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Let $G$ be a finite group and $p$ be an odd prime. It's known by work of Quillen that the minimal primes of $H^{2\ast}(G;\mathbb{F}_p)$ are in one-to-one correspondence with the maximal elementary abelian p-subgroups of $G$. Explicitely, if $E \le G$ is maximal elementary abelian, than the corresponding minimal prime is

$$\mathfrak{p}_E = \operatorname{ker}\lbrace\hspace{1pt} H^{2\ast}(G;\mathbb{F}_p) \to H^{2\ast}(E;\mathbb{F}_p) \to H^{2\ast}(E;\mathbb{F}_p)/\sqrt{0} \hspace{1pt}\rbrace$$

where the first map is restriction and the second is the natural epimorphism. Since $H^{2\ast}(G;\mathbb{F}_p)$ is noetherian, minimal primes are associated. Therefore, I wonder:

Is there an explicit description of an element $x \in H^{2\ast}(G;\mathbb{F}_p)$ such that $\mathfrak{p}_E = \operatorname{Ann}(x)$ ?

By "explicit" I also mean transfers, power operations, chern classes, etc. of some element that can be described in a concrete manner.

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1 Answer 1

I believe not; all information will be in Carlson's huge text Cohomology Rings of Finite Groups (check out pg324). There is an algorithm to finding such an element, but is currently just a search through 'likely candidates', noting that $\mathfrak{p}$ must be the annihilator of some generator of the ideal $Ann(\mathfrak{p})$. A full section in Chapter 17 is dedicated to associated primes in cohomology... unfortunately I do not have the book at the moment so I don't know if it will tell you anything useful that you don't already know.

Now in the case that $G$ is a p-group of rank 2 and $H^\ast(G,k)$ is not Cohen-Macaulay, we can construct a homogeneous element $\eta\in H^\ast(G,k)$ such that $H^\ast(G,k)/Ann(\eta)$ has Krull dimension one. The gist of it is: Find a particular homogeneous set of parameters $f_1,f_2$ for $H^\ast(G,k)$. As the ring is not Cohen-Macaulay, there is a $\gamma\in H^\ast(G,k)$ such that the induced class in $H^\ast(G,k)/(f_1)$ is annihilated by $f_2$. Then $\eta$ lies in $H^\ast(G,k)\cdot\gamma$. (See Theorem 12.6.1 of said text).

For a particular example: the semidihedral 2-group $G$ of order 16. Its cohomology ring is $k[z,y,x,w]/(zy,y^3,yx,z^2w+x^2)$, and the associated prime $(z,y,x)$ is the annihilator of $y^2$.

Jon Carlson and Dave Benson are the two experts on this stuff.

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