Homological conjectures for finite dimensional commutative algebras

Question

$\DeclareMathOperator\Ext{Ext}\DeclareMathOperator\Hom{Hom}$>Question: What are some (open) homological conjectures that are also relevent for finite dimensional commutative algebras over a field $K$?

I just know two such conjectures, namely:

The first Tachikawa conjecture: $A$ is non-Gorenstein if and only if there exists an $i \geq 1$ with $\Ext_A^i(D(A),A) \neq 0$ where $D(A)=\Hom_K(A,K)$.

The second Tachikawa conjecture: When $A$ is Gorenstein, then a module $M$ is projective if and only if $\Ext_A^i(M,M) =0$ for all $i \geq 1$.

Answer 1

Here's one of my favourites. It's about homology of complexes of free modules.

Let $E$ be an elementary abelian $p$-group of rank $r$, let $k$ be a field of characteristic $p$. The group algebra $kE$ is then a truncated polynomial ring $k[t_1,\dots,t_r]/(t_1^p,\dots,t_r^p)$. Let $F$ be a bounded complex of finitely generated free $kE$-modules, which is not exact. An old conjecture states that the homology of $F$ has total dimension at least $2^r$. Srikanth Iyengar and Mark Walker have found counterexamples to this, and to some closely related conjectures, for every $p > 2$, with $r \geqslant 8$. But it is unknown whether the conjecture is true for $p=2$. In this case the conjecture is due to Gunnar Carlsson. A proof or counterexample for $p=2$ would have implications in several neighbouring subjects.

Note that when $p=2$, $kE$ is just an exterior algebra. The same conjecture can be made about exterior algebras, and again Iyengar and Walker construct counterexamples over any field not of characteristic two.