The Cohen–Fischman semiprime smash product question and Jacobson-radical stability for Hopf actions on PI-algebras

Two linked open questions from Linchenko and Montgomery (2007) on radical stability under finite-dimensional Hopf algebra actions. (A) The Cohen–Fischman question (1986): if a finite-dimensional semisimple Hopf algebra $H$ over a field acts on a semiprime algebra $A$, must the smash product $A\#H$ be semiprime? This is proved when $A$ is Noetherian (Skryabin–van Oystaeyen 2006) and when $A$ satisfies a polynomial identity (Linchenko–Montgomery 2007), and by a duality lemma it is equivalent to stability of the prime radical $P(A)$ under every action of a finite-dimensional cosemisimple Hopf algebra; the case of an arbitrary semiprime $A$ remains open. (B) If a finite-dimensional cosemisimple Hopf algebra acts on an arbitrary PI-algebra $A$, is the Jacobson radical $J(A)$ stable under the action? This is known when $A$ is affine or algebraic over the base field and when $A$ is Noetherian, but open for general PI-algebras. Both questions remain open: no proof or counterexample is known for either.

Background

A Hopf algebra $H$ over a field $k$ is an algebra equipped with a comultiplication $\Delta$, a counit $\varepsilon$, and an antipode $S$, simultaneously making the dual space $H^*$ a Hopf algebra; the basic examples are group algebras $kG$ and enveloping algebras of Lie algebras. An $H$-module algebra is an associative algebra $A$ on which $H$ acts linearly and compatibly with multiplication, $h\cdot(ab)=(h_{(1)}\cdot a)(h_{(2)}\cdot b)$ where $\Delta(h)=h_{(1)}\otimes h_{(2)}$. The smash product $A\#H$ is then the vector space $A\otimes_k H$ with multiplication twisted by the action,

$$(a\#h)(b\#l)=a(h_{(1)}\cdot b)\#h_{(2)}l,$$

for $H=kG$ this recovers the skew group ring of a finite group action. A ring is semiprime if it has no nonzero nilpotent (two-sided) ideal, equivalently if its prime radical $P(A)$ (the intersection of its prime ideals, also called the lower nilradical) is zero; the Jacobson radical $J(A)$ is the intersection of the annihilators of all simple left $A$-modules and contains $P(A)$, with equality for artinian rings but not in general. A PI-algebra is one satisfying a polynomial identity, such as a commutative algebra or a matrix algebra over a commutative ring. A finite-dimensional Hopf algebra is semisimple if it is semisimple as an algebra, and cosemisimple if its dual $H^*$ is semisimple; it is called involutive if $S^2=\mathrm{id}$. In characteristic $0$, Larson and Radford proved that a finite-dimensional Hopf algebra is semisimple if and only if it is cosemisimple, and then it is involutive; in prime characteristic these equivalences fail in general. An ideal $I$ of an $H$-module algebra is $H$-stable if $H\cdot I\subseteq I$, and $A$ is $H$-semiprime if it has no nonzero nilpotent $H$-stable ideals; every semiprime algebra is $H$-semiprime.

For finite group actions the analogue of the main question is classical: Fisher and Montgomery (1978) and Lorenz and Passman (1980) proved that the skew group ring of a finite group acting on a semiprime ring is semiprime. Cohen and Fischman (1986) asked whether the same holds for semisimple Hopf algebras: is $A\#H$ semiprime whenever $H$ is finite-dimensional semisimple and $A$ is a semiprime $H$-module algebra? This became known as the Cohen–Fischman conjecture, and it has resisted a general answer for four decades. The confirmed cases all impose conditions on $A$: Linchenko, Montgomery and Small (2005) handled affine PI-algebras in characteristic $0$; Skryabin and van Oystaeyen (2006), proving a Goldie-type theorem for $H$-semiprime algebras, settled all Noetherian $A$ (if $H$ is semisimple and $A$ is Noetherian and $H$-semiprime, then $A\#H$ is semiprime); and Linchenko and Montgomery (2007) settled all PI-algebras (if $H$ is semisimple and $A$ is an $H$-semiprime PI-algebra, then $A\#H$ is semiprime). The same 2007 paper proved the dual statement that for cosemisimple $H$ the prime radical of any PI $H$-module algebra is $H$-stable, and made the duality precise (their Lemma 2.6): for a fixed finite-dimensional $H$, the prime radical of every $H$-module algebra is $H$-stable if and only if $A\#H^*$ is semiprime for every $H^*$-semiprime $H^*$-module algebra $A$. Hence a positive answer to the Cohen–Fischman question for all semisimple Hopf algebras is equivalent to stability of prime radicals under all finite-dimensional cosemisimple actions, so the semiprimeness question and the radical-stability question are two faces of one problem.

The Jacobson radical is the subtler of the two radicals: for PI-algebras that are affine or algebraic over the base field, Amitsur–Kaplansky theory gives $J(A)=P(A)$, and Linchenko and Montgomery deduced that $J(A)$ is $H$-stable for cosemisimple $H$; but for an arbitrary PI-algebra $J(A)$ can strictly exceed $P(A)$, and they stated that stability of $J(A)$ under cosemisimple actions on an arbitrary PI-algebra remains open. Skryabin (2015) later proved $J(A)$ $H$-stability for all Noetherian $H$-module algebras under finite-dimensional cosemisimple $H$, removing the PI hypothesis but keeping a chain condition. Outside the Noetherian and PI worlds, neither quotient-ring techniques nor PI structure theory is available; this is exactly where question (A) stands open, whereas question (B) is open for PI-algebras that are not Noetherian, affine over $k$, or algebraic over $k$.

Problem Statement

Two linked questions, posed in Linchenko–Montgomery (2007) as the open part of the Cohen–Fischman program; the common objective is a complete radical-stability and semiprimeness theory for finite-dimensional (co)semisimple Hopf algebra actions. Neither question implies the other.

(A) Let $H$ be a finite-dimensional semisimple Hopf algebra over a field $k$ and let $A$ be a semiprime $H$-module algebra. Is the smash product $A\#H$ semiprime? By the duality lemma of Linchenko–Montgomery, a positive answer for all such $H$ is equivalent to: for every finite-dimensional cosemisimple Hopf algebra acting on any algebra $B$, the prime radical $P(B)$ is $H$-stable. The answer is known to be positive when $A$ is Noetherian or satisfies a polynomial identity, so the question is open precisely for semiprime algebras that are neither Noetherian nor PI, in any characteristic.

(B) Let $H$ be a finite-dimensional cosemisimple Hopf algebra over a field $k$ acting on a PI-algebra $A$, with no further hypotheses on $A$. Is the Jacobson radical $J(A)$ $H$-stable? This is known when $A$ is Noetherian, affine over $k$, or algebraic over $k$, and open for arbitrary PI-algebras.

References

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