Intrinsic criteria for equality of the graded and ordinary Jacobson radical of finite groupoid graded algebras beyond trace invertibility

For a K-algebra A graded by a finite groupoid $\mathcal{G}$, Flôres–Morgado–Paques (2019) proved that the graded Jacobson radical $J_{\mathcal{G}}(A)$ is contained in the ordinary Jacobson radical $J(A)$, and that equality holds under the additional hypothesis that the trace element $tr_{\beta}(1)$ of the canonical action $\beta$ on the smash product $A\#K\mathcal{G}^*$ is invertible. The open problem is to find intrinsic, purely grading-theoretic conditions that ensure equality $J_{\mathcal{G}}(A)=J(A)$ — in particular, whether invertibility of $tr_{\beta}(1)$ is necessary for equality or can be strictly weakened. No such intrinsic conditions are known.

Background

A groupoid is a category in which every morphism is invertible; unlike a group, the composite $gh$ of two morphisms need only be defined when source and target match. A K-algebra $A$ is graded by a finite groupoid $\mathcal{G}$ if $A=\bigoplus_{g\in\mathcal{G}}A_g$ as a vector space and $A_gA_h\subseteq A_{gh}$ whenever the composite $gh$ is defined; this specializes to the classical notion of a group-graded ring when $\mathcal{G}$ is a group, and captures, for example, algebras built from partial actions and étale groupoids. In the unital framework of the source paper and its companion Della Flora–Flôres–Morgado–Tamusiunas (2022), when $\mathcal{G}$ is finite each identity component $A_e$ is unital with identity element $1_e$ and $1_A=\sum_{e\in\mathcal{G}_0}1_e$; correspondingly, $1_{d(g)}$ in the trace element below denotes the identity of the component $A_{d(g)}$ at the source $d(g)$ of $g$. The graded Jacobson radical $J_{\mathcal{G}}(A)$ is the largest homogeneous ideal of $A$ all of whose homogeneous elements are quasiregular (an element $x$ is quasiregular if $1-x$ is invertible, or has a quasi-inverse in the non-unital case). For group gradings, the question of how $J_{\mathcal{G}}(A)$ compares with the ordinary Jacobson radical $J(A)$ goes back to Bergman: in general only an inclusion and a nilpotency statement hold, and equality requires extra hypotheses. Flôres–Morgado–Paques transported this Bergman-type question to finite groupoid gradings through the duality theory of groupoid graded algebras: $K\mathcal{G}^*$ denotes the dual of the groupoid algebra (a weak Hopf algebra, i.e. a Hopf-like bialgebra whose comultiplication of the unit and counit of the product are weaker than in Hopf algebras), and $A\#K\mathcal{G}^*$ is the smash product for the canonical $K\mathcal{G}^*$-module algebra structure on $A$ given by the grading itself ($v_g\cdot a=a_g$ for the dual basis $\{v_g\}$), while $\beta$ denotes the canonical action of $\mathcal{G}$ on the smash product $A\#K\mathcal{G}^*$ arising from this duality, as in Flôres–Morgado–Paques (2019). They constructed a Morita context (a pair of bimodules with pairwise inverse up-to-ideal maps) linking the degree-identity component $A_1=\bigoplus_e A_e$ with $A\#K\mathcal{G}^*$, characterized semiprimality of $A\#K\mathcal{G}^*$ in terms of graded semiprimality and nondegeneracy of $A$, and proved: $J_{\mathcal{G}}(A)\subseteq J(A)$ always, with equality $J_{\mathcal{G}}(A)=J(A)$ when the trace element $tr_{\beta}(1)=\sum_{g\in\mathcal{G}}\beta_g(1_{d(g)})$ is invertible, where $d(g)$ is the domain (source) of $g$. Their Maschke-type splitting theorem also requires exactly this invertibility. What is missing is any intrinsic — purely grading-theoretic, smash-product-free — understanding of when equality holds.

Problem Statement

Let $\mathcal{G}$ be a finite groupoid and let $A$ be a unital $K$-algebra graded by $\mathcal{G}$, with $\beta$ the canonical action of $\mathcal{G}$ on the smash product $A\#K\mathcal{G}^*$ and trace element $tr_{\beta}(1)=\sum_{g\in\mathcal{G}}\beta_g(1_{d(g)})$. Flôres–Morgado–Paques proved that $J_{\mathcal{G}}(A)\subseteq J(A)$ always holds, and that equality $J_{\mathcal{G}}(A)=J(A)$ holds whenever $tr_{\beta}(1)$ is invertible. Determine, intrinsically, conditions that ensure $J_{\mathcal{G}}(A)=J(A)$ — expressed in terms of the $\mathcal{G}$-graded structure of $A$ and the combinatorics of $\mathcal{G}$ alone and without reference to the smash product $A\#K\mathcal{G}^*$ or to trace invertibility. In particular, settle whether invertibility of $tr_{\beta}(1)$ is necessary for equality, or whether strictly weaker intrinsic grading-theoretic conditions already ensure it. An explicit finite groupoid graded algebra for which equality holds while $tr_{\beta}(1)$ is not invertible (refuting necessity) or for which the inclusion $J_{\mathcal{G}}(A)\subsetneq J(A)$ is strict (delimiting the boundary) illustrates the required conditions.

References

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