logo CRAS
Comptes Rendus. Mathématique
Algebra
Gillespie’s questions and Grothendieck duality
Comptes Rendus. Mathématique, Volume 359 (2021) no. 5, pp. 593-607.

Gillespie posed two questions in [Front. Math. China 12 (2017) 97-115], one of which states that “for what rings R do we have K(AC)=K(R-Inj)?”. We give an answer to such a question. As applications, we obtain a new homological approach that unifies some well-known conditions of rings such that Krause’s recollement holds, and give an example to show that there exists a Gorenstein injective module which is not Gorenstein AC-injective. We also improve Neeman’s angle of view to the Grothendieck duality for derived categories of modules from the case of left Noether and right coherent rings such that all flat left modules have finite projective dimension to the case of left and right coherent rings.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.198
Classification: 18G35,  18G05,  18G20
Junpeng Wang 1; Zhongkui Liu 1; Gang Yang 2

1 Department of Mathematics, Northwest Normal University, Lanzhou 730070, People’s Republic of China
2 Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, People’s Republic of China
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{CRMATH_2021__359_5_593_0,
     author = {Junpeng Wang and Zhongkui Liu and Gang Yang},
     title = {Gillespie{\textquoteright}s questions and {Grothendieck} duality},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {593--607},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {5},
     year = {2021},
     doi = {10.5802/crmath.198},
     language = {en},
}
TY  - JOUR
TI  - Gillespie’s questions and Grothendieck duality
JO  - Comptes Rendus. Mathématique
PY  - 2021
DA  - 2021///
SP  - 593
EP  - 607
VL  - 359
IS  - 5
PB  - Académie des sciences, Paris
UR  - https://doi.org/10.5802/crmath.198
DO  - 10.5802/crmath.198
LA  - en
ID  - CRMATH_2021__359_5_593_0
ER  - 
%0 Journal Article
%T Gillespie’s questions and Grothendieck duality
%J Comptes Rendus. Mathématique
%D 2021
%P 593-607
%V 359
%N 5
%I Académie des sciences, Paris
%U https://doi.org/10.5802/crmath.198
%R 10.5802/crmath.198
%G en
%F CRMATH_2021__359_5_593_0
Junpeng Wang; Zhongkui Liu; Gang Yang. Gillespie’s questions and Grothendieck duality. Comptes Rendus. Mathématique, Volume 359 (2021) no. 5, pp. 593-607. doi : 10.5802/crmath.198. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.198/

[1] Driss Bennis Rings over which the class of Gorenstein flat modules is closed under extensions, Commun. Algebra, Volume 37 (2009) no. 3, pp. 855-868 | DOI | MR | Zbl

[2] Driss Bennis; Najib Mahdou Strongly Gorenstein projective, injective and flat modules, J. Pure Appl. Algebra, Volume 210 (2007) no. 2, pp. 437-445 | DOI | MR | Zbl

[3] Daniel Bravo; James Gillespie Absolutely clean, level, and Gorenstein AC-injective complexes, Commun. Algebra, Volume 44 (2016) no. 5, pp. 2213-2233 | DOI | MR | Zbl

[4] Daniel Bravo; James Gillespie; Mark Hovey The stable module category of a general ring (2014) (https://arxiv.org/abs/1405.5768)

[5] Xiao-Wu Chen Homotopy equivalences induced by balanced pairs, J. Algebra, Volume 324 (2010) no. 10, pp. 2718-2731 | DOI | MR | Zbl

[6] Nanqing Ding; Yuanlin Li; Lixin Mao Strongly Gorenstein flat modules, J. Aust. Math. Soc., Volume 86 (2009) no. 3, pp. 323-338 | DOI | MR | Zbl

[7] Ioannis Emmanouil On the finiteness of Gorenstein homological dimensions, J. Algebra, Volume 372 (2012), pp. 376-396 | DOI | MR | Zbl

[8] Ioannis Emmanouil On pure acyclic complexes, J. Algebra, Volume 465 (2016), pp. 190-213 | DOI | MR | Zbl

[9] Edgar E. Enochs; Overtoun M. G. Jenda Gorenstein injective and projective modules, Math. Z., Volume 220 (1995) no. 4, pp. 611-633 | DOI | MR | Zbl

[10] Edgar E. Enochs; Overtoun M. G. Jenda Relative Homological Algebra, de Gruyter Expositions in Mathematics, 30, Walter de Gruyter, 2000 | MR | Zbl

[11] Edgar E. Enochs; Overtoun M. G. Jenda; Blas Torrecillas Gorenstein flat modules, J. Nanjing Univ., Math. Biq., Volume 10 (1993) no. 1, pp. 1-9 | MR | Zbl

[12] Sergio Estrada; James Gillespie The projective stable category of a coherent scheme, Proc. R. Soc. Edinb., Sect. A, Math., Volume 149 (2019) no. 1, pp. 15-43 | DOI | MR | Zbl

[13] Sergio Estrada; Alina Iacob; Holly Zolt Acyclic complexes and Gorenstein rings, Algebra Colloq., Volume 27 (2020) no. 3, pp. 575-586 | DOI | MR | Zbl

[14] Nan Gao; Pu Zhang Gorenstein derived categories, J. Algebra, Volume 323 (2010) no. 7, pp. 2041-2057 | MR | Zbl

