Is being a Frobenius algebra a rare condition for local algebras?

Question

Let $U_{r,l,q}$ be the set of finite dimensional local algebras $A$ over a finite field with $q$ elements such that $J/J^2$ is $r$-dimensional for a number $r \geq 2$ and such that $J^l=0$ for the Jacobson radical $J$ (=unique maximal ideal) of $A$ for some $l \geq 2$. Let $W_{r,l,q}$ be the subset of $U_{r,l,q}$ consisting of those algebras that are additionally Frobenius algebras (or equivalently Gorenstein algebras). Let $u_{r,l,q}$ be the cardinality of $U_{r,l,q}$ and $w_{r,l,q}$ the cardinality of $W_{r,l,q}$. Question:

Is it true that $\frac{w_{r,l,q}}{u_{r,l,q}}$ goes to zero for $l \rightarrow \infty $ for any $r,q$? Can one even give asymptotics for spcific values of $r$ and $q$?

What if we restrict to commutative algebras?

Answer 1

Here is an emphatically "no-brainer", and partial, answer, just for the commutative case.

In the commutative case each such algebra $A$ is a quotient of a polynomial ring $R = k[x_1,\dotsc,x_r]$ ($k$ is a field), $A = R/I$, where, for $A$ to be local, $I$ is contained in a unique maximal ideal $\mathfrak{m}$. Up to translation in $\operatorname{Spec}(R) \cong k^r$ we might as well assume $\mathfrak{m}=(x_1,\dotsc,x_r)$. The condition $\dim J/J^2 = r$ means that $I$ contains no linear forms. (We don't assume $I$ is homogeneous.) The assumption $J^l = 0$ means that $\mathfrak{m}^l \subseteq I$.

Here is a crude lower bound for the family $U$: we can just take ideals $I$ such that $\mathfrak{m}^l \subseteq I \subset \mathfrak{m}^{l-1}$. Then ideals $I$ correspond to (arbitrary) proper subspaces of $\mathfrak{m}^{l-1} / \mathfrak{m}^l$, which is a vector space of dimension $\binom{r+l-2}{r-1}$. For each $t$ we get a family of algebras in $U$ corresponding to $t$-dimensional subspaces of the quotient, equivalently, points in the Grassmannian $G(t,\mathfrak{m}^{l-1}/\mathfrak{m}^l)$. The biggest of these is when $t = \frac{1}{2} \binom{r+l-2}{r-1}$, in which case the Grassmannian has dimension $\frac{1}{4} \binom{r+l-2}{r-1}^2$, which is asymptotically of the order $l^{2r-2}$. This is a lower bound for the dimension of the family $U$, rather than the cardinality you asked for.

What about $W$? There is a bijective correspondence between ideals $I$ such that $I \subseteq \mathfrak{m}$ and $R/I$ is a local Gorenstein ring, and nonzero polynomials $f \in R$, up to scalar multiple. See Theorem 21.6 in Commutative algebra by Eisenbud. The condition $J^l = 0$ in $A = R/I$ is equivalent to $\deg f \leq l-1$. The polynomials of degree $ \leq l-1$ are a vector space of dimension $\binom{r+l-1}{r}$, which is $O(l^r)$. (Minus $1$, for the scalar multiple; doesn't matter, asymptotically.)

So, if I haven't made too many horrible mistakes, then $\frac{\dim W}{\dim U}$ is $O(l^{-r+2})$, which goes to $0$ as $l \to \infty$, for $r>2$.

Both $W$ and our very crude lower bound for $U$ are Grassmannians ($W$ is actually a projective space). Over a finite field $k$ with $q$ elements, a Grassmannian of dimension $t$ has $O(q^t)$ points (very crude approximation). So $$ \frac{w}{u} \approx \frac{q^{l^r}}{q^{l^{2r-2}}} = q^{l^r - l^{2r-2}} . $$ This goes to $0$ as $l \to \infty$, for $r > 2$.

Well, I thought I had it also for $r=2$, but I don't. Sorry.