Testing whether a module generates $K_0(\mbox{mod-}A)$

Question

Given a representation-finite (connected) quiver algebra $A$ and a module $M$.

Is there a good way to test whether the set $\{ [N] \mid N \in \mathrm{add}(M) \}$ generates $K_0(\mbox{mod-}A)$?

Can this be done using the GAP-package QPA?

Answer 1

Suppose that $M = \oplus_{i=1}^n M_i$ with $M_i$ being indecomposable and assume that $M_i \not\simeq M_j$ for $i\neq j$ (that is, $M$ is basic) over a finite dimensional algebra $A$. Define a matrix $K$ with rows equal to the dimension vector of $M_i$ for $i = 1,2,\ldots,n$. Then $M$ generates Grothendieck group of $A$ if and only if the Smith normal form of $K$ over $\mathbb{Z}$ is of the form $\begin{bmatrix} I \\ \hline O\end{bmatrix}$ where $I$ is an identity matrix. So we could do the following in QPA:

gap> A := NakayamaAlgebra(Rationals, [3,4,3,3]); 
<Rationals[<quiver with 4 vertices and 4 arrows>]/
<two-sided ideal in <Rationals[<quiver with 4 vertices and 4 arrows>]>, (3 generators)>>
gap> P := IndecProjectiveModules(A);             
[ <[ 1, 1, 1, 0 ]>, <[ 1, 1, 1, 1 ]>, <[ 1, 0, 1, 1 ]>, <[ 1, 1, 0, 1 ]> ]
gap> I := IndecInjectiveModules(A);              
[ <[ 1, 1, 1, 1 ]>, <[ 1, 1, 0, 1 ]>, <[ 1, 1, 1, 0 ]>, <[ 0, 1, 1, 1 ]> ]
gap> M := Concatenation(P,I);                   
[ <[ 1, 1, 1, 0 ]>, <[ 1, 1, 1, 1 ]>, <[ 1, 0, 1, 1 ]>, <[ 1, 1, 0, 1 ]>, <[ 1, 1, 1, 1 ]>, 
  <[ 1, 1, 0, 1 ]>, <[ 1, 1, 1, 0 ]>, <[ 0, 1, 1, 1 ]> ]
gap> K := List( M, m -> DimensionVector( m ) ); 
[ [ 1, 1, 1, 0 ], [ 1, 1, 1, 1 ], [ 1, 0, 1, 1 ], [ 1, 1, 0, 1 ], [ 1, 1, 1, 1 ], [ 1, 1, 0, 1 ], 
  [ 1, 1, 1, 0 ], [ 0, 1, 1, 1 ] ]
gap> SNK := SmithNormalFormIntegerMat( K );
[ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ], 
  [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ] ]
gap> Display(SNK);
[ [  1,  0,  0,  0 ],
  [  0,  1,  0,  0 ],
  [  0,  0,  1,  0 ],
  [  0,  0,  0,  1 ],
  [  0,  0,  0,  0 ],
  [  0,  0,  0,  0 ],
  [  0,  0,  0,  0 ],
  [  0,  0,  0,  0 ] ]