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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)].

##### 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.)
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