REVIEW 2 major objections 4 minor 56 references
Coderived and contraderived categories for a cotorsion pair, flat-type cotorsion pairs, and relative periodicity
T0 review · 2 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash
Pith's one-line read A cotorsion pair is of flat type exactly when two relative periodicity properties hold.
desk verdict Positselski's cotorsion-pair machine is new and mostly solid; the dependency gap in the quasi-coherent-sheaf applications is what needs referee attention. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing identity is the kernel A∩B of the cotorsion pair. Because A has enough injectives and B enough projectives, both injectives of A and projectives of B coincide with A∩B; hence the homotopy category Hot(A∩B) is the common recipient of the coderived and contraderived categories. The proof mechanism is the construction of four hereditary complete cotorsion pairs inside the category of complexes Com(K), from the pairs (Ac_bco(A), Com(B)), (Com(A), Ac_bctr(B)), (Ac(A), DG(B)), and (DG(A), Ac(B)); these let the author transfer completeness and homotopy-category equivalences from K to Com(K). In the sandwiched setting, the outer very-flat/flat pair supplies fixed identifications amo
What would settle it
In the module setting, take a ring homomorphism R→A and construct an acyclic complex of A/R-flaprojective modules whose cocycles are flat as A-modules; if one cocycle is not A/R-flaprojective, condition (10) of Corollary 10.5 fails and, by the theorem, so do all the equivalent derived-category conditions.
Extended reading notes
Core claim
The paper's central claim is Corollary 4.8: for any hereditary complete cotorsion pair (A,B) generated by a set of objects in a Grothendieck category, there are natural triangulated equivalences D_bco(A) ≃ Hot(A∩B) ≃ D_bctr(B). Here A is the left ('somewhat projective') class and B the right ('somewhat injective') class, and A∩B is their kernel. This makes the coderived category of the left class and the contraderived category of the right class literally the same homotopy category. The paper then feeds this identification into nested pairs of cotorsion pairs and obtains Theorem 9.9: in the sandwiched very-flat/flat setting, ten conditions are equivalent, including the flat-type condition, s
Load-bearing premise
The headline examples presuppose that the flat cotorsion pairs in the module and sheaf categories are already of flat type—claims cited to the preprint [41] and the book manuscript [36] rather than proved here—and the whole machine requires each cotorsion pair to be hereditary, complete, and generated by a set.
Editorial extensions
If this is right
- Any hereditary complete set-generated cotorsion pair gives a single triangulated category Hot(A∩B) that serves as both coderived and contraderived category, so computations can be done in the kernel.
- Nested cotorsion pairs give an adjunction between coderived categories on the left and contraderived categories on the right, realized by a Quillen adjunction of abelian model structures.
- For sandwiched pairs, proving the middle pair is of flat type is the same as proving the two periodicity properties; each of the ten conditions can be checked in whichever formulation is easiest.
- The flaprojective conjecture is equivalent to any of the derived-category equivalences in Corollary 10.5, so a counterexample to any one is a counterexample to all.
- The relatively cotorsion conjecture is likewise equivalent to a list of derived-category equivalences, and the flat/projective periodicity half holds automatically by prior theorems.
Reading between the lines
- The paper leaves implicit that the recollement of Theorem 9.8 quantifies the failure of flat type: when the middle pair is not flat type, the Verdier quotient Hot(F)∩Ac_bco(Flat)/Ac_bco(F) and its mirror are the obstruction, so periodicity failure is not only a module-theoretic phenomenon but also a nonzero derived-category invariant.
- A testable extension: in the scheme setting, if one can construct an acyclic complex of cotorsion quasi-coherent sheaves with a non-cotorsion cocycle, Theorem 9.9 would force the sandwiched pair to violate one of the listed equivalences; the sheaf case is therefore a concrete place to stress-test the periodicity-to-equivalence bridge.
