Algebraic & Geometric Topology

Gorenstein duality for real spectra

J P C Greenlees and Lennart Meier

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/agt.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Following Hu and Kriz, we study the C2–spectra BPn and E(n) that refine the usual truncated Brown–Peterson and the Johnson–Wilson spectra. In particular, we show that they satisfy Gorenstein duality with a representation grading shift and identify their Anderson duals. We also compute the associated local cohomology spectral sequence in the cases n=1 and 2.

Article information

Source
Algebr. Geom. Topol., Volume 17, Number 6 (2017), 3547-3619.

Dates
Received: 13 July 2016
Revised: 17 January 2017
Accepted: 1 February 2017
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1510841515

Digital Object Identifier
doi:10.2140/agt.2017.17.3547

Mathematical Reviews number (MathSciNet)
MR3709655

Zentralblatt MATH identifier
06791657

Subjects
Primary: 55P91: Equivariant homotopy theory [See also 19L47] 55U30: Duality
Secondary: 55P43: Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.) 55Q91: Equivariant homotopy groups [See also 19L47]

Keywords
Anderson duality Gorenstein duality real K-theory real bordism real Brown-Peterson spectra real Johnson-Wilson theories

Citation

Greenlees, J P C; Meier, Lennart. Gorenstein duality for real spectra. Algebr. Geom. Topol. 17 (2017), no. 6, 3547--3619. doi:10.2140/agt.2017.17.3547. https://projecteuclid.org/euclid.agt/1510841515


Export citation

References

  • D W Anderson, Universal coefficient theorems for K-theory, mimeographed notes, Univ. of California, Berkeley (1969)
  • S Araki, Orientations in $\tau $–cohomology theories, Japan. J. Math. 5 (1979) 403–430
  • S Araki, M Murayama, $\tau $–cohomology theories, Japan. J. Math. 4 (1978) 363–416
  • M F Atiyah, $K$–theory and reality, Quart. J. Math. Oxford Ser. 17 (1966) 367–386
  • R Banerjee, On the $ER(2)$–cohomology of some odd-dimensional projective spaces, Topology Appl. 160 (2013) 1395–1405
  • E H Brown, Jr, M Comenetz, Pontrjagin duality for generalized homology and cohomology theories, Amer. J. Math. 98 (1976) 1–27
  • R R Bruner, J P C Greenlees, Connective real $K$–theory of finite groups, Mathematical Surveys and Monographs 169, Amer. Math. Soc., Providence, RI (2010)
  • D Dugger, An Atiyah–Hirzebruch spectral sequence for $KR$–theory, $K$-Theory 35 (2005) 213–256
  • W G Dwyer, J P C Greenlees, S Iyengar, Duality in algebra and topology, Adv. Math. 200 (2006) 357–402
  • A D Elmendorf, I Kriz, M A Mandell, J P May, Rings, modules, and algebras in stable homotopy theory, Math. Surv. Monogr. 47, Amer. Math. Soc., Providence, RI (1997)
  • J P C Greenlees, J P May, Completions in algebra and topology, from “Handbook of algebraic topology” (I M James, editor), North-Holland, Amsterdam (1995) 255–276
  • J P C Greenlees, J P May, Generalized Tate cohomology, Mem. Amer. Math. Soc. 543, Amer. Math. Soc., Providence, RI (1995)
  • J P C Greenlees, V Stojanoska, Anderson and Gorenstein duality, preprint (2017)
  • D Heard, V Stojanoska, $K$–theory, reality, and duality, J. K-Theory 14 (2014) 526–555
  • M A Hill, The equivariant slice filtration: a primer, Homology Homotopy Appl. 14 (2012) 143–166
  • M A Hill, M J Hopkins, D C Ravenel, On the nonexistence of elements of Kervaire invariant one, Ann. of Math. 184 (2016) 1–262
  • P Hu, On Real-oriented Johnson–Wilson cohomology, Algebr. Geom. Topol. 2 (2002) 937–947
  • P Hu, I Kriz, Real-oriented homotopy theory and an analogue of the Adams–Novikov spectral sequence, Topology 40 (2001) 317–399
  • P C Kainen, Universal coefficient theorems for generalized homology and stable cohomotopy, Pacific J. Math. 37 (1971) 397–407
  • N Kitchloo, W S Wilson, Unstable splittings for real spectra, Algebr. Geom. Topol. 13 (2013) 1053–1070
  • N Kitchloo, W S Wilson, The $ER(n)$–cohomology of $BO(q)$ and real Johnson–Wilson orientations for vector bundles, Bull. Lond. Math. Soc. 47 (2015) 835–847
  • P S Landweber, Conjugations on complex manifolds and equivariant homotopy of $MU$, Bull. Amer. Math. Soc. 74 (1968) 271–274
  • G Laures, M Olbermann, $\mathit{TMF}_0(3)$–characteristic classes for string bundles, Math. Z. 282 (2016) 511–533
  • V Lorman, The real Johnson–Wilson cohomology of $\mathbb{C`P}^\infty$, Topology Appl. 209 (2016) 367–388
  • M A Mandell, J P May, Equivariant orthogonal spectra and $S$–modules, Mem. Amer. Math. Soc. 755, Amer. Math. Soc., Providence, RI (2002)
  • H R Margolis, Spectra and the Steenrod algebra, North-Holland Mathematical Library 29, North-Holland, Amsterdam (1983)
  • L Meier, M A Hill, The $C_2$–spectrum $Tmf_1(3)$ and its invertible modules, preprint (2015)
  • N Ricka, Equivariant Anderson duality and Mackey functor duality, Glasg. Math. J. 58 (2016) 649–676
  • S Schwede, Symmetric spectra, book draft (2012) \setbox0\makeatletter\@url http://tinyurl.com/SchSS12 {\unhbox0
  • S Schwede, Lectures on equivariant stable homotopy theory, lecture notes (2016) \setbox0\makeatletter\@url http://tinyurl.com/SchESHT16 {\unhbox0
  • V Stojanoska, Duality for topological modular forms, Doc. Math. 17 (2012) 271–311
  • S M J Tilson, Power operations in the Künneth and $C_2$–equivariant Adams spectral sequences with applications, PhD thesis, Wayne State University, Detroit, MI (2013) \setbox0\makeatletter\@url http://search.proquest.com/docview/1459447634 {\unhbox0
  • S Tilson, Power operations in the Künneth spectral sequence and commutative $H\mathbb{F}_p$–algebras, preprint (2016)

Corrections

  • J P C Greenlees, Lennart Meier. Correction to the article Gorenstein duality for real spectra. Algebr. Geom. Topol., Volume 18, Number 5 (2018), 3129-3131.