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

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?

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 ] ]