Comptes Rendus
Algebra, Algebraic geometry
Tensor weight structures and t-structures on the derived categories of schemes
Comptes Rendus. Mathématique, Volume 361 (2023), pp. 877-888.

We give a condition which characterises those weight structures on a derived category which come from a Thomason filtration on the underlying scheme. Weight structures satisfying our condition will be called c -weight structures. More precisely, for a Noetherian separated scheme X, we give a bijection between the set of compactly generated c -weight structures on D(QcohX) and the set of Thomason filtrations of X. We achieve this classification in two steps. First, we show that the bijection [12, Theorem 4.10] restricts to give a bijection between the set of compactly generated c -weight structures and the set of compactly generated tensor t-structures. We then use our earlier classification of compactly generated tensor t-structures to obtain the desired result. We also study some immediate consequences of these classifications in the particular case of the projective line. We show that in contrast to the case of tensor t-structures, there are no non-trivial tensor weight structures on D b (Coh k 1 ).

On dégage une condition qui caractérise les structures de poids sur une catégorie dérivée qui proviennent d’une filtration de Thomason sur le schéma sous-jacent. Les structures de poids satisfaisant notre condition s’appelleront des c -structures de poids. Plus précisément, pour tout schéma séparé et noethérien X, nous construisons une bijection entre l’ensemble des c -structures de poids à engendrement compact sur D(QcohX) et l’ensemble des filtrations de Thomason sur X. La construction se fait en deux étapes. On montre d’abord que la bijection de [12, Theorem 4.10] donne par restriction une bijection entre l’ensemble des c -structures de poids à engendrement compact et l’ensemble des t-structures tensorielles à engendrement compact. Nous utilisons ensuite notre classification précédente des c -structures de poids à engendrement compact pour arriver au résultat. Nous étudions aussi quelques conséquences immédiates dans le cas particulier de la droite projective. Nous montrons que, contrairement au cas des t-structures tensorielles, il n’y a pas de structure de poids tensorielle non-triviale sur D b (Coh k 1 ).

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.450
Classification: 14F08, 18G80

Umesh V. Dubey 1; Gopinath Sahoo 1

1 Harish-Chandra Research Institute, A CI of Homi Bhabha National Institute, Chhatnag Road, Jhunsi, Prayagraj 211019, India
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{CRMATH_2023__361_G5_877_0,
     author = {Umesh V. Dubey and Gopinath Sahoo},
     title = {Tensor weight structures and t-structures on the derived categories of schemes},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {877--888},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.450},
     language = {en},
}
TY  - JOUR
AU  - Umesh V. Dubey
AU  - Gopinath Sahoo
TI  - Tensor weight structures and t-structures on the derived categories of schemes
JO  - Comptes Rendus. Mathématique
PY  - 2023
SP  - 877
EP  - 888
VL  - 361
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.450
LA  - en
ID  - CRMATH_2023__361_G5_877_0
ER  - 
%0 Journal Article
%A Umesh V. Dubey
%A Gopinath Sahoo
%T Tensor weight structures and t-structures on the derived categories of schemes
%J Comptes Rendus. Mathématique
%D 2023
%P 877-888
%V 361
%I Académie des sciences, Paris
%R 10.5802/crmath.450
%G en
%F CRMATH_2023__361_G5_877_0
Umesh V. Dubey; Gopinath Sahoo. Tensor weight structures and t-structures on the derived categories of schemes. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 877-888. doi : 10.5802/crmath.450. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.450/

[1] Leovigildo Alonso Tarrío; Ana Jeremías López; Manuel Saorín Compactly generated t-structures on the derived category of a Noetherian ring, J. Algebra, Volume 324 (2010) no. 3, pp. 313-346 | DOI | MR | Zbl

[2] Alexander A. Beĭlinson; Joseph Bernstein; Pierre Deligne Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) (Astérisque), Volume 100, Société Mathématique de France, 1982, pp. 5-171 | MR | Zbl

[3] Roman Bezrukavnikov Perverse coherent sheaves (after Deligne) (2000) | arXiv

[4] Mikhail V. Bondarko Weight structures vs. t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general), J. K-Theory, Volume 6 (2010) no. 3, pp. 387-504 | DOI | MR | Zbl

[5] Mikhail V. Bondarko; Vladimir A. Sosnilo On constructing weight structures and extending them to idempotent completions, Homology Homotopy Appl., Volume 20 (2018) no. 1, pp. 37-57 | DOI | MR | Zbl

[6] Stephen U. Chase Direct products of modules, Trans. Am. Math. Soc., Volume 97 (1960), pp. 457-473 | DOI | MR | Zbl

[7] Umesh V. Dubey; Gopinath Sahoo Compactly generated tensor t-structures on the derived category of a Noetherian scheme (2022) | arXiv

[8] Alexey L. Gorodentsev; S. A. Kuleshov; Alexei N. Rudakov t-stabilities and t-structures on triangulated categories, Izv. Ross. Akad. Nauk, Ser. Mat., Volume 68 (2004) no. 4, pp. 117-150 | DOI | MR

[9] Mark Hovey; John H. Palmieri; Neil P. Strickland Axiomatic stable homotopy theory, Mem. Am. Math. Soc., Volume 128 (1997) no. 610, p. x+114 | DOI | MR | Zbl

[10] Henning Krause; Greg Stevenson The derived category of the projective line, Spectral structures and topological methods in mathematics (EMS Series of Congress Reports), European Mathematical Society, 2019, pp. 275-297 | DOI | MR | Zbl

[11] David Pauksztello Compact corigid objects in triangulated categories and co-t-structures, Cent. Eur. J. Math., Volume 6 (2008) no. 1, pp. 25-42 | DOI | MR | Zbl

[12] Jan Šťovíček; David Pospíšil On compactly generated torsion pairs and the classification of co-t-structures for commutative noetherian rings, Trans. Am. Math. Soc., Volume 368 (2016) no. 9, pp. 6325-6361 | DOI | MR | Zbl

[13] Robert W. Thomason The classification of triangulated subcategories, Compos. Math., Volume 105 (1997) no. 1, pp. 1-27 | DOI | MR | Zbl

Cited by Sources:

Comments - Policy