# Does this algebra have finite global dimension ? (Human vs computer)

Usually computers can calculate the global dimension of a finite dimensional quiver algebra much faster than humans. But in this case a high end computer (calculating for 3 weeks) was not able to determine whether this algebra has global dimension 3 or not.

Let $$A=K\langle a,b\rangle/I$$ with $$I$$ the ideal generated by $$\langle a^2, ab+b^2-aba, ab^2, bab, ab+b^2+b^2a, b^3\rangle$$ over a field $$K$$ of characteristic not two. Let $$D=\operatorname{Hom}_K(-,K)$$ the natural duality.

This algebra is a local non-Gorenstein algebra that was found by Jan Geuenich as a rare algebra with $$\operatorname{Ext}_A^1(D(A),A)=0$$. Let $$\tau_2 := \tau \Omega^1$$. Now let $$M:=A \oplus D(A) \oplus \tau_2(D(A)) \oplus \tau_2^2(D(A)) \oplus \tau_2^3(D(A))$$ and $$B:=\operatorname{End}_A(M)$$.

The module $$M$$ has vector space dimension 33 and the algebra $$B$$ has vector space dimension 165.

It can be shown that $$M$$ is a precluster tilting object in the sense of Iyama and Solberg - Auslander-Gorenstein algebras and precluster tilting and that the algebra has dominant dimension equal to the Gorenstein dimension equal to three. But the computer was not able to determine whether $$B$$ has finite global dimension (the global dimension is either 3 or infinite).

Thus the question:

Does $$B$$ have finite global dimension?

In case the answer is positive it would be the first 2-cluster tilting object for a local algebra in history! (at least to my knowledge)

I can think of two ways to determine the answer. The first is to check whether $$M$$ is a 2-cluster tilting object directly but $$A$$ is representation-infinite and one needs good knowledge of the module category of $$A$$ for that. The other way would be to calculate the quiver and relations of $$B$$ but this looks like a cruel torture when even a high end computer can not do it. So I hope there might be a good trick. $$B$$ has Cartan determinant 1, which makes it look like the global dimension could really be finite.

A positive answer would also answer this old question: Cluster-tilting object for a local non-selfinjective algebra .

• I don’t understand “The other way would be to calculate the endomorphism ring of B”. Typo? – Jeremy Rickard Jan 14 '19 at 13:58
• @JeremyRickard I meant the quiver and relations of $B$. That is what also the computer can not do. – Mare Jan 14 '19 at 14:02

Let $$M = P \oplus I \oplus \tau_2 \oplus \tau_2^2 \oplus \tau_2^3$$, where the notation is the obvious one. One way of computing the global dimension of $$B=\operatorname{End}(M)$$ is to find the projective resolution of all the simple $$B$$-modules. The simple $$B$$-modules are given by for each indecomposable direct summand $$M_i$$ of $$M$$ finding a "radical map" $$M_i\xrightarrow{f} M(i)$$ such that the cokernel of the induced map $$\operatorname{Hom}(M(i),M)\xrightarrow{\operatorname{Hom}(f,M)} \operatorname{Hom}(M_i,M)$$ is a simple $$B$$-module, where $$M(i)$$ is in $$\operatorname{add}M$$. All the maps from all the indecomposable direct summands of $$M$$ different from $$M_i$$ are "radical maps". Hence a left approximation $$f'\colon M_i\to \widehat{M_i}^{M_i}$$ by all the indecomposable direct summands of $$M$$ different from $$M_i$$ is part of $$f$$. In some cases it might be everything, for instance when cokernel of the induced map $$\operatorname{Hom}(f',M)$$ is one dimensional. Then we would have $$\operatorname{Hom}(\widehat{M_i}^{M_i},M) \to \operatorname{Hom}(M_i,M) \to S_{M_i}\to 0,$$ where $$S_{M_i}$$ is the simple $$B$$-module associated to the indecomposable projective module $$\operatorname{Hom}(M_i,M)$$. We can continue this projective resolution by finding a left $$\operatorname{add}M$$-approximation $$f_2\colon \operatorname{Coker}(f')\to M^{\operatorname{Coker}(f')}$$ of $$\operatorname{Coker}(f')$$. Before doing this we can remove any direct summands of $$\operatorname{Coker}(f')$$ isomorphic to an indecomposable direct summand of $$M$$. Then if $$K_2 = \operatorname{Coker}(f_2)$$ is in $$\operatorname{add}M$$, then $$\operatorname{pd}S_{M_i}\leq 3$$. Carrying out these computations, as far as I can see, one gets that $$\operatorname{pd}S_{M_i}$$ is equal to $$3$$ for $$M_i = P, \tau_2, \tau_2^2,\tau_2^3$$ and equal to $$2$$ for $$M_i = I$$.