- The two motivating conjectures are dual: each imports periodicity on one side from known theorems, so the open content is a single missing periodicity statement on the other side. Proving either conjecture would immediately certify flat type for the corresponding relative cotorsion pair, and vice versa.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper develops a general machine for derived categories of the second kind associated to cotorsion pairs in Grothendieck categories. Its foundational result, Corollary 4.8, gives natural triangulated equivalences D^bco(A) ≃ Hot(A∩B) ≃ D^bctr(B) for any hereditary complete cotorsion pair (A,B) generated by a set. For nested cotorsion pairs it constructs induced functors and a Quillen adjunction (Theorem 5.3). It then introduces very-flat-type and flat-type cotorsion pairs, proves several triangulated equivalences for flat-type pairs (Theorems 8.2 and 8.5), and establishes the main structural theorem (Theorem 9.9/Theorem 0.3): for a sandwiched pair (F,C) between a very-flat-type and a flat-type pair, flat-type behavior is equivalent to a list of triangulated equivalences and to two periodicity properties. Applications to relative flat/projective modules and relative cotorsion modules are formulated as Conjectures 0.1 and 0.2, with equivalent reformulations in Corollaries 10.5 and 11.3.
Significance. If the results are correct, this is a valuable contribution: it provides a clean, uniform framework in which flat-type behavior, Becker co/contraderived equivalences, and periodicity properties are shown to be the same phenomenon for sandwiched cotorsion pairs. The proof strategy is mostly explicit and internally coherent; the key adjunction/quotient chain in Sections 5 and 9 is carefully checked. Important strengths are the parameter-free statements of the main theorems, the precise listing of hypotheses, and the honest discussion of which verifications remain open (notably Example 8.10). The paper does not merely cite its own earlier work in a circular way: the imported theorems are used as stated assumptions, not as consequences of the results being proved. The main weakness is that several load-bearing verifications for the motivating examples are delegated to unpublished or self-published preprints, so the advertised scheme-level conclusions are conditional in the text as written.
major comments (2)
- [§8, Examples 8.8–8.9 and Theorem 9.9] The scheme-level application of the main theorem depends on the assertion that (X–Qcoh_flat, X–Qcoh_cta) is of flat type, which is imported from the author's book manuscript [36, Corollary 5.5.11] without proof here. Since Theorem 9.9 requires the ambient pair (Flat,Cot) to be of flat type, the advertised 'same applies to quasi-coherent sheaves over a quasi-compact semi-separated scheme' is not independently established in the manuscript. In addition, Example 8.10 explicitly states that the analogous flat-type verification for A–Qcoh 'has not been worked out yet.' This is a dependency gap rather than an internal inconsistency, but it is load-bearing for a central advertised claim. The authors should either prove the needed flat-type statement (or give a detailed proof outline) or clearly mark the scheme-level theorems as conditional on the unpublished [36, Corollary 5.5.11]. A similar co
- [§11, Proposition 11.1] The relative cotorsion conjecture is made to rest on Proposition 11.1, whose key step is the deconstructibility of the class A–Mod_R-proj_flat. The proof delegates this to [38, Proposition 2.3] and [52, Proposition 2.9(2)] rather than presenting the Hill-lemma argument. This is not necessarily a fatal issue, since both references are available, but it is another place where a nontrivial, load-bearing verification is left to the reader and to the reliability of the cited sources. In a revised version I would ask for at least a sketch of the deconstructibility argument, or a precise statement of the cited results, so that the corollary is not hostage to an unprinted argument.
minor comments (4)
- [§0.2 / Conjecture 0.2 / Corollary 11.3] Typographical errors: 'comples' in Conjecture 0.2 and 'compex' in Theorem 9.9(10b) and Corollary 11.3(10) should be 'complex.'
- [§11, definition of A/R-cotorsion] The sentence 'Ext^1_A(F, C)=0 for all R-projective flat left A-modules C' uses the letter C both for the module being tested and for the class of test modules; the second C should be F (or another letter) for clarity.
- [§5, Remark 5.4] Minor typo: 'arizing' should be 'arising'.
- [§8, Corollary 8.6] The notation D^{∅=bco}(E) is introduced only on the spot; it would help to state explicitly before the diagram that the superscript is a shorthand for 'Ac(E)=Ac^{bco}(E), hence D(E)=D^{bco}(E)'.
