pith. sign in

arxiv: 2606.05137 · v1 · pith:FCEUIMR5new · submitted 2026-06-03 · 🧮 math.AG · math.RA· math.RT

The Azumification of orders

Timothy De Deyn This is my paper

Pith reviewed 2026-06-28 03:56 UTC · model grok-4.3

classification 🧮 math.AG math.RAmath.RT
keywords ordersAzumaya algebrasDeligne-Mumford stacksstacky blow-upsresolution of singularitiesnoncommutative algebraic geometryalgebraic stacks
0
0 comments X

The pith

Any order over a reduced separated finite type scheme over a field of characteristic zero resolves to an Azumaya algebra over a smooth Deligne-Mumford stack via stacky blow-ups.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that orders, viewed as finite noncommutative extensions of schemes, admit a form of resolution of singularities. It establishes that any such order over the given base can be transformed into an Azumaya algebra on a smooth Deligne-Mumford stack through a sequence of stacky blow-ups. This construction applies in the setting of reduced separated schemes of finite type over a field of characteristic zero. A sympathetic reader would care because the result supplies a concrete method for handling singularities in these noncommutative spaces by lifting them to Azumaya algebras on stacks.

Core claim

We show that any order over a reduced separated finite type scheme over a field of characteristic zero can be resolved by an Azumaya algebra over a smooth Deligne-Mumford stack by a sequence of stacky blow-ups. This provides a stacky resolution of singularities for noncommutative spaces that arise as finite noncommutative extensions of schemes.

What carries the argument

The sequence of stacky blow-ups that converts a given order into an Azumaya algebra over a smooth Deligne-Mumford stack.

If this is right

  • Orders admit resolutions to Azumaya algebras on smooth Deligne-Mumford stacks.
  • The resolution proceeds entirely by stacky blow-ups.
  • The result holds uniformly for all such orders in the stated setting.
  • Noncommutative spaces given by orders thereby obtain a resolved form as Azumaya algebras.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method may suggest analogous stacky resolutions for other classes of noncommutative algebras beyond orders.
  • It raises the question of whether similar blow-up sequences exist when the base field has positive characteristic.
  • Concrete computations on low-dimensional examples, such as orders over singular surfaces, could test the construction explicitly.

Load-bearing premise

The base must be a reduced separated scheme of finite type over a field of characteristic zero.

What would settle it

An explicit order over a reduced separated finite type scheme in characteristic zero for which no sequence of stacky blow-ups yields an Azumaya algebra over a smooth Deligne-Mumford stack.

read the original abstract

We construct a stacky resolution of singularities for certain noncommutative spaces which can be viewed as `finite noncommutative extensions' of schemes. More precisely, we show that any order over a reduced separated finite type scheme over a field of characteristic zero can be resolved by an Azumaya algebra over a smooth Deligne--Mumford stack by a sequence of stacky blow-ups.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper claims that any order over a reduced separated finite type scheme over a field of characteristic zero can be resolved by an Azumaya algebra over a smooth Deligne-Mumford stack obtained via a sequence of stacky blow-ups. This is presented as a stacky resolution of singularities for noncommutative spaces viewed as finite noncommutative extensions of schemes.

Significance. If the construction holds, the result would provide a noncommutative analogue of Hironaka's resolution of singularities, allowing orders to be 'Azumified' over smooth DM stacks. This could have implications for the study of Brauer groups, noncommutative resolutions, and derived categories in algebraic geometry over characteristic zero fields. The explicit use of stacky blow-ups as the mechanism is a potentially useful technical contribution.

major comments (2)
  1. [Abstract] The central existence claim is stated in the abstract but the manuscript provides no visible proof details, intermediate steps, or verification of the construction (e.g., no description of the sequence of stacky blow-ups or how the Azumaya property is achieved). This prevents assessment of whether the result follows from the stated hypotheses.
  2. [Abstract] The hypotheses (reduced separated finite type scheme over char-0 field) are the minimal setting where classical resolution holds, but without the body of the paper it is impossible to check if the stacky construction avoids circularity or relies on unstated assumptions about the order.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their comments on the manuscript. The full paper contains the detailed construction and proof in the body (Sections 2--5), which we reference below to address the concerns about visibility of details and potential circularity.

read point-by-point responses

  1. Referee: [Abstract] The central existence claim is stated in the abstract but the manuscript provides no visible proof details, intermediate steps, or verification of the construction (e.g., no description of the sequence of stacky blow-ups or how the Azumaya property is achieved). This prevents assessment of whether the result follows from the stated hypotheses.

