Derived equivalences of upper-triangular ring spectra via lax limits

Gustavo Jasso

doi:10.5802/crmath.559

Research article - Algebra, Representation theory

Derived equivalences of upper-triangular ring spectra via lax limits

Gustavo Jasso ¹

¹ Lund University, Centre for Mathematical Sciences, Box 118, 22100 Lund, Sweden

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 279-285.

Abstract
Résumé

We extend a theorem of Ladkani concerning derived equivalences between upper-triangular matrix rings to ring spectra. Our result also extends an analogous theorem of Maycock for differential graded algebras. We illustrate the main result with certain canonical equivalences determined by a smooth or proper ring spectrum.

Nous étendons un théorème de Ladkani concernant les équivalences dérivées entre les anneaux à matrice triangulaire supérieure aux spectres en anneaux. Notre résultat étend également un théorème analogue de Maycock pour les algèbres différentielles graduées. Nous illustrons le résultat principal par certaines équivalences canoniques déterminés par un spectre en anneaux lisse ou propre.

Received: 2023-06-23
Revised: 2023-08-30
Accepted: 2023-08-30
Published online: 2024-05-02

DOI: 10.5802/crmath.559

Classification: 18G80
Keywords: Upper-triangular matrix ring, derived equivalences, reflection functors, ring spectrum
Mots-clés : Anneaux de matrices triangulaires supérieures, équivalences dérivées, foncteurs de réflexion, spectre en anneaux

Author's affiliations:

Gustavo Jasso ¹

¹ Lund University, Centre for Mathematical Sciences, Box 118, 22100 Lund, Sweden

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Gustavo Jasso},
     title = {Derived equivalences of upper-triangular ring spectra via lax limits},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {279--285},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.559},
     language = {en},
}

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Gustavo Jasso. Derived equivalences of upper-triangular ring spectra via lax limits. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 279-285. doi : 10.5802/crmath.559. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.559/

References
Cited by

[1] Ioseph N. Bernšteĭn; Israilʼ M. Gelʼfand; V. A. Ponomarev Coxeter functors, and Gabriel’s theorem, Usp. Mat. Nauk, Volume 28 (1973) no. 2(170), pp. 19-33 | MR

[2] Tristan Bozec; Damien Calaque; Sarah Scherotzke Relative critical loci and quiver moduli (to appear in Ann. Sci. École Norm. Sup.) | arXiv

[3] Christopher Brav; Tobias Dyckerhoff Relative Calabi–Yau structures, Compos. Math., Volume 155 (2019) no. 2, pp. 372-412 | DOI | MR | Zbl

[4] Christopher Brav; Tobias Dyckerhoff Relative Calabi–Yau structures II: shifted Lagrangians in the moduli of objects, Sel. Math., New Ser., Volume 27 (2021) no. 4, 63, 45 pages | DOI | MR

[5] Tobias Dyckerhoff; Gustavo Jasso; Tashi Walde Generalised BGP reflection functors via the Grothendieck construction, Int. Math. Res. Not. (2021) no. 20, pp. 15733-15745 | DOI | MR

[6] David Gepner; Rune Haugseng; Thomas Nikolaus Lax colimits and free fibrations in $\infty$ -categories, Doc. Math., Volume 22 (2017), pp. 1225-1266 | DOI | MR

[7] Victor Ginzburg Calabi–Yau algebras (2006) | arXiv

[8] Bernhard Keller Deformed Calabi–Yau completions, J. Reine Angew. Math., Volume 654 (2011), pp. 125-180 (with an appendix by Michel Van den Bergh) | DOI | MR

[9] Bernhard Keller; Yilin Wu Relative cluster categories and Higgs categories with infinite-dimensional morphism spaces (2023) (with an appendix Chris Fraser and Keller, Bernhard) | arXiv

[10] Maxim Kontsevich; Yan Soibelman Notes on $A_{\infty}$ -algebras, $A_{\infty}$ -categories and non-commutative geometry, Homological mirror symmetry (Lecture Notes in Physics), Volume 757, Springer, 2006, pp. 153-219 | MR

[11] Alexander Kuznetsov; Valery A. Lunts Categorical resolutions of irrational singularities, Int. Math. Res. Not. (2015) no. 13, pp. 4536-4625 | DOI | MR

[12] Sefi Ladkani Derived equivalences of triangular matrix rings arising from extensions of tilting modules, Algebr. Represent. Theory, Volume 14 (2011) no. 1, pp. 57-74 | DOI | MR

[13] Jacob Lurie Higher topos theory, Annals of Mathematics Studies, 170, Princeton University Press, 2009, xviii+925 pages | DOI | MR

[14] Jacob Lurie Higher Algebra, 2017 (available online at the author’s webpage: https://www.math.ias.edu/~lurie/papers/HA.pdf)

[15] Jacob Lurie Spectral Algebraic Geometry, 2018 (available online at the author’s website: https://www.math.ias.edu/~lurie/papers/SAG-rootfile.pdf)

[16] Daniel Maycock Derived equivalences of upper triangular differential graded algebras, Commun. Algebra, Volume 39 (2011) no. 7, pp. 2367-2387 | DOI | MR

[17] Claus Michael Ringel Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, 1099, Springer, 1984, xiii+376 pages | DOI | MR

[18] Vladimir Sosnilo Regularity of spectral stacks and discreteness of weight-hearts, Q. J. Math., Volume 73 (2022) no. 1, pp. 23-44 | DOI | MR

[19] Bertrand Toën Derived algebraic geometry, EMS Surv. Math. Sci., Volume 1 (2014) no. 2, pp. 153-240 | DOI | MR

[20] Yilin Wu Categorification of ice quiver mutation, Math. Z., Volume 304 (2023) no. 1, 11, 42 pages | DOI | MR

[21] Yilin Wu Relative cluster categories and Higgs categories, Adv. Math., Volume 424 (2023), 109040, 112 pages | DOI | MR

[22] Wai-Kit Yeung Relative Calabi–Yau completions (2016) | arXiv

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