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Eilenberg–Mac Lane spectrum of a chain complex

PDF p. 4: Shift [2j] in equation (2.1)

Statement

For every unbounded chain complex $D$ of abelian groups, the cited Eilenberg–Mac Lane construction gives a functorial pointed simplicial Omega spectrum $H D$ and natural isomorphisms from its stable homotopy groups to $H_{n} (D)$ for all integers $n$ . An Omega spectrum is a sequential spectrum whose adjoint structure maps are weak equivalences.

External input

This is the one-object $Z$ case of the cited construction and equation (B.1.9). The result and its construction are imported from the cited literature; no local reconstruction is claimed. The source defines the levels using the functor $W$ and explicitly states that the underlying spectra are Omega spectra. The model is a naive module spectrum, not the symmetric Eilenberg–Mac Lane functor distinguished in the source's Remark B.1.10.

Depends on

Stable model categories are categories of modules · Appendix B.1, pp. 38–40, especially construction preceding equation (B.1.9) and Theorem B.1.11

Used by · 1

Complex shifts and spectral homotopy indices

Dependency tree · 1 level

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text.

External dependencies

Stable model categories are categories of modules · Appendix B.1, pp. 38–40, especially construction preceding equation (B.1.9) and Theorem B.1.11

For an unbounded chain complex D of abelian groups, the Eilenberg–Mac Lane construction gives an underlying Omega spectrum whose stable homotopy groups are naturally H_n(D).

Hypotheses: Take the one-object ringoid with endomorphism ring Z; D is an unbounded chain complex of Z-modules.

Reading limits: Read the construction, equation (B.1.9), and the statement and opening proof of Theorem B.1.11 in the linked text; the full proof and other model comparisons were not independently reconstructed.

Sources

Stable model categories are categories of modules · Appendix B.1, pp. 38–40, especially construction preceding equation (B.1.9) and Theorem B.1.11