Circularity Check
No circular derivation found: the central equivalences are proved internally from stated hypotheses; heavy self-citation supplies hypotheses rather than conclusions.
full rationale
I walked the derivation chain and found no step in which a claimed prediction or first-principles result is equivalent to its input by construction. Corollary 4.8 (D^bco(A) ≃ Hot(A∩B) ≃ D^bctr(B)) is derived from Propositions 2.1–2.4, whose cotorsion-pair constructions are credited to Gillespie and are proved in the text using the Eklof–Trlifaj/St’ovíček completeness theorems. Theorem 5.3, giving the adjunction for nested pairs, is proved by explicit construction of the functors and homotopy-category quasi-isomorphisms. Theorem 9.9 does not reduce to the definition of 'flat type': although Definition 7 (flat-type cotorsion pair) is formulated via Corollary 7.8, the equivalence of conditions (1) and (10) is established by a genuine argument routing through the recollement/adjunction structure (Lemmas 9.1–9.5, Theorem 9.8); the converse direction checks Lemma 7.4(1) and 7.5(1) using the flat-type hypothesis on (Flat,Cot). The periodicity properties in (10) are not assumed in the definition of flat-type for the sandwiched pair, and the proof that (1) implies (10) passes through the triangulated equivalences rather than being true by fiat. The main load-bearing dependencies are imported hypotheses: the flat-type property of R-Mod_flat (Example 8.8) is supported by Neeman [33] and Bazzoni–Cortés-Izurdiaga–Estrada [2] together with the paper’s Corollaries 4.6–4.7, so it is grounded in external periodicity theorems; the flat-type property of X-Qcoh_flat^cta (Example 8.9) is cited to the unpublished manuscript [36, Cor. 5.5.11], and Example 8.10 explicitly states that the analogous verification for A-Qcoh 'has not been worked out yet.' These are dependency gaps and unrefereed self-citations that make the scheme-level applications conditional, but they do not make the derivation circular: the cited statements are parameter-free and their assumptions do not include the target results of the present paper. No fitted parameter is renamed as a prediction, no known result is merely re-coordinatized, and no uniqueness claim is imported from the authors’ prior work to force a choice. The central results are self-contained conditional theorems; the score reflects the heavy reliance on the author’s own unpublished manuscripts and the explicit open verification in Example 8.10, not any identified circular step.
Assumptions & free parameters
assumptions (8)
- standard math Eklof-Trlifaj theorem: a cotorsion pair generated by a set of objects in a locally presentable abelian category is complete, and its left class is Fil(S)^⊕
- standard math Eklof lemma: the class ⊥1B is closed under transfinitely iterated extensions
- domain assumption The flat cotorsion pair (R-Mod_flat, R-Mod_cot) is of flat type
- domain assumption The exact category of flat contraadjusted quasi-coherent sheaves X-Qcoh^cta_flat is of flat type
- standard math Cotorsion periodicity theorem: in an acyclic complex of cotorsion modules, the cocycles are cotorsion
- standard math Flat and projective periodicity: in an acyclic complex of projective modules with flat cocycles, the cocycles are projective
- standard math Gillespie's theorem: the two cotorsion pairs from Propositions 2.1-2.2 define a unique abelian model structure on Com(K)
- standard math Grothendieck categories have enough injectives; Com(K) is locally presentable when K is
Cite this review
Pith. "Pith review of Coderived and contraderived categories for a cotorsion pair, flat-type cotorsion pairs, and relative periodicity." pith.science (2026). https://pith.science/paper/VEGPQKMM
@misc{pith2026250907645,
author = {Pith},
title = {Pith review of: Coderived and contraderived categories for a cotorsion pair, flat-type cotorsion pairs, and relative periodicity},
year = {2026},
howpublished = {\url{https://pith.science/paper/VEGPQKMM}},
note = {Machine review of arXiv:2509.07645}
}
abstract
Given a hereditary complete cotorsion pair $(\mathsf A,\mathsf B)$ generated by a set of objects in a Grothendieck category $\mathsf K$, we construct a natural equivalence between the Becker coderived category of the left-hand class $\mathsf A$ and the Becker contraderived category of the right-hand class $\mathsf B$. We show that a nested pair of cotorsion pairs $(\mathsf A_1,\mathsf B_1)\le(\mathsf A_2,\mathsf B_2)$ provides an adjunction between the related co/contraderived categories, which is induced by a Quillen adjunction between abelian model structures. Then we specialize to the cotorsion pairs $(\mathsf F,\mathsf C)$ sandwiched between the projective and the flat cotorsion pairs in a module category, and prove that the related co/contraderived categories for $(\mathsf F,\mathsf C)$ are the same as for the projective and flat cotorsion pairs if and only if two periodicity properties hold for $\mathsf F$ and $\mathsf C$. The same applies to the cotorsion pairs sandwiched between the very flat and the flat cotorsion pairs in the category of quasi-coherent sheaves over a quasi-compact semi-separated scheme. More generally, we define and discuss cotorsion pairs of the very flat type and of the flat type in Grothendieck categories (as well as exact categories of the flat type), and work with a cotorsion pair sandwiched between one of the very flat type and one of the flat type. The motivating examples of the classes of flaprojective modules and relatively cotorsion modules for a ring homomorphism are discussed, and periodicity conjectures formulated for them.
