×

zbMATH — the first resource for mathematics

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)].

MSC:
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.)
PDF BibTeX XML Cite
Full Text: DOI
References:
[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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.