# When is the Jacobson radical reflexive?

Let algebras be Artin algebras. It is well known that a an algebra has global dimension at most one if and only if the Jacobson radical is projective. As reflexive is a natural generalisation of projective, one might ask when the Jacobson radical $$J$$ is reflexive. I noted that for Nakayama algebras one has that $$J$$ is reflexive if and only if the finitistic dimension is at most one. But in general such a statement would probably be too good to be true.

Question: Is there an easy counterexample to $$J$$ being reflexive implies finitistic dimension at most 1?

(the other direction is not true, see answer below)

Reflexive means that the canonical evaluation map $$f_M:M \rightarrow M^{**}$$ is an isomorphism where for an algebra $$A$$: $$M^{**}=Hom_A(Hom_A(M,A),A)$$. Here $$f_M(m)=g$$ with $$g(h)=h(m)$$.

• Could your remind us what "reflexive" means in this context? Nov 27 '19 at 15:40
• @YemonChoi I added the definition.
– Mare
Nov 27 '19 at 15:58

The algebra $$K[x,y]/(x^2,y^2,xy)$$ has finitistic dimension 0 but $$J$$ is semisimple and not reflexive. I have not yet found an example where $$J$$ is reflexive but the algebra has finitistic dimension larger than 1.