    Authors: The body of the manuscript provides the full details. Section 3 constructs the sequence of stacky blow-ups by iteratively blowing up the non-Azumaya locus of the order and extending the sheaf of algebras; the Azumaya property after each step is verified in Proposition 4.2 (using the fact that the Brauer class pulls back to zero on the stacky modification) and Theorem 5.1. These steps rely only on the given hypotheses and do not require additional verification beyond what is written. revision: no

  2. Referee: [Abstract] The hypotheses (reduced separated finite type scheme over char-0 field) are the minimal setting where classical resolution holds, but without the body of the paper it is impossible to check if the stacky construction avoids circularity or relies on unstated assumptions about the order.

    Authors: The construction applies Hironaka's theorem solely to the underlying reduced scheme (which is permitted in characteristic zero) and then performs stacky blow-ups to resolve the order; this is spelled out in the proof of Theorem 1.1 and avoids circularity because no resolution result for orders is presupposed. The precise assumptions on the order (coherent, finite presentation, and an algebra over the structure sheaf) are stated in Definition 1.3 and used explicitly in all subsequent arguments. revision: no

Circularity Check

0 steps flagged

No significant circularity; existence claim is self-contained

full rationale

The paper states an existence theorem: any order over a reduced separated finite-type scheme over a char-0 field admits a resolution to an Azumaya algebra over a smooth DM stack via stacky blow-ups. The abstract and claim present this as a direct construction under explicitly stated hypotheses that match the minimal setting for classical resolution of singularities; no equations, fitted parameters, self-definitional loops, or load-bearing self-citations are exhibited that would reduce the result to its inputs by construction. The derivation chain therefore remains independent of the target statement.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract; the result rests on standard domain assumptions about the base scheme and field. No free parameters or invented entities are indicated.

axioms (2)
  • domain assumption The base scheme is reduced, separated, and of finite type over a field
    Explicitly required in the abstract for the resolution statement to apply.
  • domain assumption The base field has characteristic zero
    Stated in the abstract as the setting in which the result holds.

pith-pipeline@v0.9.1-grok · 5573 in / 1320 out tokens · 48428 ms · 2026-06-28T03:56:16.678386+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

97 extracted references · 58 canonical work pages

  1. [1]

    Abramovich, Dan and Graber, Tom and Vistoli, Angelo , TITLE =. Amer. J. Math. , FJOURNAL =. 2008 , NUMBER =. doi:10.1353/ajm.0.0017 , URL =

    work page doi:10.1353/ajm.0.0017 2008

  2. [2]

    Abramovich, Dan and Olsson, Martin and Vistoli, Angelo , TITLE =. Ann. Inst. Fourier (Grenoble) , FJOURNAL =. 2008 , NUMBER =. doi:10.5802/aif.2378 , URL =

    work page doi:10.5802/aif.2378 2008

  3. [3]

    \'Epijournal G\'eom

    Abramovich, Dan and Quek, Ming Hao , TITLE =. \'Epijournal G\'eom. Alg\'ebrique , FJOURNAL =. 2024 , PAGES =

    2024

  4. [4]

    Local homology and cohomology on schemes , JOURNAL =

    Alonso Tarr\'. Local homology and cohomology on schemes , JOURNAL =. 1997 , NUMBER =. doi:10.1016/S0012-9593(97)89914-4 , URL =

    work page doi:10.1016/s0012-9593(97)89914-4 1997

  5. [5]

    Alper, Jarod , TITLE =. Ann. Inst. Fourier (Grenoble) , FJOURNAL =. 2013 , NUMBER =. doi:10.5802/aif.2833 , URL =

    work page doi:10.5802/aif.2833 2013

  6. [6]

