A formula for the projective dimension of finite dimensional algebras

Question

Let $A$ be a finite dimensional ring-indecomposable $K$-algebra that is not selfinjective for $K$ a field and let $I(A)$ denote the injective envelope of the regular module $A_A$. Define the stable $A$-module $\underline{A}$ as the cokernel of the trace of $I(A)$ in $A$. Recall here that the trace of a module $N$ in a module $M$ is the maximal submodule of $M$ generated by $N$, see for example https://www.math.uni-bielefeld.de/~ringel/opus/good-d-r.pdf at the end of page 2.

I have the guess that for a general $A$-module $M$ we have the formula $$pdim M= sup \{ i \geq 0 \mid Ext_A^i(M,\underline{A}) \neq 0 \}.$$

Question: Is this true?

This would have some nice homological applications. I can show a similar formula for the dominant dimension, so I would expect this formula to be true for the projective dimension but I have not found a proof. Also several computer experiments suggest that this question has a positive answer although the class of examples tested with the computer were rather random and not too complicated.

Answer 1

There are very easy examples (e.g., the path algebra of an $A_2$ quiver) where there is a nonzero projective module $P$ with $\operatorname{Hom}_A(P,\underline{A})=0$, so I assume you mean $\sup(\emptyset)$ to be interpreted as $0$ rather than $-\infty$.

Here's a way to get examples of a module $M$ with $\operatorname{pdim}M=\infty$ and $\operatorname{Ext}^i_A(M,\underline{A})=0$ for all $i\geq0$.

Let $B$ and $C$ be non-semisimple weakly symmetric algebras (over an algebraically closed field $K$ so that I don't need to worry) where $C$ has at least two nonisomorphic simple modules (where "weakly symmetric" means self-injective with the projective cover of each simple coinciding with its injective hull: e.g., a finite group algebra).

Let $S$ be a simple $B$-module and let $T$ be a simple $C$-module, and let $$A=\begin{pmatrix}k&S\otimes_KT\\0&B\otimes_KC\end{pmatrix},$$ a one-point extension algebra of $B\otimes_KC$.

Every indecomposable projective (right) $A$-module is also injective, other than the projective covers of the simple modules $\begin{pmatrix}0&S\otimes_KT\end{pmatrix}$ and $\begin{pmatrix}k&0\end{pmatrix}=\begin{pmatrix}k&S\otimes_KT\end{pmatrix}/\begin{pmatrix}0&S\otimes_KT\end{pmatrix}$. So the only simple $A$-modules that are composition factors of $\underline{A}$ are $\begin{pmatrix}0&S\otimes_KT\end{pmatrix}$ and $\begin{pmatrix}k&0\end{pmatrix}$.

Let $Q$ be an indecomposable projective $C$-module other than the projective cover of $T$, and let $M=\begin{pmatrix}0&S\otimes_KQ\end{pmatrix}$.

If $\cdots\to P_2\to P_1\to P_0$ is a projective resolution of $S$ as a $B$-module, then $\cdots\to P_2\otimes_KQ\to P_1\otimes_KQ\to P_0\otimes_KQ$ is a projective resolution of $S\otimes_KQ$ as an $A$-module, and since no term of this resolution involves the projective cover of $\begin{pmatrix}0&S\otimes_KT\end{pmatrix}$ or $\begin{pmatrix}k&0\end{pmatrix}$, $\operatorname{Ext}^i_A(M,\underline{A})=0$ for all $i\geq0$.

This leaves open the question of what happens if $0<\operatorname{pdim}M<\infty$.