Algebraic & Geometric Topology Volume 26, issue 5 (2026)

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ISSN (electronic): 1472-2739

ISSN (print): 1472-2747

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Brauer–Wall groups and truncated Picard spectra of $K$-theory

Jonathan Beardsley, Kiran Luecke and Jack Morava

Algebraic & Geometric Topology 26 (2026) 1749–1780

DOI: 10.2140/agt.2026.26.1749

Bibliography

1	A Amabel, A Debray, P J Haine, Differential cohomology: categories, characteristic classes, and connections, (2021) arXiv:2109.12250
2	M Ando, A J Blumberg, D Gepner, Twists of K-theory and TMF, from: "Superstrings, geometry, topology, and C^∗-algebras" (editors R S Doran, G Friedman, J Rosenberg), Proc. Sympos. Pure Math. 81, Amer. Math. Soc. (2010) 27 MR2681757
3	M Ando, A J Blumberg, D Gepner, Parametrized spectra, multiplicative Thom spectra and the twisted Umkehr map, Geom. Topol. 22 (2018) 3761 MR3890766
4	M Ando, A J Blumberg, D Gepner, M J Hopkins, C Rezk, An ∞-categorical approach to R-line bundles, R-module Thom spectra, and twisted R-homology, J. Topol. 7 (2014) 869 MR3252967
5	M Ando, M Hopkins, C Rezk, Multiplicative orientations of KO-theory and of the spectrum of topological modular forms, preprint (2010)
6	N A Baas, B I Dundas, B Richter, J Rognes, Stable bundles over rig categories, J. Topol. 4 (2011) 623 MR2832571
7	J Beardsley, On bialgebras, comodules, descent data and Thom spectra in ∞-categories, Homology Homotopy Appl. 25 (2023) 219 MR4689082
8	U Bunke, T Nikolaus, M Völkl, Differential cohomology theories as sheaves of spectra, J. Homotopy Relat. Struct. 11 (2016) 1 MR3462099
9	J Distler, D S Freed, G W Moore, Spin structures and superstrings, from: "Surveys in differential geometry, XV : Perspectives in mathematics and physics" (editors T Mrowka, S T Yau), Surv. Differ. Geom. 15, International Press (2011) 99 MR2815726
10	P Donovan, M Karoubi, Graded Brauer groups and K-theory with local coefficients, Inst. Hautes Études Sci. Publ. Math. 38 (1970) 5 MR282363
11	V Drinfeld, On a notion of ring groupoid, (2021) arXiv:2104.07090
12	S Eilenberg, S Mac Lane, On the groups H(Π,n), II : Methods of computation, Ann. of Math. (2) 60 (1954) 49 MR65162
13	A Ergus, Hopf-algebras and Hopf–Galois extensions in ∞-categories, PhD thesis, École polytechnique fédérale de Lausanne (2022)
14	D E Evans, U Pennig, Equivariant higher Dixmier–Douady theory for circle actions on UHF-algebras, Adv. Math. 410 (2022) 108745 MR4500637
15	D Freed, Bordism : old and new, lecture notes (2012)
16	D Freed, Lecture notes on k-theory and orientifolds, lecture notes (2012)
17	D S Freed, Anomalies and invertible field theories, from: "String-Math 2013" (editors R Donagi, M R Douglas, L Kamenova, M Roček), Proc. Sympos. Pure Math. 88, Amer. Math. Soc. (2014) 25 MR3330283
18	D S Freed, Short-range entanglement and invertible field theories, (2014) arXiv:1406.7278
19	D S Freed, M J Hopkins, Reflection positivity and invertible topological phases, Geom. Topol. 25 (2021) 1165 MR4268163
20	D S Freed, M J Hopkins, C Teleman, Consistent orientation of moduli spaces, from: "The many facets of geometry" (editors O García-Prada, J P Bourguignon, S Salamon), Oxford Univ. Press (2010) 395 MR2681705
