On Commutative Ring Spectra of Characteristic 2

Commutative 𝕊–algebras of prime characteristics and applications to unoriented bordism

“…Theorem 1.4. ([Wür86, Theorem 1.1], [PR85]) Suppose E is a commutative ring spectrum (up to homotopy) with char(E) = 2. Then E is multiplicatively an Eilenberg-Mac Lane spectrum (up to homotopy).…”

Commutative 2-local ring spectra

“…We shall give below our own proof of Theorem 1.3. This proof repeats the basic idea used in [7] and [11] of contrasting the commutativity of £ with the noncommutativity of some appropriate standard ring spectrum, but it is simpler and more elementary in that its principal tool is the Atiyah-Hirzebruch spectral sequence and in that it does not require the calculation of the E-homology of the spectra P(n). We shall close by pointing out that Theorems 1.1 and 1.3 are in fact equivalent statements.…”

“…Let T be the map of RP 2 " x RP 2 The decision to work with k(n) in the proof of Theorem 1.3 was purely a matter of taste. We could equally well have chosen to work with P(n), as in [7] and [11], since these spectra coincide in the range of dimensions under consideration.…”

“…We call a spectrum a generalised Eilenberg-MacLane spectrum (briefly, a GEM) if it is equivalent in the stable category [1] to a wedge sum of Eilenberg-MacLane spectra. Pazhitnov and Rudyak in [7] and Wilrgler in [11] have proved that every commutative mod 2 ring spectrum is a GEM (see Theorem 1.3 below). In this paper, we apply their result to a characterisation of GEMs that are 2-local commutative ring spectra.…”

“…The statement of the theorem of Pazhitnov, Rudyak and Wiirgler is as follows. THEOREM 1.3 [7,11]. Let E be a commutative ring spectrum such that n 0 E is a vector space over Z/2.…”