Examples of Tate cohomology rings - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T19:24:02Z http://mathoverflow.net/feeds/question/78287 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/78287/examples-of-tate-cohomology-rings Examples of Tate cohomology rings TJ 2011-10-16T20:39:23Z 2011-10-19T12:13:42Z <p>If $G$ is a finite group with periodic cohomology then the Tate cohomology ring can be easily computed to be the localization $\hat{H}^\ast(G,\mathbb{Z}) = H^\ast(G,\mathbb{Z})_{(z)}$ where $z$ is a unit of minimal positive degree. Examples are </p> <ul> <li><p>Cyclic group: </p> <p>$\hat{H}^\ast(C_n,\mathbb{Z}) = \mathbb{Z}[z,z^{-1}]/(nz), |z|=2$</p></li> <li><p>Quaternion group:</p> <p>$\hat{H}^\ast(Q_{2^n},\mathbb{Z}) = \mathbb{Z}[z,z^{-1},a,b]/(2^nz, 2a,2b), |z| =4, |a| = |b| =2$ </p></li> <li><p>Binary icosahedral group: </p> <p>$\hat{H}^\ast(I,\mathbb{Z}) = \mathbb{Z}[z,z^{-1}]/(120z), |z|=4$ </p></li> </ul> <p>The same formular holds in mod-$p$ cohomology, if the cohomology of $G$ is $p$-periodic. </p> <p><em>Question:</em> Are there computations of integral or mod-$p$ Tate cohomology rings of finite groups with non-periodic cohomology in the literature ? </p> http://mathoverflow.net/questions/78287/examples-of-tate-cohomology-rings/78290#78290 Answer by Chris Gerig for Examples of Tate cohomology rings Chris Gerig 2011-10-16T21:08:44Z 2011-10-17T10:29:16Z <p>Antonio Bellezza's PhD thesis (Pisa, 2002) computes the ring structure of the Tate cohomology for $\mathbb{Z}/p^a\times\mathbb{Z}/p^b$, and also the mod-p Tate cohomology of $\mathbb{Z}_p^2$. The title is <em>Integral Duality and the Structure of Tate Cohomology Rings</em>.</p> <p>Also, there is an unpublished/unfinished paper of Weiss, found here: <a href="http://www.math.uwo.ca/~schebolu/research/Jan/tateprop.pdf" rel="nofollow">http://www.math.uwo.ca/~schebolu/research/Jan/tateprop.pdf</a> , which computes $\hat{H}^*(\mathbb{Z}_2^r,\mathbb{Z}_2)$.</p> <p>And just to add to your current list (periodic groups): $\hat{H}^*(S_3,\mathbb{Z}_3)=\Lambda[x]\otimes\mathbb{Z}_3[z,z^{-1}]$ with $|x|=3$ and $|z|=4$.</p> http://mathoverflow.net/questions/78287/examples-of-tate-cohomology-rings/78292#78292 Answer by Fernando Muro for Examples of Tate cohomology rings Fernando Muro 2011-10-16T22:16:06Z 2011-10-16T22:16:06Z <p>Artin and Tate showed in their class field theory book that a finite group has periodic mod $p$ cohomology if and only if its Sylow $p$â€“group is either a cyclic group or a generalised quaternion group. Swan proved later that a group has periodic cohomology for all primes if and only if it acts freely on a finite CW-complex which has the homotopy type of a sphere.</p> http://mathoverflow.net/questions/78287/examples-of-tate-cohomology-rings/78307#78307 Answer by Eric Wofsey for Examples of Tate cohomology rings Eric Wofsey 2011-10-17T03:53:11Z 2011-10-17T04:22:04Z <p>The ring structure on mod $p$ Tate cohomology can be split up into 4 parts:</p> <ol> <li>Products $\hat{H}^n\otimes\hat{H}^m\to\hat{H}^{n+m}$ for $n,m\geq0$. These just come from products on ordinary cohomology.</li> <li>Products $\hat{H}^n\otimes\hat{H}^m\to\hat{H}^{n+m}$ for $n\geq0$, $m&lt;0$, and $n+m&lt;0$. These are given by cap products when we identify $\hat{H}^m$ with the dual of $H^{-m-1}$.</li> <li>Products $\hat{H}^n\otimes\hat{H}^m\to\hat{H}^{n+m}$ for $n\geq0$, $m&lt;0$, and $n+m\geq 0$.</li> <li>Products $\hat{H}^n\otimes\hat{H}^m\to\hat{H}^{n+m}$ for $n,m&lt;0$.</li> </ol> <p>In fact, for most groups, <i>all</i> products of type 3 and 4 vanish. This is true, for instance, if the center of a $p$-Sylow subgroup of $G$ has rank greater than 1: see <a href="http://www.ams.org/mathscinet-getitem?mr=1182934" rel="nofollow">this paper of Benson and Carlson</a> (they also show that all products of type 3 vanish iff all products of type 4 vanish). By Chouinard's theorem, this implies that for arbitrary non-periodic groups, all elements in negative mod $p$ Tate cohomology are nilpotent.</p> http://mathoverflow.net/questions/78287/examples-of-tate-cohomology-rings/78570#78570 Answer by Rene Schoof for Examples of Tate cohomology rings Rene Schoof 2011-10-19T12:13:42Z 2011-10-19T12:13:42Z <p>Bellezza's thesis is here</p> <p><a href="http://www.abdn.ac.uk/~mth192/html/archive/bellezza.html" rel="nofollow">http://www.abdn.ac.uk/~mth192/html/archive/bellezza.html</a></p>