21	D S Freed, M J Hopkins, C Teleman, Loop groups and twisted K-theory, I, J. Topol. 4 (2011) 737 MR2860342
22	S Galatius, U Tillmann, I Madsen, M Weiss, The homotopy type of the cobordism category, Acta Math. 202 (2009) 195 MR2506750
23	D Gepner, T Lawson, Brauer groups and Galois cohomology of commutative ring spectra, Compos. Math. 157 (2021) 1211 MR4270473
24	N Gurski, N Johnson, A M Osorno, M Stephan, Stable Postnikov data of Picard 2-categories, Algebr. Geom. Topol. 17 (2017) 2763 MR3704242
25	D Heard, V Stojanoska, K-theory, reality, and duality, J. K-Theory 14 (2014) 526 MR3349325
26	F Hebestreit, S Sagave, Homotopical and operator algebraic twisted K-theory, Math. Ann. 378 (2020) 1021 MR4163521
27	M Hill, personal communication (2022)
28	M A Hill, M J Hopkins, D C Ravenel, Equivariant stable homotopy theory and the Kervaire invariant problem, 40, Cambridge Univ. Press (2021) MR4273305
29	M Hopkins, J Lurie, On Brauer groups of Lubin–Tate spectra, I, preprint (2017)
30	M J Hopkins, I M Singer, Quadratic functions in geometry, topology, and M-theory, J. Differential Geom. 70 (2005) 329 MR2192936
31	M Joachim, Higher coherences for equivariant K-theory, from: "Structured ring spectra" (editors A Baker, B Richter), London Math. Soc. Lecture Note Ser. 315, Cambridge Univ. Press (2004) 87 MR2122155
32	N Johnson, A M Osorno, Modeling stable one-types, Theory Appl. Categ. 26 (2012) 20 MR2981952
33	M Karoubi, Bott periodicity in topological, algebraic and Hermitian K-theory, from: "Handbook of K-theory" (editors E M Friedlander, D R Grayson), Springer (2005) 111 MR2181822
34	P Kristel, M Ludewig, K Waldorf, 2-vector bundles, High. Struct. 9 (2025) 36 MR4918785
35	M L Laplaza, Coherence for distributivity, from: "Coherence in categories" (editor S MacLane), Lecture Notes in Math. 281, Springer (1972) 29 MR335598
36	J Lurie, Higher topos theory, 170, Princeton Univ. Press (2009) MR2522659
37	J Lurie, Derived algebraic geometry, XIV : Representability theorems, preprint (2012)
38	J Lurie, Higher algebra, preprint (2017)
39	J Lurie, Spectral algebraic geometry, preprint (2018)
40	A Mathew, Second homotopy group of the mod 2 Moore spectrum (2012)
41	J P May, E_∞ ring spaces and E_∞ ring spectra, 577, Springer (1977) 268 MR494077
42	D Mertsch, Geometric models of twisted K-theory based on bundle gerbes and algebra bundles, PhD thesis, Universität Greifswald (2020)
43	E k M Moutuou, Graded Brauer groups of a groupoid with involution, J. Funct. Anal. 266 (2014) 2689 MR3158706
44	E M Parker, The Brauer group of graded continuous trace C^∗-algebras, Trans. Amer. Math. Soc. 308 (1988) 115 MR946434
45	C Redden, String structures and canonical 3-forms, Pacific J. Math. 249 (2011) 447 MR2782680
46	C Rezk, When are homotopy colimits compatible with homotopy base change?, preprint (2014)
47	J Rognes, Galois extensions of structured ring spectra : stably dualizable groups, 898, Amer. Math. Soc. (2008) MR2387923
48	G Segal, Equivariant K-theory, Inst. Hautes Études Sci. Publ. Math. 34 (1968) 129 MR234452
49	C A Weibel, An introduction to homological algebra, 38, Cambridge Univ. Press (1994) MR1269324