Gorenstein duality for real spectra

Abstract

we study the C 2 -spectra BPRhni and ER.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 D 1 and 2.

The representation-ring-graded local cohomology spectral sequence for BPℝ⟨3⟩

“…where ǫ is a small homological correction, just as happens with torsion in integral Poincaré duality). This was described for n = 1 (like a 4-manifold) and n = 2 (like a 12-manifold) in [13], and we describe it here for n = 3 (like a 28-manifold).…”

“…As described in [13,Subsection 12.F and Remark 13.3], one way of thinking about the duality is that we start with a diagonal BB δ , take local cohomology, usually in a single degree, to reach H i J (BB δ ). As an abelian group, this is usually either all torsionfree or annihilated by 2, so that when Anderson-dualized it contributes to a single diagonal NB δ ′ .…”

“…so that given orientability and the hypothesis on Eilenberg-MacLane spectra if R is Gorenstein of shift V then it also has Gorenstein duality of shift W = V + γ [13]. The purpose of the present paper is to study this duality in a case established in [13].…”

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The purpose of this paper is to examine the calculational consequences of the duality proved by the first author and Meier in [13], making them explicit in one more example. We give an explicit description of the behaviour of the spectral sequence in the title, with pictures.

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“…where the differentials are of the standard homological form d i : E i n,V −→ E i n−i,V +i−1 changing the total RO(Q)-degree by adding the integer −1. The result was illustrated in [13] by working out the exact behaviour of the LCT for n = 1 and n = 2.…”