Finitistic dimension conjecture for quadratic algebras The finitistic dimension of a finite dimensional algebra is defined as the supremum of all projective dimensions of modules having finite projective dimension. The finitistic dimension conjecture says that the finitistic dimension is always finite.

Question: Is it known whether the finitistic dimension conjecture is true for quadratic finite dimensional quiver algebras?

 A: Every quadratic algebra is a quotient of one of the following algebras: https://arxiv.org/pdf/1706.07022.pdf if it has relations defined by setting certain paths to zero. So the periodic resolutions mentioned in https://arxiv.org/abs/1810.06750 show that the supremum of projective dimensions of the Finitistic Dimension Conjecture can only decrease to finite resolutions when adding more relations to such algebras.
The other option is algebras obtained by adding commutativity relations. This will be solved in a followup comment. The reasoning is to show the projective dimension is preserved or at least bounded in one direction or the other (can't increase or decrease by too much) by the gentle algebra it was obtained from when going from gentle to skew-gentle.
A: Projective dimension is defined as the length of a minimal projective resolution of a module. As an example, take the $\mathbb{A}_3$ quadratic algebra:
$$\bullet \to_a \bullet \to_b \, ,$$
with the relation $ba = 0$. Label the simple modules represented by vertices from left to right by (1), (2), and (3). The projective resolutions are:
(1) ← P_1 ← P_2 ← P_3=(3)  
(2) ← P_2 ← (3)  
(3)

Note: Quiver path algebras with no relations (and therefore non-quadratic) are called "hereditary" and the minimal projective resolutions are always of length 1. Quadratic algebras will only ever have minimal resolutions of length 2. Infinite dimensional algebras with relations will often have infinite length projective resolutions, and some may even pe periodic.