Here is a copy of the GAP-session computing this (note that this is using a new function LeftApproximationByAddM added to QPA today) for $$P$$. The computations for the other direct summands are similar.

gap> Q := Quiver( 1, [[1,1,"a"],[1,1,"b"]] );
<quiver with 1 vertices and 2 arrows>
gap> kQ := PathAlgebra( GF( 3 ), Q );
<GF(3)[<quiver with 1 vertices and 2 arrows>]>
gap> AssignGeneratorVariables( kQ );
#I  Assigned the global variables [ v1, a, b ]
gap> $, b^3 ]; [ (Z(3)^0)*a^2, (Z(3)^0)*a*b+(Z(3)^0)*b^2+(Z(3))*a*b*a, (Z(3)^0)*a*b^2, (Z(3)^0)*b*a*b, (Z(3)^0)*a*b+(Z(3)^0)*b^2+(Z(3)^0)*b^2*a, (Z(3)^0)*b^3 ] gap> A := kQ/relations; <GF(3)[<quiver with 1 vertices and 2 arrows>]/ <two-sided ideal in <GF(3)[<quiver with 1 vertices and 2 arrows>]>, (6 generators)>> gap> P := IndecProjectiveModules(A)[1]; <[ 6 ]> gap> I := IndecInjectiveModules(A)[1]; <[ 6 ]> gap> OI := NthSyzygy(I,1); <[ 6 ]> gap> tau2 := DTr(OI,1); Computing step 1... <[ 8 ]> gap> tau22 := DTr(NthSyzygy(tau2, 1),1); Computing step 1... <[ 5 ]> gap> tau23 := DTr(NthSyzygy(tau22, 1),1); Computing step 1... <[ 8 ]> gap> M := DirectSumOfQPAModules([P,I,tau2,tau22,tau23]); <[ 33 ]> gap> N := DirectSumOfQPAModules([I,tau2,tau22,tau23]); <[ 27 ]> gap> U := P; <[ 6 ]> gap> f := LeftApproximationByAddM(U,N); <<[ 6 ]> ---> <[ 54 ]>> gap> test := List(HomOverAlgebra(Range(f),M), s -> f*s);; gap> n := Length(vectors[1]); 198 gap> V := Subspace(GF(3)^n, vectors); <vector space over GF(3), with 67 generators> gap> Dimension(V); 32 gap> Length( HomOverAlgebra(U,M) ); 33 gap> K1 := CoKernel(f); <[ 48 ]> gap> CommonDirectSummand(P,K1); false gap> CommonDirectSummand(I,K1); [ <[ 6 ]>, <[ 0 ]>, <[ 6 ]>, <[ 42 ]> ] gap> K1 := last[4]; <[ 42 ]> gap> CommonDirectSummand(I,K1); [ <[ 6 ]>, <[ 0 ]>, <[ 6 ]>, <[ 36 ]> ] gap> K1 := last[4]; <[ 36 ]> gap> CommonDirectSummand(I,K1); false gap> CommonDirectSummand(tau2,K1); [ <[ 8 ]>, <[ 0 ]>, <[ 8 ]>, <[ 28 ]> ] gap> K1 := last[4]; <[ 28 ]> gap> CommonDirectSummand(tau2,K1); false gap> CommonDirectSummand(tau22,K1); [ <[ 5 ]>, <[ 0 ]>, <[ 5 ]>, <[ 23 ]> ] gap> K1 := last[4]; <[ 23 ]> gap> CommonDirectSummand(tau22,K1); [ <[ 5 ]>, <[ 0 ]>, <[ 5 ]>, <[ 18 ]> ] gap> K1 := last[4]; <[ 18 ]> gap> CommonDirectSummand(tau22,K1); false gap> CommonDirectSummand(tau23,K1); [ <[ 8 ]>, <[ 0 ]>, <[ 8 ]>, <[ 10 ]> ] gap> K1 := last[4]; <[ 10 ]> gap> CommonDirectSummand(tau23,K1); false gap> f2 := LeftApproximationByAddM( K1, M ); K2 := CoKernel(f2); <<[ 10 ]> ---> <[ 99 ]>> gap> K2 := CoKernel(f2); <[ 89 ]> gap> CommonDirectSummand(P,K2); [ <[ 6 ]>, <[ 0 ]>, <[ 6 ]>, <[ 83 ]> ] gap> K2 := last[4]; <[ 83 ]> gap> CommonDirectSummand(P,K2); [ <[ 6 ]>, <[ 0 ]>, <[ 6 ]>, <[ 77 ]> ] gap> K2 := last[4]; <[ 77 ]> gap> CommonDirectSummand(P,K2); false gap> CommonDirectSummand(I,K2); [ <[ 6 ]>, <[ 0 ]>, <[ 6 ]>, <[ 71 ]> ] gap> K2 := last[4]; <[ 71 ]> gap> CommonDirectSummand(I,K2); false gap> CommonDirectSummand(tau2,K2); [ <[ 8 ]>, <[ 0 ]>, <[ 8 ]>, <[ 63 ]> ] gap> K2 := last[4]; <[ 63 ]> gap> CommonDirectSummand(tau2,K2); [ <[ 8 ]>, <[ 0 ]>, <[ 8 ]>, <[ 55 ]> ] gap> K2 := last[4]; <[ 55 ]> gap> CommonDirectSummand(tau2,K2); [ <[ 8 ]>, <[ 0 ]>, <[ 8 ]>, <[ 47 ]> ] gap> K2 := last[4]; <[ 47 ]> gap> CommonDirectSummand(tau2,K2); false gap> CommonDirectSummand(tau22,K2); [ <[ 5 ]>, <[ 0 ]>, <[ 5 ]>, <[ 42 ]> ] gap> K2 := last[4]; <[ 42 ]> gap> CommonDirectSummand(tau22,K2); [ <[ 5 ]>, <[ 0 ]>, <[ 5 ]>, <[ 37 ]> ] gap> K2 := last[4]; <[ 37 ]> gap> CommonDirectSummand(tau22,K2); [ <[ 5 ]>, <[ 0 ]>, <[ 5 ]>, <[ 32 ]> ] gap> K2 := last[4]; <[ 32 ]> gap> CommonDirectSummand(tau22,K2); false gap> CommonDirectSummand(tau23,K2); [ <[ 8 ]>, <[ 0 ]>, <[ 8 ]>, <[ 24 ]> ] gap> K2 := last[4]; <[ 24 ]> gap> CommonDirectSummand(tau23,K2); [ <[ 8 ]>, <[ 0 ]>, <[ 8 ]>, <[ 16 ]> ] gap> K2 := last[4]; <[ 16 ]> gap> CommonDirectSummand(tau23,K2); [ <[ 8 ]>, <[ 0 ]>, <[ 8 ]>, <[ 8 ]> ] gap> K2 := last[4]; <[ 8 ]> gap> CommonDirectSummand(tau23,K2); [ <[ 8 ]>, <[ 0 ]>, <[ 8 ]>, <[ 0 ]> ]  I hope that these comments are helpful. The QPA-team. • Thanks. Do I understand it correctly that like this it would not be possible to check for a general generator-cogenerator$M$over an algebra$A$whether$End_A(M)$has finite global dimension, but one has to be lucky and all the cokernel of the induced map$Hom_A(f',M)$has to be 1-dimensional? (this condition should be equivalent to the condition that for every simple$B$-module$S$, one has a minimal projective presentation$P_1 \rightarrow P_0 \rightarrow S \rightarrow 0$such that$P_0$is not a direct summand of$P_1$and should be rather rare.) – Mare Jan 15 '19 at 10:29 • So who won? Human or computer? – Jeremy Rickard Jan 15 '19 at 11:23 • I agree with your comments. Finding the quiver of$\operatorname{End}(M)\$ is not so expensive. But determining the relations are more costly. Here is what you can do to find the quiver: – Oeyvind Solberg Jan 15 '19 at 11:29
• endo := EndOverAlgebra(M); radendo := RadicalOfAlgebra(endo); radendo2 := ProductSpace(radendo,radendo); f := NaturalHomomorphismByIdeal(endo,radendo2); idemps := IdempotentsForDecomposition(endo); B := BasisVectors(Basis(radendo));; I := Ideal(Range(f), List(B, b->ImageElm(f, b))); mat := List( [1..5], i -> List([1..5], j -> Dimension( Subspace( Range(f), ImageElm(f,idemps[i])*BasisVectors(Basis(I))*ImageElm(f,idemps[j]))))); Display(mat); – Oeyvind Solberg Jan 15 '19 at 11:29
• @JeremyRickard This time it is probably a draw. – Mare Jan 15 '19 at 12:22