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Given a finite dimensional algebra $A$ with Jacobson radical $J$. Is the global dimension of $A$ equal to the injective dimension of $J$? I can prove this for algebras with finite global dimension and in general it should be equivalent to the inequality $injdim(J) \geq injdim(J^2)-1$, see my answer.

Given a finite dimensional algebra $A$ with Jacobson radical $J$. Is the global dimension of $A$ equal to the injective dimension of $J$?

Together with Xiao-Wu Chen and Srikanth Iyengar we proved this now more generally for any semi-perfect noetherian ring $A$.

Given a finite dimensional algebra $A$ with Jacobson radical $J$. Is the global dimension of $A$ equal to the injective dimension of $J$? I can prove this for algebras with finite global dimension and in general it should be equivalent to the inequality $injdim(J) \geq injdim(J^2)-1$, see my answer.

See also https://mathoverflow.net/questions/288745/injective-dimensions-of-radical-powers .

Given a finite dimensional algebra $A$ with Jacobson radical $J$. Is the global dimension of $A$ equal to the injective dimension of $J$? I can prove this for algebras with finite global dimension and in general it should be equivalent to the inequality $injdim(J) \geq injdim(J^2)-1$, see my answer.

The following might be well known and/or embarrassingly simple. The motivation is the well known result by Auslander that the projective dimension of the Jacobson radical equal the global dimension minus one. Let $A$ be a finite dimensional algebra with Jacboson radical $J$ and radical series $0=J^n \subseteq J^{n-1} \subseteq ... \subseteq J$.

Are the following true and/or well known:

1. Is the global dimension of $A$ equal to the injective dimension of $J$?

2.Is the sequence $id(J^i)$ of injective dimensions of $J^i$ weakly decreasing for algebras with finite global dimension (or general algebras)?

3. Is the sequence $pd(J^i)$ of projective dimensions of $J^i$ weakly decreasing for algebras with finite global dimension (or general algebras)?

I did tests for some quiver algebras and found no counter example, but those algebras had rather special properties. For example I found a proof of 1. for higher Auslander algebras.

Given a finite dimensional algebra $A$ with Jacobson radical $J$.  Is the global dimension of $A$ equal to the injective dimension of $J$? I can prove this for algebras with finite global dimension and in general it should be equivalent to the inequality $injdim(J) \geq injdim(J^2)-1$, see my answer.

See also https://mathoverflow.net/questions/288745/injective-dimensions-of-radical-powers .