Volume 18, issue 2 (2014)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 21
Issue 6, 3191–3810
Issue 5, 2557–3190
Issue 4, 1931–2555
Issue 3, 1285–1930
Issue 2, 647–1283
Issue 1, 1–645

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Subscriptions
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Author Index
To Appear
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Brauer groups and étale cohomology in derived algebraic geometry

Benjamin Antieau and David Gepner

Geometry & Topology 18 (2014) 1149–1244
Bibliography
1 M Ando, A J Blumberg, D Gepner, Parameterized spectra, multiplicative Thom spectra, and the twisted Umkehr map, arXiv:1112.2203
2 B Antieau, D Gepner, J M Gómez, Actions of Eilenberg–Mac Lane spaces on $K$–theory spectra and uniqueness of twisted $K$–theory, to appear in Trans. Amer. Math. Soc.
3 D Arinkin, D Gaitsgory, Singular support of coherent sheaves, and the geometric Langlands conjecture, arXiv:1201.6343
4 M Artin, D Mumford, Some elementary examples of unirational varieties which are not rational, Proc. London Math. Soc. (3) 25 (1972) 75 MR0321934
5 M Auslander, O Goldman, The Brauer group of a commutative ring, Trans. Amer. Math. Soc. 97 (1960) 367 MR0121392
6 G Azumaya, On maximally central algebras, Nagoya Math. J. 2 (1951) 119 MR0040287
7 A Baker, A Lazarev, Topological Hochschild cohomology and generalized Morita equivalence, Algebr. Geom. Topol. 4 (2004) 623 MR2100675
8 A Baker, B Richter, M Szymik, Brauer groups for commutative $S$–algebras, J. Pure Appl. Algebra 216 (2012) 2361 MR2927172
9 D Ben-Zvi, J Francis, D Nadler, Integral transforms and Drinfeld centers in derived algebraic geometry, J. Amer. Math. Soc. 23 (2010) 909 MR2669705
10 M Van den Bergh, Three-dimensional flops and noncommutative rings, Duke Math. J. 122 (2004) 423 MR2057015
11 P Berthelot, A Grothendieck, L Illusie, Théorie des intersections et théorème de Riemann–Roch, Lecture Notes in Math. 225, Springer (1971) MR0354655
12 A J Blumberg, D Gepner, G Tabuada, A universal characterization of higher algebraic $K$–theory, Geom. Topol. 17 (2013) 733 MR3070515
13 M Bökstedt, A Neeman, Homotopy limits in triangulated categories, Compositio Math. 86 (1993) 209 MR1214458
14 A Bondal, M van den Bergh, Generators and representability of functors in commutative and noncommutative geometry, Mosc. Math. J. 3 (2003) 1, 258 MR1996800
15 F Borceux, E Vitale, Azumaya categories, Appl. Categ. Structures 10 (2002) 449 MR1937232
16 A K Bousfield, D M Kan, Homotopy limits, completions and localizations, Lecture Notes in Math. 304, Springer (1972) MR0365573
17 K S Brown, S M Gersten, Algebraic $K\!$–theory as generalized sheaf cohomology, from: "Algebraic $K\!$–theory, I: Higher $K\!$–theories", Lecture Notes in Math. 341, Springer (1973) 266 MR0347943
18 D C Cisinski, G Tabuada, Symmetric monoidal structure on non-commutative motives, J. $K$–Theory 9 (2012) 201 MR2922389
19 F R DeMeyer, The Brauer group of affine curves, from: "Brauer groups", Springer (1976) 16 MR0437537
20 C Deninger, A proper base change theorem for nontorsion sheaves in étale cohomology, J. Pure Appl. Algebra 50 (1988) 231 MR938616
21 J W Duskin, The Azumaya complex of a commutative ring, from: "Categorical algebra and its applications" (editor F Borceux), Lecture Notes in Math. 1348, Springer (1988) 107 MR975963
22 D Edidin, B Hassett, A Kresch, A Vistoli, Brauer groups and quotient stacks, Amer. J. Math. 123 (2001) 761 MR1844577
23 A D Elmendorf, I Kriz, M A Mandell, J P May, Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs 47, Amer. Math. Soc. (1997) MR1417719
24 H Fausk, Picard groups of derived categories, J. Pure Appl. Algebra 180 (2003) 251 MR1966659
25 O Gabber, Some theorems on Azumaya algebras, from: "The Brauer group" (editors M Kervaire, M Ojanguren), Lecture Notes in Math. 844, Springer (1981) 129 MR611868
26 D Gepner, T Lawson, Brauer groups and Galois cohomology of commutative ring spectra, in preparation
