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 $(p-1)!$.

3more comments