Are there any connective $E\_1$ rings $R$ over $\mathbb{F}\_p$ satisfying the following?


1. $\pi\_\*(R)$ is a finite dimensional $\mathbb{F}\_p$ vector space
2. $R$ is compact as a module over $R \otimes R^{op}$
3. $R$ is not concentrated in degree 0


Motivation: I am looking for a smooth proper category with negative Hochschild cohomology and someone suggested $\mathit{Perf}(R)$ with $R$ satisfying the above if it existed. In fact they suggested to look for a finite CW complex $X$ with
$H\_\*(\Omega X,\mathbb{F}\_p)$ finite over $\mathbb{F}\_p$ (then we can take $R=H\_\*(\Omega X,\mathbb{F}\_p)$) but I have not found any of those either.


Edit: As Shaul Barkan explained to me, such an example cannot come from a space X as described above, because the conditions will imply that $H\_\*(\Omega X,\mathbb{F}\_p)$ is in degree 0.



Are there strictly connective smooth proper algebras over $\mathbb{F}_p$?
Here's an example: consider the $\infty$-category $Fun(\Delta^1,Perf(\mathbb F\_p))$. It has two canonical generators $A= \mathbb F\_p\to \mathbb F\_p$ and $B=\mathbb F\_p\to 0$; and I claim that $R= End(A\oplus \Sigma B)$ is an example of what you're looking for (derived endomorphisms - more generally, everything here is "derived").


Note that $\hom(A,B) = \mathbb F\_p$, $\hom(B,A)=0$, $\hom(A,A) = \mathbb F\_p, \hom(B,B)= \mathbb F\_p$.


In particular, $End(A\oplus\Sigma B) = \hom(A,\Sigma B)\oplus \hom(A,A)\oplus \hom(\Sigma B,A)\oplus \hom(\Sigma B,\Sigma B)$.


Note that $\hom(\Sigma B,\Sigma B)\simeq \hom(B,B)$, $\hom(\Sigma B,A) = 0$, so really $End(A\oplus \Sigma B)= \hom(A,A)\oplus \hom(B,B)\oplus \Sigma\hom(A,B)\simeq \mathbb F\_p\oplus\mathbb F\_p\oplus\Sigma\mathbb F\_p$.


In particular, $End(A\oplus\Sigma B)$ is connective and perfect over $\mathbb F\_p$, so it remains to argue that it is smooth. But smoothness is *Morita invariant*, i.e. it only depends on the category of modules.


Now, $A,B$ are generators of $Fun(\Delta^1,Perf(\mathbb F\_p))$, so $Perf(End(A\oplus\Sigma B))\simeq Fun(\Delta^1,Perf(\mathbb F\_p))$ by the Schwede-Shipley theorem (I'm using implicitly here that $Ind(C^{\Delta^1}) = Ind(C)^{\Delta^1}$ for any stable $C$), so it suffices to argue that $Fun(\Delta^1,Perf(\mathbb F\_p))$ is smooth.


Now, for an indexing diagram $C$, $Fun(C,Mod\_R)^\omega$ is smooth over $R$ if and only if $R[map\_C(-,-)]$ is compact in $Fun(C^{op}\times C,Mod\_R)$, and one can show that $map(R[map\_C(-,-)], F)\simeq \int\_{c\in C}F$, the *end* of $F$. It follows that $Fun(C,Mod\_R)^\omega$ is smooth if and only if $\int\_{c\in C}: Fun(C^{op}\times C,Mod\_R)\to Mod\_R$ preserves filtered colimits.


Recall that ends are a special case of limits over $Tw(C)$, so it suffices (but is not equivalent, as far as I know) for $\lim\_{Tw(C)}$ to preserve filtered colimits. In this case, however, $Tw(\Delta^1)$ is a *finite* $\infty$-category, so this stronger condition is satisfied, and $Fun(\Delta^1,Perf(\mathbb F\_p))$ is smooth, hence so is $R$.


