Full projective resolutions for path algebras in GAP

Question

I am trying to do various computations on path algebras using GAP. One such computation is obtaining projective resolutions of simples modules (and possibly other modules). The command in the GAP manual is just "ProjectiveResolution(M)" to compute the projective resolution for a module $M$. However, when I display this projective resolution, I only ever see the first two steps of the projective resolution: the module $M$ appears with an arrow pointing to it from the projective module in the projective cover for $M$, but there should (generally speaking) be more modules in the projective resolution for $M$.

The following picture shows a projective resolution with two steps (there should be 4). The right most module (other than the $0$ on the right) is the module in question (the fifth simple module in this case), and the left most module is the projective covering module for the module in question.

Is there a way to display all projectives in the projective resolution of a module $M$ in GAP?

Edit: I am using the QPA package, and the original code I used is the following (which includes the definition of the quiver):

Q := Quiver( 5, [ [1,2,"u"], [2,1,"e"], [2,3,"a"], [3,4,"b"], [4,2,"l"], [4,5,"c"], [5,4,"d"] ]);
Display(Q);
kQ := PathAlgebra(Rationals, Q);
Display(kQ);
AssignGeneratorVariables(kQ);
relations := [a*b*c*d*l*e*u*a*b-e*u*a*b*c*d, u*e, l*a, b*l, d*c];
A := kQ/relations;
Display(Dimension(A));
projectives := IndecProjectiveModules(A);
simples := SimpleModules(A);

Display(ProjDimensionOfModule(simples[1], 100));
Display(ProjDimensionOfModule(simples[2], 100));
Display(ProjDimensionOfModule(simples[3], 100));
Display(ProjDimensionOfModule(simples[4], 100));
Display(ProjDimensionOfModule(simples[5], 100));

Display(ProjectiveResolution(simples[5]));

Note: the numbers appearing in the first picture (i.e.: $9,8,6,5,2$) are the projective dimensions of the modules I'm looking at. In particular, the $2$ is the projective dimension of the module in my question (it is the fifth simple module), so I know the projective resolution in question is finite.

Answer 1

The function ProjectiveResolution returns a Complex. You can look up in the QPA manual how to compute and access its differentials etc. Here's an example

P := ProjectiveResolution(simples[5]);

P2 := ObjectOfComplex(P, 2);
d2 := DifferentialOfComplex(P, 2);