27 P G Goerss, J F Jardine, Simplicial homotopy theory, Progress in Mathematics 174, Birkhäuser (1999) MR1711612
28 R Gordon, A J Power, R Street, Coherence for tricategories, Mem. Amer. Math. Soc. 558 (1995) MR1261589
29 A Grothendieck, Éléments de géométrie algébrique, I: Le langage des schémas, Inst. Hautes Études Sci. Publ. Math. (1960) 228 MR0217083
30 A Grothendieck, Éléments de géométrie algébrique, IV: Étude locale des schémas et des morphismes de schémas IV, Inst. Hautes Études Sci. Publ. Math. (1967) 361 MR0238860
31 A Grothendieck, Le groupe de Brauer, I: Algèbres d'Azumaya et interprétations diverses, from: "Dix exposés sur la cohomologie des schémas", North-Holland (1968) 46 MR0244269
32 A Grothendieck, Le groupe de Brauer, II: Théorie cohomologique, from: "Dix exposés sur la cohomologie des schémas", North-Holland (1968) 67 MR0244270
33 A Grothendieck, Le groupe de Brauer, III: Exemples et compléments, from: "Dix exposés sur la cohomologie des schémas", North-Holland (1968) 88 MR0244271
34 M J Hopkins, M Mahowald, H Sadofsky, Constructions of elements in Picard groups, from: "Topology and representation theory" (editors E M Friedlander, M E Mahowald), Contemp. Math. 158, Amer. Math. Soc. (1994) 89 MR1263713
35 N Johnson, Azumaya objects in triangulated bicategories, to appear in Journal of Homotopy and Related Structures
36 A J de Jong, A result of Gabber, preprint
37 M Kapranov, Noncommutative geometry based on commutator expansions, J. Reine Angew. Math. 505 (1998) 73 MR1662244
38 A Lazarev, Homotopy theory of $A_\infty$ ring spectra and applications to $M\mathrm{U}$–modules, $K$–Theory 24 (2001) 243 MR1876800
39 M Lieblich, Moduli of twisted sheaves and generalized Azumaya algebras, PhD thesis, Mass. Inst. of Tech. (2004)
40 J Lurie, Derived algebraic geometry, PhD thesis, Mass. Inst. of Tech. (2004)
41 J Lurie, Higher topos theory, Annals of Mathematics Studies 170, Princeton Univ. Press (2009) MR2522659
42 J Lurie, Derived algebraic geometry VII: Spectral schemes (2011)
43 J Lurie, Derived algebraic geometry XI: Descent theorems (2011)
44 J Lurie, Derived algebraic geometry XIV: Representability theorems (2012)
45 J Lurie, Higher algebra (2012)
46 H Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics 8, Cambridge Univ. Press (1989) MR1011461
47 A Neeman, The connection between the $K\!$–theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel, Ann. Sci. École Norm. Sup. 25 (1992) 547 MR1191736
48 A Neeman, The Grothendieck duality theorem via Bousfield's techniques and Brown representability, J. Amer. Math. Soc. 9 (1996) 205 MR1308405
49 P Pandit, Moduli problems in derived noncommutative geometry, PhD thesis, University of Pennsylvania (2011)
50 S Schwede, The stable homotopy category is rigid, Ann. of Math. 166 (2007) 837 MR2373374
51 S Schwede, B Shipley, A uniqueness theorem for stable homotopy theory, Math. Z. 239 (2002) 803 MR1902062
52 S Schwede, B Shipley, Stable model categories are categories of modules, Topology 42 (2003) 103 MR1928647
53 C Simpson, Algebraic (geometric) $n$–stacks, arXiv:absalg-geom/9609014
54 M Szymik, Brauer spaces for commutative rings and structured ring spectra, arXiv:1110.2956
55 R W Thomason, The classification of triangulated subcategories, Compositio Math. 105 (1997) 1 MR1436741
56 R W Thomason, T Trobaugh, Higher algebraic $K\!$–theory of schemes and of derived categories, from: "The Grothendieck Festschrift, Vol. III" (editors P Cartier, L Illusie, N M Katz, G Laumon, K A Ribet), Progr. Math. 88, Birkhäuser (1990) 247 MR1106918
57 B Toën, Derived Azumaya algebras and generators for twisted derived categories, Invent. Math. 189 (2012) 581 MR2957304
58 B Toën, M Vaquié, Moduli of objects in dg–categories, Ann. Sci. École Norm. Sup. 40 (2007) 387 MR2493386
59 B Toën, G Vezzosi, Homotopical algebraic geometry, II: Geometric stacks and applications, Mem. Amer. Math. Soc. 193 (2008) MR2394633
60 E M Vitale, The Brauer and Brauer–Taylor groups of a symmetric monoidal category, Cahiers Topologie Géom. Différentielle Catég. 37 (1996) 91 MR1394505