In mathematics, specifically algebraic topology, there is a distinguished class of spectra called Eilenberg–Maclane spectra for any Abelian group pg 134. Note, this construction can be generalized to commutative rings as well from its underlying Abelian group. These are an important class of spectra because they model ordinary integral cohomology and cohomology with coefficients in an abelian group. In addition, they are a lift of the homological structure in the derived category of abelian groups in the homotopy category of spectra. In addition, these spectra can be used to construct resolutions of spectra, called Adams resolutions, which are used in the construction of the Adams spectral sequence. Definition For a fixed abelian group let denote the set of Eilenberg–MacLane spaces with the adjunction map coming from the property of loop spaces of Eilenberg–Maclane spaces: namely, because there is a homotopy equivalencewe can construct maps from the adjunction giving the desired structure maps of the set to get a spectrum. This collection is called the Eilenberg–Maclane spectrum of pg 134. Properties Using the Eilenberg–Maclane spectrum we can define the notion of cohomology of a spectrum and the homology of a spectrum pg 42. Using the functorwe can define cohomology simply asNote that for a CW complex , the cohomology of the suspension spectrum recovers the cohomology of the original space . Note that we can define the dual notion of homology aswhich can be interpreted as a "dual" to the usual hom-tensor adjunction in spectra. Note that instead of , we take for some Abelian group , we recover the usual (co)homology with coefficients in the abelian group and denote it by . Mod-p spectra and the Steenrod algebra For the Eilenberg–Maclane spectrum there is an isomorphismfor the p-Steenrod algebra . Tools for computing Adams resolutions One of the quintessential tools for computing stable homotopy groups is the Adams spectral sequence. In order to make this construction, the use of Adams resolutions are employed. These depend on the following properties of Eilenberg–Maclane spectra. We define a generalized Eilenberg–Maclane spectrum as a finite wedge of suspensions of Eilenberg–Maclane spectra , soNote that for and a spectrum so it shifts the degree of cohomology classes. For the rest of the article for some fixed abelian group Equivalence of maps to K Note that a homotopy class represents a finite collection of elements in . Conversely, any finite collection of elements in is represented