Tensor weight structures and t-structures on the derived categories of schemes

Umesh V. Dubey; Gopinath Sahoo

doi:10.5802/crmath.450

Algèbre, Géométrie algébrique

Tensor weight structures and t-structures on the derived categories of schemes
[Structures de poids tensorielles et

t

-structures sur les catégories dérivées des schémas]

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

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 877-888.

Résumé
Abstract

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 $\otimes^{c}$ -structures de poids. Plus précisément, pour tout schéma séparé et noethérien $X$ , nous construisons une bijection entre l’ensemble des $\otimes^{c}$ -structures de poids à engendrement compact sur $D (Qcoh X)$ 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 $\otimes^{c}$ -structures de poids à engendrement compact et l’ensemble des t-structures tensorielles à engendrement compact. Nous utilisons ensuite notre classification précédente des $\otimes^{c}$ -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 $D^{b} (Coh ℙ_{k}^{1})$ .

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 $\otimes^{c}$ -weight structures. More precisely, for a Noetherian separated scheme $X$ , we give a bijection between the set of compactly generated $\otimes^{c}$ -weight structures on $D (Qcoh X)$ 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 $\otimes^{c}$ -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 $D^{b} (Coh ℙ_{k}^{1})$ .

Reçu le : 2022-10-14
Révisé le : 2022-11-25
Accepté le : 2022-11-25
Publié le : 2023-07-18

DOI : 10.5802/crmath.450

Classification : 14F08, 18G80

Affiliations des auteurs :

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

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     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},
}

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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/

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