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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?

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2 Answers 2

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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. 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.

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  • $\begingroup$ To clarify, string modules over any quotient of one of the algebras mentioned above by zero relations has at worst an infinite periodic resolution. In particular, once the projective cover of the socle is covered in a resolution, the resolution of s string module becomes infinite periodic for what Ringel calls "complete gentle algebras". Any quotient of a complete gentle algebra by zero relations will then have at worst an infinite periodic resolution. If the projective modules are finite dimensional (i.e. finite dimensional algebra obtained via a quotient) then the resolutions become finite. $\endgroup$
    – Amelie Schreiber
    Commented Aug 14, 2022 at 17:31
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    $\begingroup$ Why is every quadratic algebra a quotient of one of the algebras that you refer to? If I understand correctly, the algebras you refer to are all special biserial algebras (which are tame), but a quadratic algebra can be wild. $\endgroup$
    – Jeremy Rickard
    Commented Aug 14, 2022 at 19:18
  • $\begingroup$ define quadratic algebra for me $\endgroup$
    – Amelie Schreiber
    Commented Aug 16, 2022 at 18:15
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    $\begingroup$ I think the definition that the OP has in mind for a "quadratic finite dimensional quiver algebra" is an algebra given by a quiver and relations, where the relations are quadratic (i.e., linear combinations of length two paths). $\endgroup$
    – Jeremy Rickard
    Commented Aug 16, 2022 at 18:29
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    $\begingroup$ The same definition that I said in my last comment that I thought the OP intended. I’m curious to know what definition you intended? Maybe there are multiple unrelated definitions in use. $\endgroup$
    – Jeremy Rickard
    Commented Aug 16, 2022 at 19:55
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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.

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    $\begingroup$ Quiver path algebras with no relations are quadratic. And quadratic algebras can certainly have minimal projective resolutions of length greater than two: for example, the $A_n$ quiver with arrows in the same direction, with relations saying all paths of length two are zero, has a simple module with projective dimension $n-1$. $\endgroup$
    – Jeremy Rickard
    Commented Jul 2, 2022 at 5:31

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