[15] Juan R. García Rozas Covers and Envelopes in the Category of Complexes of Modules, CRC Research Notes in Mathematics, 407, Chapman & Hall/CRC, 1999 | MR | Zbl

[16] James Gillespie The flat model structure on Ch(R), Trans. Am. Math. Soc., Volume 356 (2004) no. 8, pp. 3369-3390 | DOI | MR | Zbl

[17] James Gillespie Model Structures on Modules over Ding-Chen rings, Homology Homotopy Appl., Volume 12 (2010) no. 1, pp. 61-73 | DOI | MR | Zbl

[18] James Gillespie Gorenstein complexes and recollements from cotorsion pairs, Adv. Math., Volume 291 (2016), pp. 859-911 | DOI | MR | Zbl

[19] James Gillespie Models for homotopy categories of injectives and Gorenstein injectives, Commun. Algebra, Volume 45 (2017) no. 6, pp. 2520-2545 | DOI | MR | Zbl

[20] James Gillespie On Ding injective, Ding projective, and Ding flat modules and complexes, Rocky Mt. J. Math., Volume 47 (2017) no. 8, pp. 2641-2673 | MR | Zbl

[21] James Gillespie On the homotopy category of AC-injective complexes, Front. Math. China, Volume 12 (2017) no. 1, pp. 97-115 | DOI | MR | Zbl

[22] James Gillespie AC-Gorenstein rings and their stable module categories, J. Aust. Math. Soc., Volume 107 (2019) no. 2, pp. 181-198 | DOI | MR | Zbl

[23] Henrik Holm Gorenstein homological dimensions, J. Pure Appl. Algebra, Volume 189 (2004) no. 1-3, pp. 167-193 | DOI | MR | Zbl

[24] Srikanth Iyengar; Henning Krause Acyclicity versus total acyclicity for complexes over noetherian rings, Doc. Math., Volume 11 (2006), pp. 207-240 | MR | Zbl

[25] Peter Jørgensen The homotopy category of complexes of projective modules, Adv. Math., Volume 193 (2005) no. 1, pp. 223-232 | DOI | MR | Zbl

[26] Ellen Kirkman; James Kuzmanovich On the global dimension of fibre products, Pac. J. Math., Volume 134 (1988) no. 1, pp. 121-132 | DOI | MR | Zbl

[27] Henning Krause The stable derived category of a Noetherian scheme, Compos. Math., Volume 141 (2005) no. 5, pp. 1128-1162 | DOI | MR | Zbl

[28] Zhongkui Liu; Chunxia Zhang Gorenstein injective complexes of modules over Noetherian rings, J. Algebra, Volume 321 (2009) no. 5, pp. 1546-1554 | DOI | MR | Zbl

[29] Bo Lu; Zhenxing Di Gorenstein cohomology of N-complexes, J. Algebra Appl., Volume 19 (2020) no. 9, 2050174, 14 pages | MR | Zbl

[30] Lixin Mao; Nanqing Ding Gorenstein FP-injective and Gorenstein flat modules, J. Algebra, Volume 7 (2008) no. 4, pp. 497-506 | MR | Zbl

[31] Daniel Murfet The mock homotopy category of projectives and Grothendieck duality (2007) (available at www.therisingsea.org) (Ph. D. Thesis)

[32] Amnon Neeman The connection between the K-theory localisation theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel, Ann. Sci. Éc. Norm. Supér., Volume 25 (1992), pp. 547-566 | DOI | Numdam | Zbl

[33] Amnon Neeman The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Am. Math. Soc., Volume 9 (1996) no. 1, pp. 205-236 | DOI | MR | Zbl

[34] Amnon Neeman The homotopy category of flat modules, and Grothendieck duality, Invent. Math., Volume 174 (2008) no. 2, pp. 255-308 | DOI | MR | Zbl

[35] Jan Št’ovíček On purity and applications to coderived and singularity categories (2014) (https://arxiv.org/abs/1412.1615)

[36] J. P. Wang; Zhongkui Liu; Xiaoyan Yang A negative answer to a question of Gillespies, Sci. China, Math. (2018), pp. 1121-1130

[37] Gang Yang Gorenstein projective, injective and flat complexes, Acta Math. Sin., Volume 54 (2011) no. 3, pp. 451-460 | MR | Zbl

[38] Gang Yang; Sergio Estrada Characterizations of Ding injective complexes, Bull. Malays. Math. Sci. Soc., Volume 43 (2020) no. 3, pp. 2385-2398 | DOI | MR | Zbl

[39] Gang Yang; Zhongkui Liu; Li Liang Model structures on categories of complexes over Ding-Chen rings, Commun. Algebra, Volume 41 (2013) no. 1, pp. 50-69 | DOI | MR | Zbl

[40] Xiaoyan Yang; Zhongkui Liu Strongly Gorenstein projective, injective and flat modules, J. Algebra, Volume 320 (2008) no. 7, pp. 2659-2674 | DOI | MR | Zbl

[41] Xiaoyan Yang; Zhongkui Liu Gorenstein projective, injective, and flat complexes, Commun. Algebra, Volume 39 (2011) no. 5, pp. 1705-1721 | DOI | MR | Zbl

Cited by Sources: