Depth and detection for Noetherian unstable algebras

Heard, Drew

doi:10.1090/tran/8157

Depth and detection for Noetherian unstable algebras
HTML articles powered by AMS MathViewer

by Drew Heard PDF

Trans. Amer. Math. Soc. 373 (2020), 7429-7454 Request permission

Abstract:

For a connected Noetherian unstable algebra $R$ over the mod $p$ Steenrod algebra, we prove versions of theorems of Duflot and Carlson on the depth of $R$, originally proved when $R$ is the mod $p$ cohomology ring of a finite group. This recovers the aforementioned results, and also proves versions of them when $R$ is the mod $p$ cohomology ring of a compact Lie group, a profinite group with Noetherian cohomology, a Kac–Moody group, a discrete group of finite virtual cohomological dimension, as well as for certain other discrete groups. More generally, our results apply to certain finitely generated unstable $R$-modules. Moreover, we explain the results in the case of the $p$-local compact groups of Broto, Levi, and Oliver, as well as in the modular invariant theory of finite groups.

References

J. Aguadé, C. Broto, and D. Notbohm, Homotopy classification of spaces with interesting cohomology and a conjecture of Cooke. I, Topology 33 (1994), no. 3, 455–492. MR 1286926, DOI 10.1016/0040-9383(94)90023-X
Michael Aschbacher, Radha Kessar, and Bob Oliver, Fusion systems in algebra and topology, London Mathematical Society Lecture Note Series, vol. 391, Cambridge University Press, Cambridge, 2011. MR 2848834, DOI 10.1017/CBO9781139003841
Tobias Barthel, Natàlia Castellana, Drew Heard, and Gabriel Valenzuela, Stratification and duality for homotopical groups, Adv. Math. 354 (2019), 106733, 61. MR 3989930, DOI 10.1016/j.aim.2019.106733
Carlos Broto and Hans-Werner Henn, Some remarks on central elementary abelian $p$-subgroups and cohomology of classifying spaces, Quart. J. Math. Oxford Ser. (2) 44 (1993), no. 174, 155–163. MR 1222371, DOI 10.1093/qmath/44.2.155
Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
Carles Broto and Nitu Kitchloo, Classifying spaces of Kac-Moody groups, Math. Z. 240 (2002), no. 3, 621–649. MR 1924024, DOI 10.1007/s002090100391
Carles Broto, Ran Levi, and Bob Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc. 16 (2003), no. 4, 779–856. MR 1992826, DOI 10.1090/S0894-0347-03-00434-X
Carles Broto, Ran Levi, and Bob Oliver, Discrete models for the $p$-local homotopy theory of compact Lie groups and $p$-compact groups, Geom. Topol. 11 (2007), 315–427. MR 2302494, DOI 10.2140/gt.2007.11.315
M. P. Brodmann and R. Y. Sharp, Local cohomology, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 136, Cambridge University Press, Cambridge, 2013. An algebraic introduction with geometric applications. MR 3014449
C. Broto and S. Zarati, Nil-localization of unstable algebras over the Steenrod algebra, Math. Z. 199 (1988), no. 4, 525–537. MR 968318, DOI 10.1007/BF01161641
Dorra Bourguiba and Said Zarati, Depth and the Steenrod algebra, Invent. Math. 128 (1997), no. 3, 589–602. With an appendix by J. Lannes. MR 1452433, DOI 10.1007/s002220050152
James C. Cameron. On the Duflot filtration for equivariant cohomology rings and applications to group cohomology, arXiv:1711.05832 (2017).
Jon F. Carlson, Depth and transfer maps in the cohomology of groups, Math. Z. 218 (1995), no. 3, 461–468. MR 1324540, DOI 10.1007/BF02571916
Jon F. Carlson, Problems in the calculation of group cohomology, Computational methods for representations of groups and algebras (Essen, 1997) Progr. Math., vol. 173, Birkhäuser, Basel, 1999, pp. 107–120. MR 1714605
Andrew Chermak, Fusion systems and localities, Acta Math. 211 (2013), no. 1, 47–139. MR 3118305, DOI 10.1007/s11511-013-0099-5
Jon F. Carlson, Lisa Townsley, Luis Valeri-Elizondo, and Mucheng Zhang, Cohomology rings of finite groups, Algebra and Applications, vol. 3, Kluwer Academic Publishers, Dordrecht, 2003. With an appendix: Calculations of cohomology rings of groups of order dividing 64 by Carlson, Valeri-Elizondo and Zhang. MR 2028960, DOI 10.1007/978-94-017-0215-7
H. E. A. Eddy Campbell and David L. Wehlau, Modular invariant theory, Encyclopaedia of Mathematical Sciences, vol. 139, Springer-Verlag, Berlin, 2011. Invariant Theory and Algebraic Transformation Groups, 8. MR 2759466, DOI 10.1007/978-3-642-17404-9
W. G. Dwyer, H. R. Miller, and C. W. Wilkerson, Homotopical uniqueness of classifying spaces, Topology 31 (1992), no. 1, 29–45. MR 1153237, DOI 10.1016/0040-9383(92)90062-M
J. Duflot, Depth and equivariant cohomology, Comment. Math. Helv. 56 (1981), no. 4, 627–637. MR 656216, DOI 10.1007/BF02566231
W. G. Dwyer and C. W. Wilkerson, A cohomology decomposition theorem, Topology 31 (1992), no. 2, 433–443. MR 1167181, DOI 10.1016/0040-9383(92)90032-D
W. G. Dwyer and C. W. Wilkerson, Homotopy fixed-point methods for Lie groups and finite loop spaces, Ann. of Math. (2) 139 (1994), no. 2, 395–442. MR 1274096, DOI 10.2307/2946585
W. G. Dwyer and C. W. Wilkerson, Kähler differentials, the $T$-functor, and a theorem of Steinberg, Trans. Amer. Math. Soc. 350 (1998), no. 12, 4919–4930. MR 1621741, DOI 10.1090/S0002-9947-98-02373-3
Alex Gonzalez, Finite approximations of $p$-local compact groups, Geom. Topol. 20 (2016), no. 5, 2923–2995. MR 3556352, DOI 10.2140/gt.2016.20.2923
Hans-Werner Henn, Commutative algebra of unstable $K$-modules, Lannes’ $T$-functor and equivariant mod-$p$ cohomology, J. Reine Angew. Math. 478 (1996), 189–215. MR 1409058, DOI 10.1515/crll.1996.478.189
Hans-Werner Henn, Centralizers of elementary abelian $p$-subgroups and mod-$p$ cohomology of profinite groups, Duke Math. J. 91 (1998), no. 3, 561–585. MR 1604171, DOI 10.1215/S0012-7094-98-09121-9
Hans-Werner Henn, Unstable modules over the Steenrod algebra and cohomology of groups, Group representations: cohomology, group actions and topology (Seattle, WA, 1996) Proc. Sympos. Pure Math., vol. 63, Amer. Math. Soc., Providence, RI, 1998, pp. 277–300. MR 1603179, DOI 10.1090/pspum/063/1603179
Hans-Werner Henn, A variant of the proof of the Landweber Stong conjecture, Group representations: cohomology, group actions and topology (Seattle, WA, 1996) Proc. Sympos. Pure Math., vol. 63, Amer. Math. Soc., Providence, RI, 1998, pp. 271–275. MR 1603175, DOI 10.1090/pspum/063/1603175
Hans-Werner Henn, Jean Lannes, and Lionel Schwartz, Localizations of unstable $A$-modules and equivariant mod $p$ cohomology, Math. Ann. 301 (1995), no. 1, 23–68. MR 1312569, DOI 10.1007/BF01446619
Srikanth B. Iyengar, Graham J. Leuschke, Anton Leykin, Claudia Miller, Ezra Miller, Anurag K. Singh, and Uli Walther, Twenty-four hours of local cohomology, Graduate Studies in Mathematics, vol. 87, American Mathematical Society, Providence, RI, 2007. MR 2355715, DOI 10.1090/gsm/087
A. Jeanneret and A. Osse, The $K$-theory of $p$-compact groups, Comment. Math. Helv. 72 (1997), no. 4, 556–581. MR 1600146, DOI 10.1007/s000140050034
Jean Lannes, Cohomology of groups and function spaces, 1986. Unpublished manuscript.
Jean Lannes, Sur les espaces fonctionnels dont la source est le classifiant d’un $p$-groupe abélien élémentaire, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 135–244 (French). With an appendix by Michel Zisman. MR 1179079
Ran Levi and Assaf Libman, Existence and uniqueness of classifying spaces for fusion systems over discrete $p$-toral groups, J. Lond. Math. Soc. (2) 91 (2015), no. 1, 47–70. MR 3338608, DOI 10.1112/jlms/jdu062
Guido Mislin, Cohomologically central elements and fusion in groups, Algebraic topology (San Feliu de Guíxols, 1990) Lecture Notes in Math., vol. 1509, Springer, Berlin, 1992, pp. 294–300. MR 1185979, DOI 10.1007/BFb0087519
Dietrich Notbohm, Depth and homology decompositions, arXiv:0905.4635 (2009).
Mara D. Neusel and Larry Smith, Invariant theory of finite groups, Mathematical Surveys and Monographs, vol. 94, American Mathematical Society, Providence, RI, 2002. MR 1869812, DOI 10.1086/342122
M. Poulsen, Depth, detection and associated primes in the cohomology of finite groups (an introduction to Carlson’s depth conjecture), Master’s thesis, University of Copenhagen URL:, 2007. Available at http://web.math.ku.dk/~moller/students/mortenP.pdf.
Geoffrey Powell, Unstable $K$-modules and the nilpotent filtration, 2007. Available online at http://www.math.univ-angers.fr/~powell/documents/2007/kmod.pdf.
Lluis Puig, Frobenius categories, J. Algebra 303 (2006), no. 1, 309–357. MR 2253665, DOI 10.1016/j.jalgebra.2006.01.023
Daniel Quillen, The spectrum of an equivariant cohomology ring. I, II, Ann. of Math. (2) 94 (1971), 549–572; ibid. (2) 94 (1971), 573–602. MR 298694, DOI 10.2307/1970770
D. L. Rector, Noetherian cohomology rings and finite loop spaces with torsion, J. Pure Appl. Algebra 32 (1984), no. 2, 191–217. MR 741965, DOI 10.1016/0022-4049(84)90051-3
Lionel Schwartz, Unstable modules over the Steenrod algebra and Sullivan’s fixed point set conjecture, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1994. MR 1282727
Burt Totaro, Group cohomology and algebraic cycles, Cambridge Tracts in Mathematics, vol. 204, Cambridge University Press, Cambridge, 2014. MR 3185743, DOI 10.1017/CBO9781139059480