Differential graded Brauer groups
DOI:
https://doi.org/10.33044/revuma.4034Abstract
We consider central simple $K$-algebras which happen to be differential graded $K$-algebras. Two such algebras $A$ and $B$ are considered equivalent if there are bounded complexes of finite-dimensional $K$-vector spaces $C_A$ and $C_B$ such that the differential graded algebras $A\otimes_K \mathrm{End}_K^\bullet(C_A)$ and $B\otimes_K \mathrm{End}_K^\bullet(C_B)$ are isomorphic. Equivalence classes form an abelian group, which we call the dg Brauer group. We prove that this group is isomorphic to the ordinary Brauer group of the field $K$.
Downloads
References
S. T. Aldrich and J. R. García Rozas, Exact and semisimple differential graded algebras, Comm. Algebra 30 no. 3 (2002), 1053–1075. DOI MR Zbl
H. Cartan, DGA-algèbres et DGA-modules, Sémin. Henri Cartan, tome 7, no. 1 (1954–1955), exp. no. 2, 1–9. Available at https://www.numdam.org/item/SHC_1954-1955__7_1_A2_0/.
S. Dăscălescu, B. Ion, C. Năstăsescu, and J. Rios Montes, Group gradings on full matrix rings, J. Algebra 220 no. 2 (1999), 709–728. DOI MR Zbl
P. Gupta, Y. Kaur, and A. Singh, Differential central simple algebras, in Geometry, groups and mathematical philosophy, Contemp. Math. 811, American Mathematical Society, Providence, RI, 2025, pp. 113–121. DOI MR
I. N. Herstein, Noncommutative rings, The Carus Mathematical Monographs 15, Mathematical Association of America, distributed by Wiley, New York, 1968. MR Zbl
B. Keller, Derived categories and tilting, in Handbook of tilting theory, London Math. Soc. Lecture Note Ser. 332, Cambridge University Press, Cambridge, 2007, pp. 49–104. DOI MR
D. Orlov, Finite-dimensional differential graded algebras and their geometric realizations, Adv. Math. 366 (2020), article no. 107096. DOI MR Zbl
The Stacks project authors, The Stacks project, 2025, https://stacks.math.columbia.edu.
M. K. Turbow, Structure theory of graded central simple algebras, Ph.D. thesis, University of Georgia, 2016.
F. Van Oystaeyen and Y. Zhang, The Brauer group of a braided monoidal category, J. Algebra 202 no. 1 (1998), 96–128. DOI MR Zbl
C. T. C. Wall, Graded Brauer groups, J. Reine Angew. Math. 213 (1964), 187–199. DOI MR Zbl
A. Zimmermann, Differential graded orders, their class groups and idèles, [v1] 2023, [v3] 2024. arXiv:2310.06340v3 [math.RA].
Downloads
Published
Issue
Section
License
Copyright (c) 2025 Alexander Zimmermann
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal. The Journal may retract the paper after publication if clear evidence is found that the findings are unreliable as a result of misconduct or honest error.
