This can be seen as an attempt for a mini Polymath project on homological properties of (local) finite dimensional algebras. You only need to know what a finite dimensional algebra is and have GAP to contribute. You can win nice prizes (see the end of the question). A local quiver algebra is just an algebra of the form $A=K\langle x_1,...,x_p\rangle/I$ where there are at least $p \geq 2$ variables (we avoid trivial cases) and $K\langle x_1,...,x_p\rangle$ is the non-commutative polynomial ring in $p$ variables and $I$ is an ideal with $J^r \subseteq I \subseteq J^2$ where $J=\langle x_1,...,x_p\rangle$ and some $r \geq 2$. Let $D(A)=\operatorname{Hom}_K(A,K)$ and define $e_i:=\dim_K(\operatorname{Ext}_A^i(D(A),A))$ for $i \geq 1$. $A$ is called Frobenius in case $A \cong D(A)$ or equivalently $A$ has a simple socle. We are only interested in non-Frobenius algebras in the following and assume $A$ is non-Frobenius.
You can enter such an algebra and calculate $e_i$ for $i=1,..,n$ as follows in the GAP-package QPA (a specific example, that should be easy to modify to enter your favorite local algebra):
n:=3;Q := Quiver(1, [[1,1,"a"],[1,1,"b"]]);KQ := PathAlgebra(GF(3), Q);AssignGeneratorVariables(KQ);rels := [a^2, ab+b^2-aba, ab^2, bab, ab+b^2+b^2a, b^3];A := KQ/rels;IsSelfinjectiveAlgebra(A);P:=IndecProjectiveModules(A)[1];;I:=IndecInjectiveModules(A)[1];;W:=[Size(ExtOverAlgebra(I,P)[2])];;for i in [1..n-1] do Append(W,[Size(ExtOverAlgebra(NthSyzygy(I,i),P)[2])]);od;W;IsSelfinjectiveAlgebra(A);
(you need to write a * between the variables in the relations to enter it in QPA, which is sadly not shown in the Mathoverflow output using the code ) The output at the end is $[ 0, 4, 4 ]=[e_1,e_2,e_3]$ and a "false" (meaning that your algebra is not Frobenius). The general output will be $[e_1,...,e_n]$ where $n$ is the first specified parameter in the input in case your algebra is indeed finite dimensional (in case it is not finite dimensional you should get an error and in case it is selfinjective you will see a "true" at the end in which case your example is not of the needed form). You can specify the number of variables in the input Q := Quiver(1, [[1,1,"a"],[1,1,"b"]]), here we have 2 variables a and b. And you can specify the relations in the input rels := [a^2, ab+b^2-aba, ab^2, bab, ab+b^2+b^2a, b^3]; . Thus the algebra here is $K\langle a,b\rangle/\langle a^2, ab+b^2-aba, ab^2, bab, ab+b^2+b^2a, b^3\rangle$. You can easily specify the parameter $n$ (taking $n=3$ will be enough for some first experiments) and define your own algebra.
Here are the questions/prizes:
- (Main prize): Find such an algebra $A$ with $e_i=0$ for all $i \geq 1$. You then disproved the second Tachikawa conjecture, the Nakayama conjecture and the finitistic dimension conjecture (and several other conjectures), which are the most important homological conjectures for finite dimensional algebras.
A discussion of the finitistic dimensions can be found in https://link.springer.com/article/10.1007/BF02100610 where also another related conjecture was disproven.
- (Second prize): Find an infinite class of algebras $A_t$ with $e_i=0$ for $i=1,...,a_t$ ($a_t$ depending on $A$), where $a_t$ gets arbitrary large. You then disproved the Yamagata conjecture.
For Yamagata's conjecture, see for example https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/rigidity-dimension-of-algebras/6A8979068A79714D91394B746C42BA31 .
Find a commutative such $A$ with $e_1=0$. No such example is known, so you answer an open question.
Find an $A$ with $e_1=e_2=0$. No such example is known, so you answer an open question.
Find such an algebra $A$ having a d-cluster tilting module for some $d \geq 2$ (for this you need $e_1=e_2=...=e_{d-1}=0$). Prize (by me): 50 Euros for the first new (not isomorphic to $K\langle a,b\rangle/\langle a^2, ab+b^2-aba, ab^2, bab, ab+b^2+b^2a, b^3\rangle$) such algebra.
(note that it is also not known whether there is a local Frobenius algebra having a $d$-cluster tilting module for $d \geq 2$. Not even a non-projective module $M$ with $Ext_A^1(M,M)=0$ is known for local Frobenius algebras $A$)
This is the only non-elementary question. There is no fixed method to find cluster-tilting modules but when you find a nice algebra with $e_1$=0 and complexity one it might be a good candidate. Here complexity one can for example be checked by the property of the simple module having bounded projective resolution. To do this enter this code after you entered your algebra:
S:=SimpleModules(A)[1];G:=[];for i in [1..10] do Append(G,[Dimension(NthSyzygy(S,i))]);od;G;
In case this can be done quickly by the computer and the numbers are small, it is likely you found an algebra with complexity one (extremely rare for such algebras. I think for commutative (non-Frobenius) algebras no such example might be known). The output for the algebra as above is [ 5, 7, 5, 7, 5, 7, 5, 7, 5, 7 ]. For a general local algebra the output will be a quickly increasing sequence tending to infinity.
The only example known to mankind for a d-cluster tilting module for such algebras $A$ is the algebra $K\langle a,b\rangle/\langle a^2, ab+b^2-aba, ab^2, bab, ab+b^2+b^2a, b^3\rangle$ entered above, which appeared in Does this algebra have finite global dimension ? (Human vs computer) . It was found by Jan Geuenich by searching through some (random?) relations and I found a 2-precluster tilting module and Oyvind Solberg indeed proved that it was even cluster tilting in the cited thread. The relations are very weird, but it would be strange if there is really just one local algebra with (higher) cluster tilting modules. Noone knows how to construct another example, so some creativity and experiments might help to find more such examples! The algebra has just vector space dimension 6, so maybe there are other algebras out there of small dimension that can be found using some weird relations. Note that the relations of this example can also be written in the form $rels := [a^2, (a+b)^2-(b^2+b)a]$ to give an isomorphic algebra.