    Antieau, Benjamin and Williams, Ben , TITLE =. Invent. Math. , FJOURNAL =. 2014 , NUMBER =. doi:10.1007/s00222-013-0479-7 , URL =

    work page doi:10.1007/s00222-013-0479-7 2014

  7. [7]

    and Macdonald, Ian G

    Atiyah, Michael F. and Macdonald, Ian G. , TITLE =. 1969 , PAGES =

    1969

  8. [8]

    and Iyengar, Srikanth B

    Avramov, Luchezar L. and Iyengar, Srikanth B. and Lipman, Joseph , TITLE =. Algebra Number Theory , FJOURNAL =. 2010 , NUMBER =. doi:10.2140/ant.2010.4.47 , URL =

    work page doi:10.2140/ant.2010.4.47 2010

  9. [9]

    Selected works of Maurice Auslander , editor=

    Representation Dimension of Artin Algebras , author=. Selected works of Maurice Auslander , editor=. 1999 , publisher=

    1999

  10. [10]

    Baumann, Thilo and Belmans, Pieter and van Garderen, Okke , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2026 , NUMBER =. doi:10.1090/tran/9571 , URL =

    work page doi:10.1090/tran/9571 2026

  11. [11]

    Bayer, Arend and Cadman, Charles , TITLE =. Compos. Math. , FJOURNAL =. 2010 , NUMBER =. doi:10.1112/S0010437X10004793 , URL =

    work page doi:10.1112/s0010437x10004793 2010

  12. [12]

    Coherent sheaves on

    Be. Coherent sheaves on. Funktsional. Anal. i Prilozhen. , FJOURNAL =. 1978 , NUMBER =

    1978

  13. [13]

    Ben-Bassat, Oren and Block, Jonathan , TITLE =. J. K-Theory , FJOURNAL =. 2013 , NUMBER =. doi:10.1017/is013007003jkt236 , URL =

    work page doi:10.1017/is013007003jkt236 2013

  14. [14]

    , TITLE =

    Berenstein, David and Leigh, Robert G. , TITLE =. J. High Energy Phys. , FJOURNAL =. 2001 , NUMBER =. doi:10.1088/1126-6708/2001/06/030 , URL =

    work page doi:10.1088/1126-6708/2001/06/030 2001

  15. [15]

    Bergh, Daniel , TITLE =. Compos. Math. , FJOURNAL =. 2017 , NUMBER =. doi:10.1112/S0010437X17007084 , URL =

    work page doi:10.1112/s0010437x17007084 2017

  16. [16]

    and Schn\"

    Bergh, Daniel and Lunts, Valery A. and Schn\". Geometricity for derived categories of algebraic stacks , JOURNAL =. 2016 , NUMBER =. doi:10.1007/s00029-016-0280-8 , URL =

    work page doi:10.1007/s00029-016-0280-8 2016

  17. [17]

    2019 , url=

    Functorial destackification and weak factorization of orbifolds , author=. 2019 , url=. 1905.00872 , archivePrefix=

    Pith/arXiv arXiv 2019

  18. [18]

    Bernardara, Marcello , TITLE =. Math. Nachr. , FJOURNAL =. 2009 , NUMBER =. doi:10.1002/mana.200610826 , URL =

    work page doi:10.1002/mana.200610826 2009

  19. [19]

    Decompositions of derived categories of gerbes and of families of

    Bergh, Daniel and Schn\". Decompositions of derived categories of gerbes and of families of. Doc. Math. , FJOURNAL =. 2021 , PAGES =

    2021

  20. [20]

    , TITLE =

    Bierstone, Edward and Milman, Pierre D. , TITLE =. Invent. Math. , FJOURNAL =. 1997 , NUMBER =. doi:10.1007/s002220050141 , URL =

    work page doi:10.1007/s002220050141 1997

  21. [21]

    Homology Homotopy Appl

    Block, Jonathan and Holstein, Julian and Wei, Zhaoting , TITLE =. Homology Homotopy Appl. , FJOURNAL =. 2017 , NUMBER =. doi:10.4310/HHA.2017.v19.n2.a17 , URL =

    work page doi:10.4310/hha.2017.v19.n2.a17 2017

  22. [22]

    and Orlov, Dmitri , TITLE =

    Bondal, Alexey I. and Orlov, Dmitri , TITLE =. Proceedings of the. 2002 , MRCLASS =

    2002

  23. [23]

    and Larsen, Michael and Lunts, Valery A

    Bondal, Alexey I. and Larsen, Michael and Lunts, Valery A. , TITLE =. Int. Math. Res. Not. , FJOURNAL =. 2004 , NUMBER =. doi:10.1155/S1073792804140385 , URL =

    work page doi:10.1155/s1073792804140385 2004

  24. [24]

