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Gorenstein duality for real spectra. (English) Zbl 1391.55009
Algebr. Geom. Topol. 17, No. 6, 3547-3619 (2017); corrigendum ibid. 18, No. 5, 3129–3131 (2018).
The paper studies Gorenstein duality in \(C_2\)-equivariant homotopy theory, a theory that has previously been studied in more detail in the non-equivariant setting [W. G. Dwyer et al., Adv. Math. 200, No. 2, 357–402 (2006; Zbl 1155.55302)]. Starting with a connective commutative \(C_2\)-ring spectrum \(R\) whose zeroth homotopy Mackey functor is \(\underline{\mathbb Z}\), the authors define \(R\) to be Gorenstein of shift \(a \in RO(C_2)\) (the representation ring of \(C_2\)) if there is an equivalence of \(R\)-modules \[ \mathrm{Hom}_R(H\underline{\mathbb Z},R) \simeq \Sigma^a H\underline{\mathbb Z}. \] This contrasts with the non-equivariant setting where the shift is only allowed to be an integer. The condition that \(R\) is Gorenstein can, in some cases, be lifted to a statement that \(R\) has Gorenstein duality, which leads to interesting conclusions about the \(RO(C_2)\)-graded homotopy groups of \(R\).
The main \(C_2\)-equivariant spectra under consideration are \(BP\mathbb R\langle n \rangle\) and \(E\mathbb R(n)\), the real spectra corresponding to truncated Brown-Peterson theory and Johnson-Wilson \(E\)-theory. It is shown that the connective spectrum \(BP\mathbb R\langle n \rangle\) is Gorenstein, and satisfies Gorenstein duality, and that \(E\mathbb R\langle n \rangle\) satisfies Gorenstein duality. In all cases, the \(RO(C_2)\)-graded shifts are identified explicitly. In the case \(n = 1\) and \(n = 2\) the authors explicitly calculate the local cohomology spectral sequence, and show the implications for the homotopy ring \(\pi_*^{C_2}(BP\mathbb R\langle n \rangle)\).
This paper also contains a wealth of other interesting information for the reader interested in \(C_2\)-equivariant homotopy theory, for example a proof that \(BP\mathbb R\langle n \rangle \) and \(E\mathbb R(n)\) are strongly even in the sense of [M. Hill and L. Meier, Algebr. Geom. Topol. 17, No. 4, 1953–2011 (2017; Zbl 1421.55002)].

55P91 Equivariant homotopy theory in algebraic topology
55Q91 Equivariant homotopy groups
55U30 Duality in applied homological algebra and category theory (aspects of algebraic topology)
55P43 Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.)
Full Text: DOI
[1] ; Araki, Japan. J. Math., 5, 403, (1979)
[2] ; Araki, Japan. J. Math., 4, 363, (1978)
[3] 10.1093/qmath/17.1.367 · Zbl 0146.19101
[4] 10.1016/j.topol.2013.05.017 · Zbl 1408.55002
[5] 10.2307/2373610 · Zbl 0325.55008
[6] 10.1090/surv/169
[7] 10.1007/s10977-005-1552-9 · Zbl 1109.14024
[8] 10.1016/j.aim.2005.11.004 · Zbl 1155.55302
[9] ; Elmendorf, Rings, modules, and algebras in stable homotopy theory. Math. Surv. Monogr., 47, (1997)
[10] 10.1016/B978-044481779-2/50008-0
[11] 10.1090/memo/0543
[12] 10.1017/is014007001jkt275 · Zbl 1325.55003
[13] 10.4310/HHA.2012.v14.n2.a9 · Zbl 1403.55003
[14] 10.4007/annals.2016.184.1.1 · Zbl 1366.55007
[15] 10.2140/agt.2002.2.937 · Zbl 1025.55003
[16] 10.1016/S0040-9383(99)00065-8 · Zbl 0967.55010
[17] 10.2140/pjm.1971.37.397 · Zbl 0211.32704
[18] 10.2140/agt.2013.13.1053 · Zbl 1270.55003
[19] 10.1112/blms/bdv057 · Zbl 1328.55006
[20] 10.1090/S0002-9904-1968-11917-2 · Zbl 0181.26801
[21] 10.1007/s00209-015-1551-3 · Zbl 1337.55008
[22] 10.1016/j.topol.2016.06.018 · Zbl 1348.55007
[23] 10.1090/memo/0755
[24] ; Margolis, Spectra and the Steenrod algebra. North-Holland Mathematical Library, 29, (1983) · Zbl 0552.55002
[25] 10.1017/S0017089515000397 · Zbl 1350.55012
[26] ; Stojanoska, Doc. Math., 17, 271, (2012)
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