. Illinois J. Math., 51(1):209–236, 2007.
  31. Shiji Lyu. On some properties of birational derived splinters. arXiv preprint arXiv:2210.03193, 2022.
  32. Amnon Neeman. Strong generators in 𝐃perf⁢(X)superscript𝐃perf𝑋\mathbf{D}^{\mathrm{perf}}(X)bold_D start_POSTSUPERSCRIPT roman_perf end_POSTSUPERSCRIPT ( italic_X ) and 𝐃cohb⁢(X)subscriptsuperscript𝐃𝑏coh𝑋\mathbf{D}^{b}_{\mathrm{coh}}(X)bold_D start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_coh end_POSTSUBSCRIPT ( italic_X ). Ann. Math. (2), 193(3):689–732, 2021.
  33. Noah Olander. Resolutions, bounds, and dimensions for derived categories of varieties. Columbia University, 2022.
  34. Noah Olander. Ample line bundles and generation time. J. Reine Angew. Math., 800:299–304, 2023.
  35. Dmitri Orlov. Remarks on generators and dimensions of triangulated categories. Mosc. Math. J., 9(1):143–149, 2009.
  36. Generating the bounded derived category and perfect ghosts. Bull. Lond. Math. Soc., 44(2):285–298, 2012.
  37. Dmitrii Pirozhkov. Rouquier dimension of some blow-ups. Eur. J. Math., 9(2):13, 2023. Id/No 45.
  38. Raphaël Rouquier. Dimensions of triangulated categories. J. K𝐾Kitalic_K-Theory, 1(2):193–256, 2008.
  39. The Stacks Project Authors. Stacks Project. https://stacks.math.columbia.edu, 2023.
  40. Higher algebraic K-theory of schemes and of derived categories.
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