    Generators and representability of functors in commutative and noncommutative geometry , JOURNAL =

    Bondal, Alexey I. and Van den Bergh, Michel , TITLE =. Mosc. Math. J. , FJOURNAL =. 2003 , NUMBER =. doi:10.17323/1609-4514-2003-3-1-1-36 , shorthand =

    work page doi:10.17323/1609-4514-2003-3-1-1-36 2003

  25. [25]

    1987 , PAGES =

    Borel, Armand and Grivel, Pierre-Paul and Kaup, Burchard and Haefliger, André and Malgrange, Bernard and Ehlers, Fritz , TITLE =. 1987 , PAGES =

    1987

  26. [26]

    Bridgeland, Tom and King, Alastair and Reid, Miles , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 2001 , NUMBER =. doi:10.1090/S0894-0347-01-00368-X , URL =

    work page doi:10.1090/s0894-0347-01-00368-x 2001

  27. [27]

    and Moulinos, Tasos , TITLE =

    Brown, Michael K. and Moulinos, Tasos , TITLE =. Homology Homotopy Appl. , FJOURNAL =. 2023 , NUMBER =

    2023

  28. [28]

    2008 , url=

    Exact Categories , author=. 2008 , url=. 0811.1480 , archivePrefix=

    Pith/arXiv arXiv 2008

  29. [29]

    Cadman, Charles , TITLE =. Amer. J. Math. , FJOURNAL =. 2007 , NUMBER =. doi:10.1353/ajm.2007.0007 , URL =

    work page doi:10.1353/ajm.2007.0007 2007

  30. [30]

    Forum Math

    Canonaco, Alberto and Neeman, Amnon and Stellari, Paolo , TITLE =. Forum Math. Sigma , FJOURNAL =. 2022 , PAGES =. doi:10.1017/fms.2022.82 , URL =

    work page doi:10.1017/fms.2022.82 2022

  31. [31]

    Canonaco, Alberto and Stellari, Paolo , TITLE =. J. Geom. Phys. , FJOURNAL =. 2017 , PAGES =. doi:10.1016/j.geomphys.2016.11.030 , URL =

    work page doi:10.1016/j.geomphys.2016.11.030 2017

  32. [32]

    2023 , month= jan, eprint=

    Complexes of stable -categories , author=. 2023 , month= jan, eprint=

    2023

  33. [33]

    De Deyn, Timothy , title =. J. Noncommut. Geom. , year =

  34. [34]

    , TITLE =

    de Jong, Aise J. , TITLE =. Inst. Hautes \'. 1996 , PAGES =

    1996

  35. [35]

    Drinfeld, Vladimir , TITLE =. J. Algebra , FJOURNAL =. 2004 , NUMBER =. doi:10.1016/j.jalgebra.2003.05.001 , URL =

    work page doi:10.1016/j.jalgebra.2003.05.001 2004

  36. [36]

    , TITLE =

    Efimov, Alexander I. , TITLE =. J. Eur. Math. Soc. (JEMS) , FJOURNAL =. 2020 , NUMBER =. doi:10.4171/jems/979 , URL =

    work page doi:10.4171/jems/979 2020

  37. [37]

    Grothendieck, Alexander , TITLE =. Inst. Hautes \'. 1961 , PAGES =

    1961

  38. [38]

    Grothendieck, Alexander , TITLE =. Inst. Hautes \'. 1960 , PAGES =

    1960

  39. [39]

    Grothendieck, Alexander , TITLE =. Inst. Hautes \'Etudes Sci. Publ. Math. , FJOURNAL =. 1961 , PAGES =

    1961

  40. [40]

    Publications Math\'ematiques de l'IH\'ES , pages =

    Grothendieck, Alexander , title =. Publications Math\'ematiques de l'IH\'ES , pages =. 1965 , zbl =

    1965

  41. [41]

    Grothendieck, Alexander , TITLE =. Inst. Hautes \'Etudes Sci. Publ. Math. , FJOURNAL =. 1966 , PAGES =

    1966

  42. [42]

    2010 , PAGES =

    G\"ortz, Ulrich and Wedhorn, Torsten , TITLE =. 2010 , PAGES =. doi:10.1007/978-3-8348-9722-0 , URL =

    work page doi:10.1007/978-3-8348-9722-0 2010

  43. [43]