Reference graph
Works this paper leans on
-
[41]
L. Positselski. Philosophy of contraherent cosheaves. Electronic preprintarXiv:2311.14179 [math.AG]
-
[1]
Ad´ amek, J
J. Ad´ amek, J. Rosick´ y. Locally presentable and accessible categories. London Math. Society Lecture Note Series 189, Cambridge University Press, 1994
1994
-
[2]
S. Bazzoni, M. Cort´ es-Izurdiaga, S. Estrada. Periodic modules and acyclic complexes.Algebras and Represent. Theory23, #5, p. 1861–1883, 2020.arXiv:1704.06672 [math.RA]
arXiv 2020
-
[3]
S. Bazzoni, M. Hrbek, L. Positselski. Fp-projective periodicity.Journ. of Pure and Appl. Algebra 228, #3, article ID 107497, 24 pp., 2024.arXiv:2212.02300 [math.CT]
work page Pith review arXiv 2024
-
[4]
H. Becker. Models for singularity categories.Advances in Math.254, p. 187–232, 2014. arXiv:1205.4473 [math.CT] 56
work page Pith review arXiv 2014
-
[5]
D. J. Benson, K. R. Goodearl. Periodic flat modules, and flat modules for finite groups.Pacific Journ. of Math.196, #1, p. 45–67, 2000
work page 2000
- [6]
- [7]
Show all 56 references
-
[8]
L. W. Christensen, S. Estrada, P. Thompson. The stable category of Gorenstein flat sheaves on a noetherian scheme.Proc. of the Amer. Math. Soc.149, #2, p. 525–538, 2021. arXiv:1904.07661 [math.AC]
2021 arXiv
-
[9]
L. W. Christensen, H. Holm. The direct limit closure of perfect complexes.Journ. of Pure and Appl. Algebra219, #3, p. 449–463, 2015.arXiv:1301.0731 [math.RA]
2015 arXiv
-
[10]
Colpi, K
R. Colpi, K. R. Fuller. Tilting objects in abelian categories and quasitilted rings.Trans. of the Amer. Math. Soc.359, #2, p. 741–765, 2007
2007
-
[11]
ˇCoupek, J
P. ˇCoupek, J. ˇSt ’ov ´ ıˇ cek. Cotilting sheaves on Noetherian schemes.Math. Zeitschrift296, #1–2, p. 275–312, 2020.arXiv:1707.01677 [math.AG]
2020 arXiv
-
[12]
A. I. Efimov, L. Positselski. Coherent analogues of matrix factorizations and relative singu- larity categories.Algebra and Number Theory9, #5, p. 1159–1292, 2015.arXiv:1102.0261 [math.CT]
2015 arXiv
-
[13]
P. C. Eklof, J. Trlifaj. How to make Ext vanish.Bull. of the London Math. Soc.33, #1, p. 41–51, 2001
2001
-
[14]
E. Enochs. Flat covers and flat cotorsion modules.Proc. of the Amer. Math. Soc.92, #2, p. 179–184, 1984
1984
-
[15]
Enochs, S
E. Enochs, S. Estrada. Relative homological algebra in the category of quasi-coherent sheaves. Advances in Math.194, #2, p. 284–295, 2005
2005
-
[16]
Estrada, J
S. Estrada, J. Gillespie, S. Odaba¸ si. K-flatness in Grothendieck categories: application to quasi- coherent sheaves.Collectanea Math.76, #2, p. 435–454, 2025.arXiv:2306.04816 [math.AG]
2025 arXiv
-
[17]
Estrada, A
S. Estrada, A. Sl´ avik. Quillen equivalent models for the derived category of flats and the reso- lution property.Journ. Australian Math. Soc.110, #3, p. 302–320, 2021.arXiv:1708.05913 [math.AT]
