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Below I will describe a mod $p$ Tate resolution for the symmetric group $\Sigma_p$, i.e. a $\mathbb{Z}$-graded periodic acyclic chain complex $C^*$ of finitely generated modules over $\mathbb{F}_p[\Sigma_p]$ such that each $C^i$ is both injective and projective, and $Z^0(C^*)=\mathbb{F}_p$. I suspect that this must appear in the literature somewhere, but I do not think that I have seen it; so I am asking here for references.

The construction is as follows. Let $X$ be a set of order $p$ with $\Sigma_p$ acting in the obvious way. Put $V=\mathbb{F}_p[X]$, so $V$ has an obvious $\Sigma_p$-action and an obvious equivariant isomorphism $V\to V^*$. Let $A^*$ be the exterior algebra generated by $V$. Put $e=\sum_{x\in X}x\in V$, define $f\colon A^i\to A^{i+1}$ by $f(x)=ex$, and let $g\colon A^{i+1}\to A^i$ be adjoint to $f$ with respect to the evident inner products. (Equivalently, $g$ is interior multiplication by $e$.) Note that when $0<i<p$ the module $A^i$ is induced from $\Sigma_i\times\Sigma_{p-i}$, which has order coprime to $p$, so $A^i$ is injective and projective over $\mathbb{F}_p[\Sigma_p]$. The complex $C^*$ will have the form $$ \dotsb \to A^1\to A^2 \to \dotsb \to A^{p-2} \to A^{p-1} \to A^{p-1}\to A^{p-2} \to \dotsb \to A^2\to A^1\to A^1\to A^2\to A^3 \to \dotsb $$ The differentials $A^i\to A^{i+1}$ are given by $f$, and the differentials $A^{i+1}\to A^i$ are given by $g$. The differentials $A^1\to A^1$ are given by $fg$, and the differentials $A^{p-1}\to A^{p-1}$ are given by $gf$. The grading is arranged so that $C^{-1}=C^0=A^1$ and $d\colon C^i\to C^{i+1}$. With these conventions, $H^*((C^*)^{\Sigma_p})$ is naturally identified with the Tate cohomology ring $\widehat{H}^*(\Sigma_p;\mathbb{F}_p)=\mathbb{F}_p[x^{\pm 1}]\otimes E[a]$, where $|x|=2p-2$ and $|a|=-1$. This has a natural ring structure, so one might hope to have a corresponding equivariant differential graded ring structure on $C^*$. Also, $C^n$ is naturally dual to $C^{-n-1}$, so a product structure would also give a coproduct $C^n\to\prod_iC^{i-1}\otimes C^{n-i}$ (note that it is necessary to use the $\prod$ and not $\bigoplus$ here).

However, I think that it is not possible to give $C^*$ an associative product compatible with the differential. Probably there is some weaker operadic structure; I would be interested in any references discussing that question. I would also be interested in parallel constructions in spectra rather than chain complexes.

I also remark that all of this works integrally even if $p$ is not prime. We still get a cochain complex that is nonequivariantly contractible and has $Z^0(C^*)=\mathbb{Z}$. Each term $C^n$ is induced from a partition subgroup $\Sigma_i\times\Sigma_{p-i}$, so is relatively projective in a sense that is important for many applications, but it is not actually projective unless we invert $(p-1)!$.

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  • $\begingroup$ This is an interesting construction. I don't think I've seen it in the literature, but it may be hidden in the work of Gordan James. When you talk of associative product, are you sure you don't mean associative coproduct? Even in the case $p=3$ I think this resolution does not support one. There is an interesting question associated with this: in general, is there a strictly associative coproduct on a resolution of $\mathbb{F}_p$ with the same polynomial growthrate as the minimal resolution? In your case, that would mean an eventually periodic resolution with strictly associative coproduct. $\endgroup$ Commented Jun 1, 2023 at 21:49
  • $\begingroup$ By the way, I seem to be talking about the projective half of the Tate resolution. $\endgroup$ Commented Jun 1, 2023 at 21:52
  • $\begingroup$ Let $G$ be a finite group, and $H\subseteq G$ a Sylow $p$-subgroup. Is there a simple relation between Tate cohomologies $\mathbb F_p^{tG}$ and $\mathbb F_p^{tH}$? $\endgroup$
    – Z. M
    Commented Jun 2, 2023 at 6:24
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    $\begingroup$ @DaveBenson I have added some comments about grading conventions and expectations about products vs coproducts $\endgroup$ Commented Jun 2, 2023 at 10:20
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    $\begingroup$ @Z.M this should probably be a separate question, but the short answer is: yes, there is a stable elements formula just as for group cohomology, obtained analogously from the Mackey functor structure. $\endgroup$ Commented Jun 2, 2023 at 12:25

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Your resolution is the minimal Tate complex resolving the trivial module for $\mathbb{F}_p\Sigma_p$. Each of the exterior powers of the natural permutation module is indecomposable and projective. The images of the maps are exactly the non-projective Specht modules for $\mathbb{F}_p\Sigma_p$. Each of the modules in your sequence has four composition factors, apart from the "end" ones $A^1$ and $A^{p-1}$, and they have three. Here is an illustration for $p=7$. $$\cdots\to\begin{smallmatrix}\bar1\\\bar5\\\bar1\end{smallmatrix}\to\begin{smallmatrix}\bar1\\\bar5\\\bar1\end{smallmatrix}\to\begin{smallmatrix}\bar5\\\overline{10}\ \bar1\\\bar5\end{smallmatrix}\to\begin{smallmatrix}\overline{10}\\10\ \bar5\\\overline{10}\end{smallmatrix}\to\begin{smallmatrix}10\\5\ \overline{10}\\10\end{smallmatrix}\to\begin{smallmatrix}5\\1\ 10\\5\end{smallmatrix}\to\begin{smallmatrix}1\\5\\1\end{smallmatrix}\to\begin{smallmatrix}1\\5\\1\end{smallmatrix}\to \cdots$$ Here, the overlines mean tensor with the sign representation. Since there are non-trivial Massey products for $p\geqslant 3$, this means no associative product on the minimal resolution.

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