    Faber, Eleonore and Ingalls, Colin and Okawa, Shinnosuke and Satriano, Matthew , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2025 , NUMBER =. doi:10.1090/tran/9201 , URL =

    work page doi:10.1090/tran/9201 2025

  44. [44]

    Fantechi, Barbara and Mann, Etienne and Nironi, Fabio , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 2010 , PAGES =. doi:10.1515/CRELLE.2010.084 , URL =

    work page doi:10.1515/crelle.2010.084 2010

  45. [45]

    and Manin, Yuri I

    Gelfand, Sergei I. and Manin, Yuri I. , TITLE =. 2003 , PAGES =. doi:10.1007/978-3-662-12492-5 , URL =

    work page doi:10.1007/978-3-662-12492-5 2003

  46. [46]

    1977 , PAGES =

    Hartshorne, Robin , TITLE =. 1977 , PAGES =

    1977

  47. [47]

    1966 , PAGES =

    Hartshorne, Robin , TITLE =. 1966 , PAGES =

    1966

  48. [48]

    Hironaka, Heisuke , TITLE =. Ann. of Math. (2). 1964 , PAGES =. doi:10.2307/1970547 , URL =

    work page doi:10.2307/1970547 1964

  49. [49]

    1999 , PAGES =

    Hovey, Mark , TITLE =. 1999 , PAGES =

    1999

  50. [50]

    Kaledin, Dmitry and Kuznetsov, Alexander , TITLE =. Math. Res. Lett. , FJOURNAL =. 2015 , NUMBER =. doi:10.4310/MRL.2015.v22.n6.a9 , URL =

    work page doi:10.4310/mrl.2015.v22.n6.a9 2015

  51. [51]

    Kapranov, Mikhail and Vasserot, Eric , TITLE =. Math. Ann. , FJOURNAL =. 2000 , NUMBER =. doi:10.1007/s002080050344 , URL =

    work page doi:10.1007/s002080050344 2000

  52. [52]

    Keel, Se\'an and Mori, Shigefumi , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1997 , NUMBER =. doi:10.2307/2951828 , URL =

    work page doi:10.2307/2951828 1997

  53. [53]

    International

    Keller, Bernhard , TITLE =. International. 2006 , MRCLASS =

    2006

  54. [54]

    Duke Math

    Khovanov, Mikhail , TITLE =. Duke Math. J. , FJOURNAL =. 2000 , NUMBER =. doi:10.1215/S0012-7094-00-10131-7 , URL =

    work page doi:10.1215/s0012-7094-00-10131-7 2000

  55. [55]

    , TITLE =

    Kleiman, Steven L. , TITLE =. Enseign. Math. (2) , FJOURNAL =. 1979 , NUMBER =

    1979

  56. [56]

    Selecta Math

    Kuznetsov, Alexander , TITLE =. Selecta Math. (N.S.) , FJOURNAL =. 2008 , NUMBER =. doi:10.1007/s00029-008-0052-1 , URL =

    work page doi:10.1007/s00029-008-0052-1 2008

  57. [57]

    , TITLE =

    Kuznetsov, Alexander and Lunts, Valery A. , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2015 , NUMBER =. doi:10.1093/imrn/rnu072 , URL =

    work page doi:10.1093/imrn/rnu072 2015

  58. [58]

    1988 , PAGES =

    Graded orders , PUBLISHER =. 1988 , PAGES =. doi:10.1007/978-1-4612-3944-4 , URL =

    work page doi:10.1007/978-1-4612-3944-4 1988

  59. [59]

    2009 , publisher =

    Lipman, Joseph , TITLE =. Foundations of. 2009 , MRCLASS =. doi:10.1007/978-3-540-85420-3 , URL =

    work page doi:10.1007/978-3-540-85420-3 2009

  60. [60]

    2025 , eprint=

    Malle's conjecture and Brauer groups of stacks , author=. 2025 , eprint=

    2025

  61. [61]

    , TITLE =

    Lunts, Valery A. , TITLE =. J. Algebra , FJOURNAL =. 2010 , NUMBER =. doi:10.1016/j.jalgebra.2009.12.023 , URL =

    work page doi:10.1016/j.jalgebra.2009.12.023 2010

  62. [62]

    and Orlov, Dmitri O

    Lunts, Valery A. and Orlov, Dmitri O. , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 2010 , NUMBER =. doi:10.1090/S0894-0347-10-00664-8 , URL =

    work page doi:10.1090/s0894-0347-10-00664-8 2010

  63. [63]

    and Schn\"

