Generic forms
Ralf Fröberg, Clas Löfwall
TL;DR
The paper investigates generic presentations of graded algebras of type $t=(n;d_1,\ldots,d_r)$, analyzing how minimal Hilbert series characterize genericity across commutative, noncommutative, and Lie-type settings. It constructs universal families $A_t$ with indeterminate coefficients, establishes that generic algebras correspond to points in countable intersections of Zariski-open sets with constant Hilbert series, and proves key structural results such as a dichotomy for $\{x_i f_j\}$ in quadratic presentations. It also develops conjectures and partial results on minimal Hilbert series (notably $F_t(z)$ and the noncommutative conjecture $[1/(1-nz+rz^2)]$), and analyzes the Koszul dual: while not always generic, the dual is generic under algebraic independence of coefficients, with a sharp Koszul criterion for strictly generic noncommutative algebras. The findings illuminate the interplay between genericity, Hilbert series, and duality, with implications for homological properties and artinian thresholds in noncommutative and Lie contexts.
Abstract
We study forms $I=(f_1,\ldots,f_r)$, $°f_i=d_i$, in $F$ which is the free associative algebra $k\langle x_1,\ldots,x_n\rangle$ or the polynomial ring $k[x_1,\ldots,x_n]$, where $k$ is a field and $°x_i=1$ for all $i$. We say that $I$ has type $t=(n;d_1,\ldots,d_r)$ and also that $F/I$ is a $t$-presentation. For each prime field $k_0$ and type $t=(n;d_1,\ldots,d_r)$, there is a series which is minimal among all Hilbert series for $t$-presentations over fields with prime field $k_0$ and such a $t$-presentation is called generic if its Hilbert series coincides with the minimal one. When the field is the real or complex numbers, we show that a $t$-presentation is generic if and only if it belongs to a non-empty countable intersection $C$ of Zariski open subsets of the affine space, defined by the coefficients in the relations, such that all points in $C$ have the same Hilbert series. In the commutative case there is a conjecture on what this minimal series is, and we give a conjecture for the generic series in the non-commutative quadratic case (building on work by Anick). We prove that if $A=k\langle x_1,\ldots,x_n\rangle/(f_1,\ldots,f_r)$ is a generic quadratic presentation, then $\{ x_if_j\}$ either is linearly independent or generate $A_3$. This complements a similar theorem by Hochster-Laksov in the commutative case. Finally we show, a bit to our surprise, that the Koszul dual of a generic presentation is not generic in general. But if the relations have algebraically independent coefficients over the prime field, we prove that the Koszul dual is generic. Hereby, we give a counterexample of \cite[Proposition 4.2]{P-P}, which states a criterion for a generic non-commutative quadratic presentation to be Koszul. We formulate and prove a correct version of the proposition.