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ordinary homology spectra split

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Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Stable Homotopy theory

Higher algebra

Contents

Statement

For SS any spectrum and HAH A an Eilenberg-MacLane spectrum, then the smash product SHAS\wedge H A (the AA-ordinary homology spectrum) is non-canonically equivalent to a product of EM-spectra (hence a wedge sum of EM-spectra in the finite case).

(Adams 74, part II, lemma 6.1)

A variant for generalized (Eilenberg-Steenrod) cohomology:

Let XX be a topological space such that each of the ordinary homology groups H n(X,)H_n(X,\mathbb{Z}) is a free abelian group on genrators {h α,n} αB n\{h_{\alpha,n}\}_{\alpha \in B_n}. Write c α,nH n(X,)Hom(H n(X,),)c_{\alpha,n} \in H^n(X,\mathbb{Z}) \simeq Hom(H_n(X,\mathbb{Z}), \mathbb{Z}) for the corresponding dual basis.

Let EE be a multiplicative cohomology theory and write h n,α(τ 0E) n(X)h'_{n,\alpha} \in (\tau_{\leq 0} E)_n(X) and c n,aloha(τ 0E) n(X)c'_{n,\aloha} \in (\tau_{\leq 0} E)^n(X) for the images of these generators under the 0-truncated unit map

Hτ 0𝕊τ 0E. H \mathbb{Z} \simeq \tau_{\leq 0} \mathbb{S} \stackrel{}{\longrightarrow} \tau_{\leq 0} E \,.
Proposition

If one of the following conditions is satisfied

  • Each h n,αh'_{n,\alpha} lifts through E n(X)(τ 0E) n(X)E_n(X) \to (\tau_{\leq 0}E)_n(X);

  • each H n(X,)H_n(X,\mathbb{Z}) is finitely generated and each c n,αc'_{n,\alpha} lifts through E n(X)(τ 0E) n(X)E^n(X) \to (\tau_{\leq 0}E)^n(X),

then there are non-canonical equivalences as follows:

  1. EΣ X +n,αΣ nEEModE \wedge \Sigma^\infty X_+ \simeq \underset{n,\alpha}{\vee} \Sigma^n E \;\; \in E Mod ;

  2. [X,E]n,αΣ nE[X,E] \simeq \underset{n,\alpha}{\prod} \Sigma^{-n} E;

  3. E (X)π (E)H (X,)E_\bullet(X) \simeq \pi_\bullet(E) \otimes H_\bullet(X,\mathbb{Z})

    and

    E (X)Hom(H (X),π (E))E^\bullet(X)\simeq Hom(H_\bullet(X), \pi_\bullet(E))

(Lurie 10, lecture 4, prop. 7, Adams 74, part II, lemma 2.5)

References

Last revised on May 23, 2016 at 15:54:16. See the history of this page for a list of all contributions to it.