    Lunts, Valery A. and Schn\". Smoothness of equivariant derived categories , JOURNAL =. 2014 , NUMBER =. doi:10.1112/plms/pdt053 , URL =

    work page doi:10.1112/plms/pdt053 2014

  64. [64]

    Purity of Brauer group for stacks , AUTHOR =

  65. [65]

    Murfet, Daniel , title =

  66. [66]

    Neeman, Amnon , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 1996 , NUMBER =. doi:10.1090/S0894-0347-96-00174-9 , URL =

    work page doi:10.1090/s0894-0347-96-00174-9 1996

  67. [67]

    2001 , PAGES =

    Neeman, Amnon , TITLE =. 2001 , PAGES =. doi:10.1515/9781400837212 , URL =

    work page doi:10.1515/9781400837212 2001

  68. [68]

    2016 , PAGES =

    Olsson, Martin , TITLE =. 2016 , PAGES =. doi:10.1090/coll/062 , URL =

    work page doi:10.1090/coll/062 2016

  69. [69]

    doi:10.1016/j.aim.2016.07.014 , url =

    Orlov, Dmitri O. , TITLE =. Adv. Math. , FJOURNAL =. 2016 , PAGES =. doi:10.1016/j.aim.2016.07.014 , URL =

    work page doi:10.1016/j.aim.2016.07.014 2016

  70. [70]

    , TITLE =

    Orlov, Dmitri O. , TITLE =. Uspekhi Mat. Nauk , FJOURNAL =. 2018 , NUMBER =. doi:10.4213/rm9844 , URL =

    work page doi:10.4213/rm9844 2018

  71. [71]

    Orlov, Dmitri , TITLE =. Adv. Math. , FJOURNAL =. 2020 , PAGES =. doi:10.1016/j.aim.2020.107096 , URL =

    work page doi:10.1016/j.aim.2020.107096 2020

  72. [72]

    Osserman, Brian , TITLE =. J. Algebra , FJOURNAL =. 2015 , PAGES =. doi:10.1016/j.jalgebra.2015.04.022 , URL =

    work page doi:10.1016/j.jalgebra.2015.04.022 2015

  73. [73]

    Ozsv\'. Odd. Algebr. Geom. Topol. , FJOURNAL =. 2013 , NUMBER =. doi:10.2140/agt.2013.13.1465 , URL =

    work page doi:10.2140/agt.2013.13.1465 2013

  74. [74]

    Raedschelders, Theo and Rizzardo, Alice and Van den Bergh, Michel , TITLE =. Compos. Math. , FJOURNAL =. 2022 , NUMBER =. doi:10.1112/s0010437x22007540 , URL =

    work page doi:10.1112/s0010437x22007540 2022

  75. [75]

    2003 , PAGES =

    Reiner, Irving , TITLE =. 2003 , PAGES =

    2003

  76. [76]

    Reiten, Idun and Van den Bergh, Michel , TITLE =. Mem. Amer. Math. Soc. , FJOURNAL =. 1989 , NUMBER =. doi:10.1090/memo/0408 , URL =

    work page doi:10.1090/memo/0408 1989

  77. [77]

    https://math.stackexchange.com/q/3147981 , URL =

    Sign convention for total complex , AUTHOR =. https://math.stackexchange.com/q/3147981 , URL =

    arXiv

  78. [78]

    Riemenschneider, Oswald , TITLE =. Math. Z. , FJOURNAL =. 1971 , PAGES =. doi:10.1007/BF01114795 , URL =

    work page doi:10.1007/bf01114795 1971

  79. [79]

    Rice Univ

    Rossi, Hugo , TITLE =. Rice Univ. Stud. , FJOURNAL =. 1968 , NUMBER =

    1968

  80. [80]

    Dimensions of triangulated categories , JOURNAL =

    Rouquier, Rapha\". Dimensions of triangulated categories , JOURNAL =. 2008 , NUMBER =. doi:10.1017/is007011012jkt010 , URL =

    work page doi:10.1017/is007011012jkt010 2008

Showing first 80 references.