2021 arXiv
-
[18]
Gabriel, M
P. Gabriel, M. Zisman. Calculus of fractions and homotopy theory. Springer-Verlag, Berlin– Heidelberg–New York, 1967
1967
-
[19]
J. R. Garc ´ ıa Rozas. Covers and envelopes in the category of complexes of modules. Chapman & Hall/CRC Research Notes in Math., 407, Boca Raton, FL, 1999
1999
-
[20]
Gillespie
J. Gillespie. The flat model structure onCh(R).Trans. of the Amer. Math. Soc.356, #8, p. 3369–3390, 2004
2004
-
[21]
Gillespie
J. Gillespie. Cotorsion pairs and degreewise homological model structures.Homology, Homo- topy, and Appl.10, #1, p. 283–304, 2008
2008
-
[22]
Gillespie
J. Gillespie. How to construct a Hovey triple from two cotorsion pairs.Fundamenta Math.230, #3, p. 281–289, 2015.arXiv:1406.2619 [math.AT]
2015 arXiv
-
[23]
Gillespie
J. Gillespie. Models for mock homotopy categories of projectives.Homology, Homotopy, and Appl.18, #1, p. 247–263, 2016.arXiv:1412.4082 [math.AT]
2016 arXiv
-
[24]
G¨ obel, J
R. G¨ obel, J. Trlifaj. Approximations and endomorphism algebras of modules. Second Revised and Extended Edition. De Gruyter Expositions in Mathematics 41, De Gruyter, Berlin–Boston, 2012
2012
-
[25]
Gruson, C
L. Gruson, C. U. Jensen. Dimensions cohomologiques reli´ ees aux foncteurs lim← − (i). P. Dubreil and M.-P. Malliavin Algebra Seminar, 33rd Year,Lecture Notes Math.867, 1981, p. 234–294
1981
-
[26]
Jørgensen
P. Jørgensen. The homotopy category of complexes of projective modules.Advances in Math. 193, #1, p. 223–232, 2005.arXiv:math.RA/0312088
2005
-
[27]
Kaplansky
I. Kaplansky. Projective modules.Annals of Math.68, #2, p. 372–377, 1958. 57
1958
-
[28]
Kashiwara, P
M. Kashiwara, P. Schapira. Categories and sheaves. Grundlehren der mathematischen Wis- senschaften, 332, Springer, 2006
2006
-
[29]
B. Keller. Derived categories and their uses. In: M. Hazewinkel, Ed.,Handbook of algebra, vol. 1, 1996, p. 671–701
1996
-
[30]
H. Krause. The stable derived category of a Noetherian scheme.Compositio Math.141, #5, p. 1128–1162, 2005.arXiv:math.AG/0403526
2005
-
[31]
D. Murfet. Derived categories of quasi-coherent sheaves. Notes, October 2006. Available from http://www.therisingsea.org/notes
2006
-
[32]
A. Neeman. The derived category of an exact category.Journ. of Algebra135, #2, p. 388–394, 1990
1990
-
[33]
A. Neeman. The homotopy category of flat modules, and Grothendieck duality.Inventiones Math.174, #2, p. 255–308, 2008
2008
-
[34]
Positselski
L. Positselski. Homological algebra of semimodules and semicontramodules: Semi-infinite homological algebra of associative algebraic structures. Appendix C in collaboration with D. Rumynin; Appendix D in collaboration with S. Arkhipov. Monografie Matematyczne vol. 70, Birkh¨ a...
