Complex Cobordism and Stable Homotopy Groups of Spheres by Douglas C. Ravenel

By Douglas C. Ravenel

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2. Theorem. There is an element x ∈ M U 2 (CP ∞ ) such that M U ∗ (CP ∞ ) = M U ∗ (pt)[[x]] and M U ∗ (CP ∞ × CP ∞ ) = M U ∗ (pt)[[x ⊗ 1, 1 ⊗ x]]. 18) only in that its generators are negatively graded. The generator x is closely related to the usual generator of H 2 (CP ∞ ), which we also denote by x. The alert reader may have expected M U ∗ (CP ∞ ) to be a polynomial rather than a power series ring since H ∗ (CP ∞ ) is traditionally described as Z[x]. However, the latter is really Z[[x]] since the cohomology of an infinite complex maps onto the inverse limit of the cohomologies of its finite skeleta.

10). 5), at least for n < p − 1. 11. Morava Vanishing Theorem. 8) E1n,s = 0 for s > n2 . 6) that every sufficiently small open subgroup of Sn has the same cohomology as a free abelian group of rank n2 . 6). 19 appear in the chromatic spectral sequence. , the invariant prime ideal In = (p, v1 , . . , vn−1 )], then L/J is a submodule of N n and M n , so Ext0 (L/J) ⊂ Ext0 (N n ) ⊂ Ext0 (M n ) = E1n,0 . Recall that the Greek letter elements are images of elements in Ext0 (J) under the appropriate composition of connecting homomorphisms.

4. Theorem. 3. The height n orbit Vn is the subset defined by vi = 0 for i < n and vn = 0. Now observe that V is the set of closed points in Spec(Ln ⊗K), and Vn is the set of closed points in Spec(Ln ⊗ K), where Ln = vn−1 L/In . 5. Change-of-Rings Theorem. H ∗ (GK ; Ln ⊗ K) = H ∗ (Sn ; K). We will see in Chapter 6 that a form of this isomorphism holds over Fp as well as over K. In it the right-hand term is the cohomology of a certain Hopf algebra [called the nth Morava stabilizer algebra Σ(n)] defined over Fp , which, when tensored with Fpn , becomes isomorphic to the dual of Fpn [Sn ], the Fpn -group algebra of Sn .

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