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.