# injective dimension of envelope algebras

Let $A$ be a connected graded algebra and $A^o$ the opposite algebra of $A$. Let $A^e=A\otimes A^o$. Suppose that $A$ has a finitely generated projective resolution as a graded $A^e$-module.

My first question is that whether the graded injective dimension of $A_A$ and ${}_AA$ are finite implies the graded injective dimension of ${}_{A^e}A^e$ is finite.

Let $\mathfrak{m} = A_{\geq0}$ be the maxiamal homogeneous ideal of $A$. Let $\Gamma_{\mathfrak{m}}$ be the torsion functor at $\mathfrak{m}$.

My second question is that whether the graded injective dimension of $A_A$ and ${}_AA$ are finite implies the cohomological dimension of $\Gamma_\mathfrak{m}$ is finite.

• Do you mean that $A$ has a finitely generated projective resolution? – Dag Oskar Madsen Mar 2 '17 at 18:54
• Yes, in the category of finitely generated graded $A^e$-modules. Of course, it is equivalent to non-graded version since $A$ is connected graded. – G.-S. Zhou Jan 20 '18 at 9:30