2010 arXiv
-
[35]
Positselski
L. Positselski. Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence.Memoirs of the American Math. Society212, #996, 2011. vi+133 pp. arXiv:0905.2621 [math.CT]
2011 arXiv
-
[36]
Positselski
L. Positselski. Contraherent cosheaves on schemes. Electronic preprintarXiv:1209.2995v24 [math.CT]
-
[37]
Positselski
L. Positselski. Relative nonhomogeneous Koszul duality. Frontiers in Mathematics, Birkh¨ auser/Springer Nature, Cham, Switzerland, 2021. xxix+278 pp.arXiv:1911.07402 [math.RA]
2021 arXiv
-
[38]
Positselski
L. Positselski. An explicit self-dual construction of complete cotorsion pairs in the relative context.Rendiconti Semin. Matem. Univ. Padova149, p. 191–253, 2023.arXiv:2006.01778 [math.RA]
2023 arXiv
-
[39]
Positselski
L. Positselski. Differential graded Koszul duality: An introductory survey.Bulletin of the Lon- don Math. Society55, #4, p. 1551–1640, 2023.arXiv:2207.07063 [math.CT]
2023 arXiv
-
[40]
Positselski
L. Positselski. Local, colocal, and antilocal properties of modules and complexes over commu- tative rings.Journ. of Algebra646, p. 100–155, 2024.arXiv:2212.10163 [math.AC]
2024 arXiv
-
[42]
Positselski
L. Positselski. Roos axiom holds for quasi-coherent sheaves. Electronic preprint arXiv:2407.13651 [math.AG]
-
[43]
Positselski.D-Ω duality on the contra side
L. Positselski.D-Ω duality on the contra side. Electronic preprintarXiv:2504.18460v12 [math.AG]
-
[44]
Positselski
L. Positselski. A relative version of Bass’ theorem about finite-dimensional algebras. Electronic preprintarXiv:2507.15425 [math.RA]
-
[45]
Positselski, J
L. Positselski, J. Rosick´ y. Covers, envelopes, and cotorsion theories in locally presentable abelian categories and contramodule categories.Journ. of Algebra483, p. 83–128, 2017. arXiv:1512.08119 [math.CT]
2017 arXiv
-
[46]
Positselski, A
L. Positselski, A. Sl´ avik. Flat morphisms of finite presentation are very flat.Annali di Matem. Pura ed Appl.199, #3, p. 875–924, 2020.arXiv:1708.00846 [math.AC]
2020 arXiv
-
[47]
Positselski, J
L. Positselski, J. ˇSt ’ov ´ ıˇ cek. Derived, coderived, and contraderived categories of locally pre- sentable abelian categories.Journ. of Pure and Appl. Algebra226, #4, article ID 106883, 2022, 39 pp.arXiv:2101.10797 [math.CT]
2022 arXiv
-
[48]
Positselski, J
L. Positselski, J. ˇSt ’ov ´ ıˇ cek. Coderived and contraderived categories of locally presentable abelian DG-categories.Math. Zeitschrift308, #1, article no. 14, 70 pp., 2024.arXiv:2210.08237 [math.CT] 58
2024 arXiv
-
[49]
Positselski, J
L. Positselski, J. ˇSt ’ov ´ ıˇ cek. Flat quasi-coherent sheaves as directed colimits, and quasi- coherent cotorsion periodicity.Algebras and Represent. Theory27, #6, p. 2267–2293, 2024. arXiv:2212.09639 [math.AG]
2024 arXiv
-
[50]
platification
M. Raynaud, L. Gruson. Crit` eres de platitude et de projectivit´ e: Techniques de “platification” d’un module.Inventiones Math.13, #1–2, p. 1–89, 1971
1971
-
[51]
L. Salce. Cotorsion theories for abelian groups.Symposia Math.XXIII, Academic Press, London–New York, 1979, p. 11–32
1979
-
[52]
ˇSt ’ov ´ ıˇ cek
J. ˇSt ’ov ´ ıˇ cek. Deconstructibility and the Hill Lemma in Grothendieck categories.Forum Math. 25, #1, p. 193–219, 2013.arXiv:1005.3251 [math.CT]
2013 arXiv
-
[53]
ˇSt ’ov ´ ıˇ cek
J. ˇSt ’ov ´ ıˇ cek. Exact model categories, approximation theory, and cohomology of quasi-coherent sheaves.Advances in representation theory of algebras, p. 297–367, EMS Ser. Congr. Rep., Eur. Math. Soc., Z¨ urich, 2013.arXiv:1301.5206 [math.CT]
2013 arXiv
-
[54]
ˇSt ’ov ´ ıˇ cek
J. ˇSt ’ov ´ ıˇ cek. Derived equivalences induced by big cotilting modules.Advances in Math.263, p. 45–87, 2014.arXiv:1308.1804 [math.CT]
2014 arXiv
-
[55]
ˇSt ’ov ´ ıˇ cek
J. ˇSt ’ov ´ ıˇ cek. On purity and applications to coderived and singularity categories. Electronic preprintarXiv:1412.1615 [math.CT]
-
[56]
X. Zhu. Resolving resolution dimensions.Algebras and Represent. Theory16, #4, p. 1165–1191, 2013. Institute of Mathematics, Czech Academy of Sciences, ˇZitn´a 25, 115 67 Prague 1, Czech Republic Email address:positselski@math.cas.cz 59
2013
Reviewed August 4, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.