View source for Steenrod algebra

The graded algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s0875001.png" /> over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s0875002.png" /> of all stable cohomology operations (cf. [[Cohomology operation|Cohomology operation]]) modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s0875003.png" />. For any space ([[Spectrum of spaces|spectrum of spaces]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s0875004.png" />, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s0875005.png" /> is a module over the Steenrod algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s0875006.png" />.

The Steenrod algebra is multiplicatively generated by the Steenrod operations (cf. [[Steenrod operation|Steenrod operation]]). Thus, the Steenrod algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s0875007.png" /> is a graded associative algebra, multiplicatively generated by the symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s0875008.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s0875009.png" />, which satisfy the Adem relation:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750010.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750011.png" />, so that an additive basis (over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750012.png" />) of the Steenrod algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750013.png" /> consists of the operations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750015.png" /> (the so-called Cartan–Serre basis). Similar results are true for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750016.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750017.png" />. Furthermore,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750018.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750019.png" /> is an [[Eilenberg–MacLane space|Eilenberg–MacLane space]]. The multiplication

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750020.png" /></td> </tr></table>

induces the diagonal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750021.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750022.png" />, which is a homomorphism of algebras, and, consequently, turns <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750023.png" /> into a [[Hopf algebra|Hopf algebra]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.E. Steenrod,   D.B.A. Epstein,   "Cohomology operations" , Princeton Univ. Press  (1962)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Milnor,   "The Steenrod algebra and its dual"  ''Ann. of Math.'' , '''67'''  (1958)  pp. 150–171</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.K. Tangora,   "Cohomology operations and applications in homotopy theory" , Harper &amp; Row  (1968)</TD></TR></table>



====Comments====
The analogue of the Steenrod algebra for a cohomology theory defined by a spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750024.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750025.png" />; cf. [[Generalized cohomology theories|Generalized cohomology theories]] and [[Spectrum of spaces|Spectrum of spaces]]. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750026.png" />-term of the Adams spectral sequence, cf. [[Spectral sequence|Spectral sequence]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750027.png" /> is a purely (homological) algebra construct obtained by regarding the homology groups as modules over the Hopf algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087500/s08750028.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Dieudonné,   "A history of algebraic and differential topology 1900–1960" , Birkhäuser  (1989)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.M. Switzer,   "Algebraic topology - homotopy and homology" , Springer  (1975)  pp. Chapts. 18–19</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.F. Adams,   "Stable homotopy and generalized homology" , Univ. Chicago Press  (1974)  pp. Part III, Chapts. 12, 15</TD></TR></table>