Editing Spectrum (topology) (section)

=== Eilenberg–Maclane spectrum ===
{{Main articles|Eilenberg–Maclane spectrum}}

Consider [[singular cohomology]] <math> H^n(X;A) </math> with coefficients in an [[abelian group]] <math>A</math>. For a [[CW complex]] <math>X</math>, the group <math> H^n(X;A) </math> can be identified with the set of homotopy classes of maps from <math>X</math> to <math>K(A,n)</math>, the [[Eilenberg–MacLane space]] with homotopy concentrated in degree <math>n</math>. We write this as<blockquote><math>[X,K(A,n)] = H^n(X;A)</math></blockquote>Then the corresponding spectrum <math>HA</math> has <math>n</math>-th space <math>K(A,n)</math>; it is called the '''Eilenberg–MacLane spectrum''' of <math>A</math>. Note this construction can be used to embed any ring <math>R</math> into the category of spectra. This embedding forms the basis of spectral geometry, a model for [[derived algebraic geometry]]. One of the important properties of this embedding are the isomorphisms<blockquote><math>\begin{align}
  \pi_i( H(R/I) \wedge_R H(R/J) ) &\cong H_i\left(R/I\otimes^{\mathbf{L}}R/J\right)\\
                                  &\cong \operatorname{Tor}_i^R(R/I,R/J)
\end{align}</math></blockquote>showing the category of spectra keeps track of the derived information of commutative rings, where the smash product acts as the [[derived tensor product]]. Moreover, Eilenberg–Maclane spectra can be used to define theories such as [[topological Hochschild homology]] for commutative rings, a more refined theory than classical Hochschild homology.