Eilenberg–MacLane spectra of chain complexes

Definition 6.2.1 (Eilenberg-MacLane spectrum). Given a complex \(A \in \D (\Z )\), we define its Eilenberg-MacLane spectrum as the mapping spectrum \[ HA \quad := \quad \hom _{\D (\Z )}(\Z [0],A) \qin \Sp \] Similarly, for \(n \in \Z \) we define its \(n\)-th Eilenberg MacLane anima as \[ K(A,n) \quad := \quad \Hom _{\D (\Z )}(\Z [-n],A) \qin \An . \] These define functors \(H \colon \D (\Z ) \to \Sp \) and \(K(-,n)\colon \D (\Z ) \to \An \).

In particular, every abelian group \(A \in \Ab \) defines an Eilenberg-MacLane spectrum \(HA\) and an \(n\)-th Eilenberg MacLane anima \(K(A,n)\) by regarding it as a chain complex concentrated in degree \(0\).